Bernard Brogliato

Some Perspectives on the Analysis and Control of Complementarity Systems

Keywords: Complementarity, control, convex analysis, hybrid dynamics, measure differential inclusions, mechanical systems, nonsmooth, stability . ACS Affine complementarity system. LCS Linear complementarity system. LCP Linear complementarity problem. DES Discrete-event system. VI Variational inequality

This paper is devoted to presenting controllability and stabilizability issues associated to a class of nonsmooth dynamical systems, namely complementarity dynamical systems. The main existing results are summarized, and some possible research directions are provided. Convex analysis and complementarity problems are claimed to be the main analysis tools for control related studies. This paper mainly focuses on mechanical applications.

I. INTRODUCTION J . J. MOREAU made fundamental contributions to convex analysis and nonsmooth mechanics [START_REF] Moreau | Les liaisons unilatérales et le principe de Gauss[END_REF]- [START_REF]An introduction to unilateral dynamics[END_REF], which have had considerable influence in several branches of mechanics (discrete systems, fluid mechanics, elasto-plasticity, and friction) [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF], [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF], [START_REF] Halphen | Sur les matériaux généralisés[END_REF], [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF], and for the numerical simulation of nonsmooth mechanical systems [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF]. The reader is referred to [START_REF] Curnier | Unilateral contact: Mechanical modeling[END_REF] for an introduction to nonsmooth mechanics and to [START_REF] Pfeifer | [END_REF] for a complete panorama of the topic in applied mathematics and mechanics. In addition, nonsmooth analysis has made significant progress and has allowed researchers to solve important problems like in optimal control and calculation [START_REF] Warga | Minimizing variational curves restricted to a preassigned set[END_REF], [START_REF] Murty | Linear complementarity, linear and nonlinear programming[END_REF], by introducing tools like Clarke's generalized gradient [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF], [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], nonsmooth versions of Newton's algorithms [START_REF] Alart | Méthode de Newton généralisée en mécanique du contact[END_REF], and complementarity problems [START_REF] Billups | Complementarity problems[END_REF], [START_REF] Murty | Linear complementarity, linear and nonlinear programming[END_REF]. In this context, it is worth recalling that nonsmoothness may not only arise at the modeling level, but may also be necessary to solve some

The author is with INRIA Rhône-Alpes, ZIRST Montbonnot, 38334 Saint Ismier Cedex, France (e-mail: Bernard.Brogliato@inrialpes.fr). abstract problems. For instance minimum-time optimal control yields cost functions which are both nondifferentiable and nonconvex. Recently, mechanical systems subject to inequality (or unilateral) constraints have been the object of renewed interest in the mechanical engineering and applied mathematics scientific communities; see [START_REF] Brogliato | Nonsmooth Mechanics[END_REF], [START_REF] Brogliato | Impacts in Mechanical Systems. Analysis and Modeling[END_REF], [START_REF] Frémond | Non-Smooth Thermomechanics[END_REF], [START_REF] Ch | Set Valued Force Laws[END_REF], [START_REF] Kozlov | Billiards: A Genetic Introduction to the Dynamics of Systems With Impacts[END_REF], [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF], [START_REF] Pfeiffer | Multibody Dynamics With Unilateral Contacts[END_REF], and [START_REF] Stronge | Impact Mechanics[END_REF]. In parallel, the analysis and control of hybrid dynamical systems has become an active investigation area in the systems and control scientific community [47], [3], [START_REF][END_REF], [48], [START_REF][END_REF], [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF]. In particular, complementarity systems are an interesting class of hybrid dynamical systems [START_REF] Çamlibel | On linear passive complementarity systems[END_REF], [START_REF] Heemels | The rational complementarity problem[END_REF]- [START_REF] Heemels | Linear complementarity systems[END_REF], [START_REF]The complementarity-slackness class of hybrid systems[END_REF], [START_REF] Van Der Schaft | Complementarity modeling of hybrid systems[END_REF]. Roughly speaking, complementarity systems consist of a dynamical system that is coupled to a set of algebraic conditions through a Lagrange multiplier (also called a slack variable in nonlinear programming). They can also be seen as dynamical systems subject to a particular type of generalized nonsmooth constraint. Complementarity dynamical systems find applications in mechanical systems (multibody systemsmanipulators, bipedal robots [START_REF] Hurmuzlu | Modeling, stability and control of biped robots-A general framework[END_REF], controlled structures, tethered satellites [START_REF] Krupa | Tethered satellite systems: A new concept of space flight[END_REF], haptically augmented teleoperation, part feeding, automatic assembly, material handling systems, etc., structural mechanics, elastohydrodynamics lubrication, liquid slosh phenomena, collisions of fluids and solids, phase changes, etc. [START_REF] Frémond | Non-Smooth Thermomechanics[END_REF]), electrical circuits (nonlinear circuits with diodes, MOS transistors, operational amplifiers [START_REF] Heemels | Linear complementarity systems. A study in hybrid dynamics[END_REF], [START_REF] Vandenberghe | The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits[END_REF]), control theory (optimal control with state constraints, model predictive control [START_REF] Bemporad | On hybrid systems and closed-loop MPC systems[END_REF], variable structure systems [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF]), traffic and oligopolistic market equilibrium problems [START_REF] Billups | Complementarity problems[END_REF], [START_REF] Nagurney | Projected Dynamical Systems and Variational Inequalities With Applications[END_REF], economics (production, comsumption, the theory of option pricing, Walrasian problems [START_REF] Ferris | Engineering and Economic Applications of Complementarity Problems[END_REF]), models for biological systems (genetic networks, bacteria growth, gene regulatory networks that describe the regulatory interactions between genes and gene products in a cell [START_REF] Jong | Modeling and simulation of genetic regulatory systems: A literature survey[END_REF]), etc. This paper is devoted to the controllability and the stabilizability of complementarity dynamical systems. Similarly to the fact that nonsmooth analysis is not a straightforward extension of classical analysis [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], the control of such nonsmooth dynamical systems is by far not a direct extension of the control of smooth systems. The same can be asserted to numerical analysis and simulation [START_REF] Abadie | Dynamic simulation of rigid bodies: Modeling of frictional contact[END_REF], [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF] and bifurcation analysis [START_REF] Leine | Periodic motion and bifurcations induced by the Painlevé paradox[END_REF]. This paper does not pretend to survey the wide area of nonsmooth systems. For more complete bibliographies, the reader is referred to [START_REF] Brogliato | Nonsmooth Mechanics[END_REF] and [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF]. It rather concentrates on a specific class and mainly aims at pointing out some open problems (marked by OP 1 ) and the main features of the analyzed systems. The paper is organized as follows: In Section II the dynamics of complementarity systems is presented. Several typical examples are given in Section III, and the link with various nonsmooth dynamical systems and differential inclusions is made in Section IV. Section V is devoted to controllability issues, while Section VI focuses on stability and stabilization. Conclusions end the paper, and some technical informations are provided in Appendices A and B. Convex analysis, complementarity problems, generalized equations, impact Poincaré maps, measure differential inclusions, are advocated to be important analysis tools. Though fundamental topics in control, observability (see [START_REF] Menini | State estimation of (otherwise unobservable) linear mechanical systems through the use of nonsmooth impacts: The case of two mating gears[END_REF]) and identification problems are not discussed. Mechanical systems with unilateral constraints and friction are ubiquitous in everyday life and in industry. Moreover, mechanics has always had a leading role in Science, and many results motivated by Mechanics have found applications or extensions in other scientific fields. The main focus of this paper is, therefore, on mechanical systems, despite many other systems may be modeled in a complementarity framework (like electrical networks) as the aforementioned list shows.

II. COMPLEMENTARY DYNAMICAL SYSTEMS

A. General Dynamics

The class of finite-dimensional controlled nonsmooth dynamical systems on which we will focus in this paper can be represented by the following set of differential and algebraic equations and conditions: almost everywhere,

State

reinitialization rule [START_REF] Abadie | Dynamic simulation of rigid bodies: Modeling of frictional contact[END_REF] where is the state of the continuous dynamics, is a real number, is some measurable output signal available for feedback, is a control signal to be chosen in some admissible set , the slack variable and the signal constitute a pair of complementarity variables as indicated in the third line of [START_REF] Abadie | Dynamic simulation of rigid bodies: Modeling of frictional contact[END_REF], where is some function. The symbol means that and have to be orthogonal. The sets and , are a pair of polar convex cones, is the initial time. Let us note that the first equality in (1) will generally be satisfied only almost everywhere in the Lebesgue's measure sense, because the solution of (1) will generally not be differentiable everywhere. Following [START_REF]The complementarity-slackness class of hybrid systems[END_REF], dynamical systems as in (1) may be named complementarity systems [CS, or systems subject to generalized constraints, made of the third and fourth lines in [START_REF] Abadie | Dynamic simulation of rigid bodies: Modeling of frictional contact[END_REF]]. In order to integrate the dynamical part of (1) (i.e., the first line), one needs to calculate . It will be shown on several examples how the slack variables , are calculated at each time . Finally a state reinitialization rule will generally be needed to integrate trajectories on a time interval of strictly positive measure. The interplay between the various ingredients in (1) may be rather complicated, and this is what makes the study of such controlled dynamics challenging. Before examining some simple examples, let us briefly introduce complementarity problems (see more in Appendix B).

Complementarity conditions have been introduced by Moreau [START_REF] Moreau | Les liaisons unilatérales et le principe de Gauss[END_REF] in contact mechanics, and Karush et al. in quadratic programming [START_REF] Murty | Linear complementarity, linear and nonlinear programming[END_REF]. They yield CP of the form : the -dimensional vectors and have to belong to the sets and , respectively, and are orthogonal. If and are a pair of polar cones, then is a Cone CP [START_REF] Billups | Complementarity problems[END_REF]. If for some matrix and vector and if , then is equivalent to the Linear CP (LCP) , , , denoted as LCP .

is the LCP matrix. If the matrix is a P-matrix (i.e., all its principal minors are positive), then the LCP has a unique solution independently of [START_REF] Murty | Linear complementarity, linear and nonlinear programming[END_REF]. If for some nonlinear function , then is a nonlinear CP (NCP).

When they exist, the left and right-limits of a function at are denoted as and , respectively. Example 1: Let us consider the following complementarity systems:

(2) where is a constant and or . Consider , and let . Thus, initially so that decreases toward in finite time , i.e., . Notice that due to the complementarity conditions, one has on . In order to continue the integration for , there must exist at least one such that both the differential equation and the complementarity conditions are simultaneously satisfied. Let us propose . Then, so that on one has . One notices that if then , however if then which is impossible. Notice that in the latter case one has , consequently if , there is no nonnegative which allows to satisfy on an interval , for any . Notice also that we have written the right-limit for at because the solution we are looking for is indeed not differentiable at since whereas . Thus, if , the integration cannot be continued after . If the previous choice allows one to integrate over . Moreover, it can be shown that this is the only possible solution. Actually the proposed choice for comes from the following: assume that on a nonzero time interval. Then , from which it follows that is the solution of the LCP . This LCP always possesses a unique solution which is precisely the previous one. And if the corresponding LCP has no solution. Therefore, one sees that once the boundary has been reached, we have been able to exhibit a new slack variable such that integration of can be continued, and the complementarity conditions remain satisfied provided the LCP matrix (which is here a scalar) is positive. for some , implies that both and jump to nonnegative values at . This implies that cannot be a function, but must be a distribution involving both the Dirac distribution and its derivative . At this stage, one needs some rule to perform the calculation of the jumps in and . If this rule says that then the integration can continue on with , so that on . Using the fact that on a nonzero time interval together with , we conclude that holds. Consequently is calculated at time as the unique solution of this LCP. If the rule states that , , then bounces on the boundary at and on for some . These examples do not encapsulate all the features of complementarity systems and are not meant to be rigorous proofs for existence and uniqueness of solutions. However they do illustrate some basic peculiarities. One notices that the vector field remains continuous in , until some boundary is reached by the state . Then the multiplier is calculated and modifies the vector field of the continuous dynamics, in such a way that the integration of proceeds. It is noteworthy that the leading Markov parameter [START_REF] Sontag | Mathematical Control Theory[END_REF] of the system with output and input plays a significant role in the analysis and that it is closely related to the existence and uniqueness of solutions of some LCP constructed from the complementarity conditions and the dynamics. The well-posedness results in [START_REF] Çamlibel | On linear passive complementarity systems[END_REF], [START_REF] Heemels | The rational complementarity problem[END_REF], [START_REF] Heemels | Linear complementarity systems[END_REF], [START_REF]The complementarity-slackness class of hybrid systems[END_REF], and [START_REF] Van Der Schaft | Complementarity modeling of hybrid systems[END_REF] are based on this observation. When , , and , (1) is named a LCS. LCS have been presented for the first time in [START_REF]The complementarity-slackness class of hybrid systems[END_REF]. If is a matrix then is the unique solution of the LCP at each .

A particular feature of CS compared to other classes of hybrid systems (see [3], [47], and [START_REF][END_REF]), is that CP and convex analysis are at the core of their analysis. This is clear for both the mathematical [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF], [START_REF] Heemels | Linear complementarity systems[END_REF], [START_REF] Heemels | The rational complementarity problem[END_REF] and numerical [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF] aspects. One objective of this paper is to prove that this is also the case for control and systems analysis. As we shall see later, such systems can also be considered as differential inclusions. Models as in (1) are quite interesting because they are not too general, but yet are rich enough to potentially present a lot of challenging problems to systems and control theorists. Moreover, the engineering applications are numerous. Let us notice that the discrete-time counterpart to (1) exists [START_REF] Bemporad | On hybrid systems and closed-loop MPC systems[END_REF], [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF], but we shall focus on the continuous-time case only.

If is a solution of this NCP, then it is also a zero of the nonsmooth complementarity function . The set of piecewise linear equations is equivalent to [START_REF] Billups | Complementarity problems[END_REF]. Other complementarity functions exist. The constraint may be named a generalized constraint. Remark 1: If , , and if it follows that the third and fourth lines of (1) can be rewritten as for some complementarity function . Thus, the dynamics in (1) may be seen as a nonsmooth differential algebraic equation in this case.

B. Associated Hybrid Automaton

The time-integration of the systems in (2) suggests that there is a natural connection between the dynamics in (1) and so-called hybrid dynamical systems, i.e., dynamical systems whose evolution is the result of the interplay between continuous and discrete-event dynamics [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF], [START_REF] Heemels | Linear complementarity systems. A study in hybrid dynamics[END_REF]. In (1), the continuous part is in the first two lines, the third and fourth lines describe the discrete-event part (the modes and the conditions for transitions between modes to occur). The last line describes how transitions between modes occur. The state of the system, therefore, consists of a variable that takes its values in a continuous space, and variables that take their value in a finite set , i.e., . The DES state for CS as in (1) can be in a first instance constructed from the modes that correspond to the values of the variables , . Let us consider the case of a LCS. For each component , the complementarity conditions yield two modes: or . Hence, one has a priori . The events (or transitions) correspond to trajectories passing from one mode to another one, implying a change in the structure of the vector field . In there are such events at times and , and in at time . Let us denote the states of the DES part as , , and a trajectory (also named a path) as . A path thus consists of a finite number of DES states together with transition rules which allow the system to evolve from one state to another state . One may consider that a mode is active, if and only if has remained during a time interval of positive Lebesgue measure in this mode. The reinitialization of the state implies a discontinuity in . In general, the activation of a new mode cannot be done without a reinitialization of the state , hence, the fifth line in [START_REF] Abadie | Dynamic simulation of rigid bodies: Modeling of frictional contact[END_REF]. See, e.g., where the state has to jump at to continue the integration. However, in some cases a nondifferentiable state may suffice for a trajectory to pass from one mode to another one, see examples and in [START_REF] Alart | Méthode de Newton généralisée en mécanique du contact[END_REF]. Transitions may even occur smoothly, as in the case of a mechanical system if the constraint is attained tangentially in the phase space.

Generically event times which correspond to a trajectory passing from the mode to the mode are denoted as , and reinitialization times are denoted as , . The complexity of CS is further illustrated by simple mechanical systems, where an infinity of events can occur in finite time (finite accumulation of impact times, i.e., a particular type Zeno behavior [87, p. 10]), and several such finite accumulations may be repeated (possibly an infinity of times as

). A well-known example is the bouncing-ball dynamics , , , , , and . There is a sequence of impact times , that is followed by a switch from mode to mode at time [notice that so that this transition occurs without velocity jump]. Consequently on the system evolves almost all the time in the mode . Obviously, in some cases, a time is also a mode switching time (see , and plastic impacts in mechanics).

The distinction between the times and is not necessary from a general hybrid dynamical systems point of view, because they can all be embedded into transition or event times [87, Chs. 1 and 2]. However, it may sometimes be convenient to distinguish between state jumps and variations of the structure of vector fields since they may not have the same consequences on control properties.

Remark 2: The associated automaton has no discrete control. In other words the transitions between the modes as previously defined, are only function of the continuous part of the system. This may have important consequences on control properties.

In this paper, we will not provide many details on the well-posedness problems [existence and uniqueness of solutions for all and in a suitable function space]. Examples in (2) provide a rapid overview for some CS.

Assumption 1: There exists and such that (1) possesses a unique solution on for some , , . Depending on the ingredients in (1), solutions may be of different nature (AC, RCLBV, piecewise continuous, etc.). The least requirement is that solutions possess a right limit everywhere.

In other words assumption 1 states that there exist an initial state and a control input (possibly ) such that (1) can be integrated on a nonzero time-interval. Otherwise we consider that the model can be rejected. Clearly the space in which the solution lives, may influence control studies. In particular the input has to take its values in a suitable space , and this has consequences on controllability. Also notice that assumption 1 includes possible initial state jumps and that the solutions may be discontinuous with respect to initial data . Such evolution problems may be named prospective [START_REF]An introduction to unilateral dynamics[END_REF], because at each one looks for right limits of the solution.

C. Basic Notions From Convex Analysis

The following notions [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF], [START_REF]Fonctionnelles convexes[END_REF], [START_REF] Rockafellar | Convex Analysis[END_REF] where the normal direction is not uniquely defined [see Fig. 4 in a simple case where the sets are depicted]. Polarity is a generalization of orthogonality, for convex cones. In mechanics with bilateral holonomic constraints, the admissible velocities and the contact reaction belong to orthogonal spaces. Roughly, this generalizes to unilateral constraints by replacing these orthogonal spaces by the tangent and normal cones, respectively [START_REF]Liaisons Unilatérales Sans Frottement et Chocs Inélastiques[END_REF]. Generalizations of these notions to nonconvex sets and functions exist [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF], [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], allowing, e.g., to state that if for , for , then . A super-potential (or pseudopotential) of dissipation [START_REF]La Notion de Sur-Potentiel et les Liaisons Unilaétrales en Élastostatique[END_REF], [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF], [START_REF] Halphen | Sur les matériaux généralisés[END_REF] is a convex lsc proper mapping , , such that , where and are a pair of dual variables (in mechanics, force, and velocities). The conjugate of the lsc convex function is defined as ; see [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF] for an example. One has . Bilateral constraints with Coulomb friction can be expressed this way, as well as many other physical laws [START_REF] Frémond | Non-Smooth Thermomechanics[END_REF]. Nonconvex superpotentials can be defined using Clarke's gradient [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF]. There is also a strong apparatus of numerical algorithms associated to such analytical tools [START_REF] Alart | Méthode de Newton généralisée en mécanique du contact[END_REF], [START_REF] De Schutter | The linear dynamic complementarity problem is a special case of the extended linear complementarity problem[END_REF], [START_REF] Murty | Linear complementarity, linear and nonlinear programming[END_REF], [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF].

The conventional notation . for boundaries of sets, partial differentiation and subdifferential, is kept throughout the paper. The gradient if . If is differentiable, then .

III. EXAMPLES

Some examples of systems that may be recast into the framework of (1) are presented. In view of the list provided in the introduction, the following examples are only a few cases of CS (see [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF]Ch. 4] for other examples). We omit to indicate initial conditions. Let us notice that when solutions possess jumps, then the initial data in (1) have to be taken as a left-limit to allow for initial jumps [START_REF]Unilateral contact and dry friction in finite freedom dynamics[END_REF].

Example 2: (Mechanical system with frictionless unilateral constraint) Collision mapping.

(3)

In (3) and the other terms may be identified easily. The second line in (3) may be called the Hertz-Signorini-Moreau conditions [START_REF] Curnier | Unilateral contact: Mechanical modeling[END_REF]. According to the discussion in Section II-B, the associated automaton has states. When , then one may define , and write the complementarity relation as with and , . The dynamics in (3) encompasses manipulators performing complex robotic tasks (e.g., models for deburring tasks which have considerable importance in manufacturing industry [START_REF] Ramachandran | The role of deburring in manufacturing: A state-of-the-art survey[END_REF] and many other machining tasks) as well as controlled Lagrangian CS. Important applications also exist in aerospace (pick up tasks with autonomous robots in zero-gravity fields, tethered satellites, landing aircrafts). The first use of (3) in Control and Robotics can be found in [START_REF] Huang | Time optimal control for a robotic contour following problem[END_REF]. Models as in ( 4) including also Coulomb friction have been proved experimentally quite valuable for somewhat complex systems ( , ) in industrial applications for virtual prototyping of circuit breakers [START_REF] Abadie | Dynamic simulation of rigid bodies: Modeling of frictional contact[END_REF]. The existence and uniqueness of solutions , is guaranteed if and only if all data are piecewise analytic [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF].

If then . If on , , then , and , , on . Thus, is calculated on from the LCP , , which involves both the complementarity conditions and the continuous dynamics [similarly as for systems and in [START_REF] Alart | Méthode de Newton généralisée en mécanique du contact[END_REF]]. The LCP matrix is equal to so the LCP has at least one solution [START_REF] Moreau | Les liaisons unilatérales et le principe de Gauss[END_REF]. If , detachment from the boundary occurs if , , . If and , then jumps at , which is an impact time . At impact times is a Dirac measure and the Lagrange dynamics becomes algebraic [START_REF] Brogliato | Nonsmooth Mechanics[END_REF].

When is convex, the dynamics in (3) can be equivalently rewritten as

when (4a) (4b)
with , so that (4b) is Moreau's collision mapping [START_REF]Liaisons Unilatérales Sans Frottement et Chocs Inélastiques[END_REF]. In [START_REF] Bainov | Systems With Impulse Effects; Stability, Theory and Applications[END_REF] is the convex tangent cone to at . If this mapping reduces to Newton's law in [START_REF] Bressan | On differential systems with quadratic impulses and their application to Lagrangian mechanics[END_REF], with . A coefficient of restitution can be introduced in the mapping in (4b); see [START_REF]Unilateral contact and dry friction in finite freedom dynamics[END_REF], [START_REF] Mabrouk | A unified variational model for the dynamics of perfect unilateral constraints[END_REF], and [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]. It is crucial to recall that in ( 4) is a solution of a nonlinear equation depending on the state [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]Sec. 1.3.1] and is most often only implicitly defined. One may also replace the right-hand side of (4a) by the (more restrictive) inclusion [START_REF]Liaisons Unilatérales Sans Frottement et Chocs Inélastiques[END_REF], which incorporates the collision mapping in (4b). Actually the term can be interpreted as a displacement potential, whereas can be interpreted as a velocity potential [START_REF] Ch | Set Valued Force Laws[END_REF]. Both are associated to the unilateral constraints , and physically state the impenetrability of the bodies in contact. The notion of MDIs is introduced in Appendix A. From (4) one can obtain Moreau's second-order sweeping process [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF], i.e., the MDI [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF] where

, and the measure can be chosen as indicated in Section VI-B. This formalism is the proper extension of Lagrange equations to systems with unilateral constraints. It may be more useful than the complementarity one when dealing with existence of fixed points or stability considerations. The domain may be described as in (3) without being convex. One may then use the notion of nonconvex superpotential or generalized gradient [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF] to write (4a), see, e.g., [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]Prop. 2.4.12]. The right-hand side of the inclusion (5) encompasses both cases where is continuous [in which case ] and discontinuous (in which case for some ). At then the MDI in ( 5) becomes the classical algebraic shock dynamics.

Remark 3: Nondifferentiable points are those configurations where the normal direction to is not uniquely defined, however the normal cone exists. The formalism in (3) precludes so-called reintrant corners, or "peaks" [START_REF] Ch | Set Valued Force Laws[END_REF], but does not imply convexity of . For instance, the domain in defined as , , , is not convex, but locally around and it is convex (in the terminology of [23, def. 2.4.6], the set is regular at and ).

Example 3 (Mass Linear Spring-Damper):

The dynamics of a mass with position , that rebounds on a spring-damper system with position , and where the contact force has to remain nonnegative and satisfies a complementarity relation with the distance , is given by [START_REF] Bemporad | Observability and controllability of piecewise affine and hybrid systems[END_REF] where , the stiffness and the damping coefficients

The contact and noncontact phases correspond to and , respectively. When the graph of the piecewise-linear relation between and is depicted in Fig. 3(a). Notice that in this case , and "

" relations possess a complementarity formalism [START_REF] Heemels | Linear complementarity systems. A study in hybrid dynamics[END_REF]. This example shows that unilaterality does not imply rigidity. However, it implies impenetrability.

If on , then the complementarity condition holds on . Since , is at time the unique solution of the LCP . Transitions from noncontact to contact occur with a continuous state . Fig. 1 illustrates the cascade connection of two linear systems with transfer functions and , and with a piecewise linear interconnection between the first system's output and the second system's input . Some typical piecewise linear characteristics are depicted in Fig. 3.

Example 4 (Dead-Zone, MOS Transistor): Let us consider a dynamical system as depicted in Fig. 1 with the piecewise linear characteristic in Fig. 3(b) with . The dynamics is given by [START_REF] Bemporad | On hybrid systems and closed-loop MPC systems[END_REF] where . In [START_REF] Bemporad | On hybrid systems and closed-loop MPC systems[END_REF] , and . The three modes are and , and , . When and , Fig. 3(b) represents the characteristic of a MOS transistor (with a control voltage).

Example 5 (Saturation, Elasto-Plasticity): The dynamics of the CS in Fig. 1 where the piecewise-linear relation is as in Fig. 3(c) is given by [START_REF] Billups | Complementarity problems[END_REF] Here, and the nonfeasible mode is and . The three feasible modes are the same as in example 4.

In both examples 4 and 5, the multiplier is calculated at time as the unique solution of a LCP whose matrix is the identity, similarly as in in ( 2) with . Example 6 (Painlevé System): Coulomb friction lends itself to a representation by complementarity relations, since the graph in Fig. 3

(f) is monotone with

[so that for some convex ]. Consequently the dynamics of a slender rod subject to a unilateral point-contact with Coulomb friction (coefficient ), as in Fig. 2, is given by Newton's impact law: where , , , , , and are the center of mass coordinates and is the rod orientation, , . The vector is the generalized contact reaction and is the friction cone in generalized coordinates [12, p. 328]. In [START_REF] Bressan | On differential systems with quadratic impulses and their application to Lagrangian mechanics[END_REF], there may exist velocity jumps during sliding regimes and and may diverge to infinity in the vicinity of some critical points (the state remaining however continuous at such critical points). These two phenomena are a consequence of the coupling between complementarity conditions and Coulomb friction, which yields a very particular type of singular differential equation and may destroy the convexity of the underlying quadratic program for calculating (see Appendix B) [START_REF] Brogliato | Nonsmooth Mechanics[END_REF], [START_REF] Génot | New results on Painlevé paradoxes[END_REF]. One notes that in ( 9) is nonlinear in . Though solutions of ( 9) are such that AC and RCLBV [START_REF] Stewart | Rigid-body dynamics with friction and impacts[END_REF], uniqueness fails [START_REF] Génot | New results on Painlevé paradoxes[END_REF], and a complex behavior similar to that of example with in (2), can occur [START_REF] Leine | Periodic motion and bifurcations induced by the Painlevé paradox[END_REF]. The multiplier in [START_REF] Bressan | On differential systems with quadratic impulses and their application to Lagrangian mechanics[END_REF] does exist as a measure. Compared to (3) the coupling between Coulomb friction and unilaterality complicates the definition of ( 9) as a hybrid dynamical system since it creates new modes. As a consequence state (velocity) reinitialization may occur at impacts but also in the slidingmotion mode.

Example 7 (Square Relation): Let us consider now the system in Fig. 3 with the set-valued relation whose graph is the square characteristic in Fig. 3(d) [START_REF] Brezis | Opérateurs Maximaux Monotones[END_REF] Here, :

, , and so on for and . Notice that the characteristic does not span the whole -axis, i.e., PL . This is quite similar to the corner law characteristic PL , as in Fig. 3(g) with (the case represents adhesive contact which may be encountered with rubber support, or in micro-robotics where van der Waals forces play a role in the contact). Let us say that such graphs contain some unilateral effects. Clearly similar developments can be made for the graphs in Fig. 3(e) (fluid with two viscosity ranges [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF]) and Fig. 3(f) (rigid viscoplastic material characteristic; the case represents relaxation effects and the graph is non longer monotone, indicating a loss of dissipativity with consequences on stability); Fig. 3(h) (unilateral and adhesive effects [START_REF] Dimnet | Collisions involving solids and fluids[END_REF] where is a displacement and is a contact impulse). Example 8 (Electrical Circuit With Ideal Diode): A simple electrical circuit containing an ideal diode whose characteristic is a corner law as in Fig. 3(g) with , a current source and an inductor mounted in parallel, possesses the following dynamics:

State reinitialization rule [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF] where is the inductor current, is the voltage across the diode, is the current across the diode, is the current variable of the current source. Interestingly enough the dynamics of a yo-yo [START_REF] Lin | Yoyo dynamics: Sequence of collisions captured by a restitution effect[END_REF] and [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF] are also quite similar, since the input acts directly in the constraint . Example 9 (Generalized Nonholonomic Constraints): The dynamics of a particle in with coordinates , subject to the nonholonomic constraint [START_REF] Brogliato | Nonsmooth Mechanics[END_REF] is studied in [START_REF] Cortès | Geometric, Control, and Numerical Aspects of Nonholonomic Systems[END_REF]. It is shown that the loss of rank of when the trajectories enter some subset, implies the need for a state reinitialization to integrate the motion. Also the multiplier associated to the constraint may diverge to infinity when the trajectories approach the singular set . This is therefore an example of a system with a bilateral constraint (however, it can be easily recast in the framework of unilaterally constrained systems by expressing the equality in [START_REF] Brogliato | Nonsmooth Mechanics[END_REF] as two inequalities), that yields a behavior conformable to that of the Painlevé system in example 6.

OP 1: In view of examples 6 and 9, extend the studies on singular differential equations to link Painlevé-like problems, dynamical systems with generalized constraints, and singular systems , where rank is not constant [START_REF] Rabier | On the computation of impasse points of quasilinear differential-algebraic equations[END_REF]. Other examples of dynamical systems that can be cast in a complementarity framework can be found in [START_REF] Billups | Complementarity problems[END_REF], [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF], [START_REF] Heemels | Linear complementarity systems. A study in hybrid dynamics[END_REF], [START_REF] Heemels | The rational complementarity problem[END_REF], and [START_REF] Vandenberghe | The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits[END_REF]. From examples 3-5, 7, and 8, it follows that an important subclass of systems as in ( 1) is given by State reinitialization rule. [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF] Systems as in ( 13) may be named ACS. It may be assumed that to avoid meaningless complementarity conditions. However, as example 7 shows, is not necessarily full rank.

Remark 4: Some of the presented examples have already been thoroughly studied in the control literature without resorting to any complementarity framework (e.g., example 3 with

, or examples 4 (dead-zone), 5 with PL at the input [START_REF] Tao | Optimal and nonlinear decoupling control of systems with sandwiched backlash[END_REF]). However, on the one hand this is not true for all CS (by far). On the other hand, it is expected that the existing works, when replaced in a more general context, may provide some useful insights for the analysis of other systems. The fact that all these systems share a common structure cannot be passed over.

State reinitializations:

The examples in (2) have shown that some CS can be integrated without state jump, whereas others have discontinuous solutions. As alluded to in example 6, the fact that a discontinuous state reinitialization is needed or not, may not always be evident. Let us consider (1) disregarding state reinitialization (consequently excluding initial state jumps), and such that assumption 1 holds. By construction the system is initialized in a certain mode . The central question is: if the conditions for transition into another mode are met, can one integrate the system without reinitializing ? In other words, is continuation in another mode possible with a continuous or not? There are two main steps in this problem: determine the conditions such that state jumps are necessary, then define a rule for state reinitialization. As the mechanical systems (3) and [START_REF] Bemporad | Observability and controllability of piecewise affine and hybrid systems[END_REF] show, the relative degree between and plays a major role in the need for state jump. In (3) , so at no bounded exist to solve if . In [START_REF] Bemporad | Observability and controllability of piecewise affine and hybrid systems[END_REF], and the LCP always has a unique bounded solution, which allows the system to satisfy without state jump. Consider now [START_REF] Bemporad | On hybrid systems and closed-loop MPC systems[END_REF]. Then one can always compute a unique as a solution of a LCP , since the LCP matrix is the identity matrix so that . Thus, no state reinitialization is needed to respect the condition : a bounded suffices. This can be generalized to [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF], where the matrix should be a -matrix. A geometric approach has been presented in [START_REF] Dam | The contact problem for linear continuous-time dynamical systems: A geometric approach[END_REF] for systems , , which brings an answer to the first step. The theory is based on the use of relative degrees between and to characterize the set and builds up possible reinitialization mappings. General state reinitialization rules are proposed in [START_REF] Heemels | Linear complementarity systems[END_REF], [START_REF] Çamlibel | On linear passive complementarity systems[END_REF], [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF], and [START_REF] Cortès | Geometric, Control, and Numerical Aspects of Nonholonomic Systems[END_REF], for ACS as in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF], with , , , , . They extend Moreau's rule in [START_REF] Bainov | Systems With Impulse Effects; Stability, Theory and Applications[END_REF]. They can be formulated as quadratic programs or Cone CP, which is of great usefulness for numerical simulations. The principle of maximal dissipation (see [START_REF] Bressan | On differential systems with quadratic impulses and their application to Lagrangian mechanics[END_REF], sixth line) that is used to avoid penetration of the state into inconsistent zones for the Painlevé system in example 6, is another example of state reinitialization rule.

The choice of a suitable reinitialization map is a crucial step. Simple examples show that one has to choose it in accordance with the continuous dynamics (consider e.g., two masses which collide, moving on a straight line: linear momentum conservation precludes the use of some "imaginary" impact law such that both masses stop after any shock).

IV. RELATIONSHIPS BETWEEN CS AND OTHER HYBRID MODELS

Differential Inclusions: It has already been pointed out in example 2 that there is a strong link between CS and some kind of DI. The dynamics in ( 13) can be equivalently rewritten as a differential-algebraic inclusion State reinitialization rule.

(

) 14
The DI almost everywhere, with upper-semicontinuous, is convex and compact for all , and satisfies linear growth conditions [i.e., for some and ], is widely studied, see [START_REF] Deimling | Multivalued differential equations[END_REF]. Then AC solutions exist. In general, such DI and CS as in (1) are quite different. A case when they drastically differ is unilaterally constrained mechanical systems, since the linear growth and compactness conditions fail. Another example is in [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF], which can equivalently be rewritten as . Therefore, it can be recast into Moreau's first-order sweeping process [START_REF]Evolution problem associated with a moving convex set in a Hilbert space[END_REF], [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF], [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF], i.e., DI of the form where is a convex set depending on time. However, for certain CSs, they coincide. This is the case of the following one degree-of-freedom mechanical system with Coulomb friction (corresponding to the graph in Fig. 3(f) with )

where , and the analogy with ( 13) or ( 14) can be made. The dynamics in (15) can be equivalently written as [START_REF]Unilateral contact and dry friction in finite freedom dynamics[END_REF] (16) where is the conjugate function of the indicator and satisfies the aforementioned standard hypotheses on . It is a convex superpotential. Notice that (16) can also be cast into the framework of Filippov's systems, i.e., a special sort of DI. The Zhuravlev-Ivanov transformation can be used to transform mechanical systems as in (3) and with into Filippov's inclusions [START_REF] Brogliato | Nonsmooth Mechanics[END_REF].

OP 2: Determine the conditions on the matrices in ( 13) such that indeed ( 14) is a DI: with satisfying some standard hypotheses [28, Secs. 5 and 6], with an AC solution. Uniqueness is more complex to establish but is also an important and largely open issue [START_REF] Yu | On wellposedness of relay systems[END_REF].

Due to the numerous studies on the sweeping process [START_REF]Evolution problem associated with a moving convex set in a Hilbert space[END_REF], [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF], [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF], the following is of interest.

OP 3: Determine the conditions such that ( 14) is a first-order sweeping process, and derive well-posedness results for such ACS (in particular the nature of solutions depending on the set of admissible controllers ).

OP 3 may be a preliminary step into some cross-fertilization between mathematical studies on various types of DI ( [START_REF] Brezis | Opérateurs Maximaux Monotones[END_REF], [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF], [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF], [START_REF] Kenmochi | A class of nonlinear elliptic-parabolic equations with time-dependent constraints[END_REF], [START_REF] Deimling | Multivalued differential equations[END_REF], and the references therein), and complementarity approaches [START_REF] Çamlibel | On linear passive complementarity systems[END_REF], [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF], [START_REF] Heemels | The rational complementarity problem[END_REF]- [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF], [START_REF]The complementarity-slackness class of hybrid systems[END_REF], [START_REF] Van Der Schaft | Complementarity modeling of hybrid systems[END_REF].

Measure Differential Equations (MDE): Specific notions of solutions have been introduced for MDE of the form with RCBLV [START_REF] Yu | Instantaneous impulse response of nonlinear system[END_REF], or even with quadratic terms in [START_REF] Bressan | On differential systems with quadratic impulses and their application to Lagrangian mechanics[END_REF]. Other types of MDEs are considered in [START_REF] Bainov | Systems With Impulse Effects; Stability, Theory and Applications[END_REF] with piecewise continuous solutions, of the form if , if for some sequence . It is clear that such models and ( 1) are of different nature. Significant differences between MDE and CS are stated in [12, Sec. 1.4.2, Ch. 7], and in [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF]. Actually, the complementarity conditions in (1) play a major role in the dynamics and in all the properties of such systems. This does not prevent some analytical tools (like stability notions) to apply both to such MDE and certain CS, as pointed out in [ Consider examples 3 (with ), and 4 (with ) and 5 with no input (i.e.,

). They are PSS [such is not the case for (3)]. Certainly the definition of the vector fields on the intersections plays a role, since it is related to graphs (filling the gaps at discontinuity points) and maximality for monotone operators. This is clearly illustrated by the various definitions one may give to the signum function. OP 4 has been clarified in [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF] when the continuous dynamics is in a discrete-time representation. Related results are also in [START_REF] Çamlibel | Existence and uniqueness of solutions for a class of piecewise linear dynamical systems[END_REF]. It follows from a result in [START_REF] Vandenberghe | The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits[END_REF] that all systems with a piecewise linear connection (see Fig. 1) can be represented as an ACS in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF].

OP 5: Study conditions on the Cone CP: and on the function , for (1) to be PSS. Discover state variable changes (possibly nonsmooth, noninvertible) such that the transformed trajectories are time-continuous. How may one use conjugacy of convex functions in this setting?

More generally, there is certainly a strong need for clarifying the links between numerous models of hybrid and nonsmooth systems. This section only aims at pointing out possible paths for few of them.

For instance, let us notice in passing the following equivalences: [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF] with in in (2) and is a linear projection. The last-but-one equivalence is between a differential inclusion and a linear evolution variational inequality [START_REF] Goeleven | Variational and Hemivariational Inequalities[END_REF]. The last formalism is called a projected dynamical system (PDS) [START_REF] Nagurney | Projected Dynamical Systems and Variational Inequalities With Applications[END_REF] and the equivalence uses basic convex analysis [44, Prop. A5.3.3]. VIs are another type of dynamical system which is widely used in some branches of applied mathematics [START_REF] Goeleven | Variational and Hemivariational Inequalities[END_REF], [START_REF] Nagurney | Projected Dynamical Systems and Variational Inequalities With Applications[END_REF]. As [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF] shows, in some cases DI, VI, CS, PSS, and PDS are only different formalisms for the same system.

V. CONTROLLABILITY ISSUES

The system (1) is controllable on if for any pair of states, there is an admissible input such that . There is a unique notion of controllability for linear time-invariant systems. However, such is not the case for nonlinear systems. Many relaxed notions of controllability exist in this case, which are not in general equivalent. Roughly, the controllability properties are characterized by the reachable sets [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF] such that the evolution of (1) satisfies for all and [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF] where is a neighborhood of . It is assumed in this definition that assumption 1 holds at least on . The system is accessible from when contains a nonempty open set for any and sufficiently small . A Lagrangian system is said to be equilibrium controllable [START_REF] Lewis | Configuration controllability of simple mechanical control systems[END_REF] if for any two equilibrium points and , there exists an input , and a solution , , such that , , for all , and .

A. Example

Let us illustrate briefly on an example how the complementarity relations and the state reinitialization rule, may influence the controllability properties of a dynamical system. To this end, let us first consider the controllable dynamics . Let us add a unilateral constraint and an impact law , with . As a consequence all states with cannot be reached from any admissible state , using bounded inputs. If is RCLBV, then the states with cannot be reached neither, so that only the origin is reachable on . However, the CS ( , , and ) is equilibrium controllable. Let us now modify the controlled dynamics by adding , . Then, in , . Consequently the system may be rendered controllable only if negative jumps in are introduced on , which may not be a good idea. Otherwise, the system is accessible only. As we have seen in example in (2), the external action can be used to steer inside .

B. Admissible States and Inputs

Let us first propose the following. Definition 1: Let be a control input such that assumption 1 is satisfied. Then, is admissible for (1) at time if such that and . The set is the admissible set.

Therefore, clearly appears that one interpretation of the third and fourth lines in ( 1) is that they form a generalized constraint. Due to assumption 1, the right-limit always exists. From definition 1 admissibility applies to the right limit. One notices that admissibility does not preclude accumulations of jumps in . At times admissibility may be stated as: belongs to the domain of the state jump mapping. The notion of admissibility is a static notion to be checked at a time instant. For instance, the left limit may be nonadmissible, so that the solution has to instantaneously jump to an admissible right limit. In in (2) with , only is admissible. In with , all are admissible. In (3), has to be admissible since is AC. However one may need to consider higher derivatives of the signals to verify the admissibility. This is the case of example 2 in which one needs to differentiate twice to enable the calculation of during phases where , by constructing a suitable LCP. Consider example 5. Then can be calculated as the solution of the LCP:

, which always possesses a unique solution since the LCP matrix is positive definite (it is the identity matrix). In both cases, one sees that does not depend on . Let us consider now example 6 with . The mathematical analysis [START_REF] Stewart | Rigid-body dynamics with friction and impacts[END_REF], [START_REF] Génot | New results on Painlevé paradoxes[END_REF] shows that and can be Dirac measures so that may jump, during sliding modes and . Such discontinuities are a consequence of the coupling between the complementarity relations and Coulomb friction. They are not due to any impact between the rod's tip and the ground, in which case would be a Dirac measure. They prevent the system from violating the constraint and from penetrating into an inconsistent subspace of the state space. Such inconsistencies are not rare, and practical examples show that they do occur for reasonable values of physical parameters [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF], [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]. When and differentiating twice , one obtains that the contact force is the solution of a LCP of the form . Consequently the domain of inconsistent (see [START_REF] Brogliato | Nonsmooth Mechanics[END_REF] or [34, Fig. 2]) depends on . Moreover, if depends on , then the LCP matrix is modified also. This may have important consequences on the controllability of the system, since the states which are a priori reachable must belong to . In the latter case, one might even imagine a bounded such that . OP 6: Determine conditions on , , such that does not depend on . Start with ACS as in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF]. Classify CS for which does not modify the properties of the CP to calculate , systems for which is forbidden, systems for which may modify the shape of . OP 7: Determine the set of admissible inputs such that , or such that , where is a given subset of the state space, for all . The notion of admissibility does not at all replace nor imply the uniqueness of solutions. Actually, a state may be admissible, and be unique while nonuniqueness of solutions emanating from holds, or may not be unique while solutions are, see [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]Sec. 5.4]. However. it is a preliminary notion which may help the control designer in planning which states are a priori reachable or not. It is thus advocated that the accurate characterization of is an important step in a control study. One should always speak of controllability over (independent of ) for constrained systems, since for any admissible . Admissible sets have been characterized in [START_REF] Çamlibel | On linear passive complementarity systems[END_REF] for passive LCS and in [START_REF] Çamlibel | Existence and uniqueness of solutions for a class of piecewise linear dynamical systems[END_REF] for a special class of piecewise linear systems.

Remark 5: The notion of viability of the input [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF] also has to be considered. Let us consider and . Roughly a control input is said to be viable on , , if the trajectory . Consider again , , with , , , , see Fig. 4. If then at the orbit hits the boundary whatever . Thus, no input is viable for . Viability of the input may be crucial in some control studies [START_REF] Jimenez | Analysis of PD and nonlinear control of mechanical systems with dynamic backlash[END_REF]. Let us introduce the following notion.

Definition 2: Let for (1), with differentiable components , . Let us denote the projection of a solution on the boundary as . Let us also denote the projection of the trajectories on as . Then, if the mapping is surjective for any , the system is said to be normal controllable (NC). In other words, NC means that trajectories can be controlled in the normal direction to , see Fig. 4. The NC property implies the existence of a viable input, and the work in [START_REF] Dam | The contact problem for linear continuous-time dynamical systems: A geometric approach[END_REF] can be used to characterize it. This notion is quite consistent with viability of sets in DI theory and is natural since the orbits evolve in some admissible sets. The novelty is that the set of forbidden states may not be constant.

C. Complementarity Mechanical Systems

From the previous developments, it follows that solutions of CS possess the generic form (we drop all arguments for clarity of writing) [START_REF] Bullo | Modeling and controllability for a class of hybrid mechanical systems[END_REF] where the DES path is and each transition is a priori done through a state reinitialization mapping . The solution between two switches in a mode is denoted as , where is the transition instant, and . In case of a finite accumulation of transitions, one has in [START_REF] Bullo | Modeling and controllability for a class of hybrid mechanical systems[END_REF]. Notice that there may be several infinities of events, even in simple systems (think of a bouncing ball). From [START_REF] Bullo | Modeling and controllability for a class of hybrid mechanical systems[END_REF] it follows that the controllability of (1) depends on (what sequence?), the mappings , (do they help, bother, or are not concerned? Does bijectivity of these mappings allow one to ignore their effect on controllability?), the controllability properties in each mode in [recall the , , in (19) depend on ], and combination of all these ingredients. Some peculiarities of controllability of systems in (1) have been highlighted in the foregoing section. Let us consider (3). Then preimpact states with and , are not reachable since is right-continuous. More generally the work in [START_REF] Dam | The contact problem for linear continuous-time dynamical systems: A geometric approach[END_REF] can be used to characterize reachable subsets of . The work in [START_REF] Bullo | Modeling and controllability for a class of hybrid mechanical systems[END_REF] considers the controllability properties of a mechanical system in each mode and combines them in order to check the equilibrium controllability. However it is assumed in [START_REF] Bullo | Modeling and controllability for a class of hybrid mechanical systems[END_REF] that at any state , one can switch to any mode instantaneously, i.e., there is a nonempty set of discrete controls for the event times

. Consequently, the derived criterion applies to systems with clamping devices (bilateral constraints that can be activated or deactivated at will), rather than to mechanical systems as in (3), see Remark 2. The work in [START_REF] Goodwine | Controllability of kinematic control systems on stratified configuration spaces[END_REF] is also based on the controllability properties of a mechanical system in each mode. It applies to quasistatic systems (no drift, no collisions, inputs are velocities). The interest of this study mainly lies in the fact that it shows how to characterize the controllability properties by switching between the modes of the associated automaton. The natural notion of controllability restricted to constraint surfaces of the configuration space is also introduced in [START_REF] Goodwine | Controllability of kinematic control systems on stratified configuration spaces[END_REF]. The previous studies use the geometric structure of nonsmooth mechanical systems [START_REF] Cortès | Geometric, Control, and Numerical Aspects of Nonholonomic Systems[END_REF].

OP 8: Depending on the input matrix in (3), the unconstrained system may not be controllable [e.g., underactuated systems with , or the structure in [START_REF] Çamlibel | On linear passive complementarity systems[END_REF]]. Can the addition of complementarity conditions and state reinitialization improve its controllability properties? So what is the relation between the controllable subspaces of the dynamics in each mode, the state reinitialization rule and the set of admissible controls , so that the CS is controllable? A path for local controllability study may be to use the linearization of the solution in ( 19) which incorporates the so-called saltation matrix [START_REF] Dankowicz | Exploiting discontinuities for stabilization of recurrent motions[END_REF]. Some preliminary answers have been given for juggling systems [START_REF] Lynch | Recurrence, controllability, and stabilization of juggling[END_REF], [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF], [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF], [START_REF] Spong | Impact controllability of an air hockey puck[END_REF], which are a subclass of mechanical CS as in (3) that possess the following general form: Collision rule [START_REF] Çamlibel | On linear passive complementarity systems[END_REF] with . Examples of mechanical jugglers are running biped robots, hoppers, controlled structures, nonprehensile manipulation systems, manipulators with dynamic passive environment, systems with dynamic backlash or liquid slosh phenomena [START_REF] Jimenez | Analysis of PD and nonlinear control of mechanical systems with dynamic backlash[END_REF], tethered satellites [START_REF] Krupa | Tethered satellite systems: A new concept of space flight[END_REF], etc. Provided the dynamics is controllable, the unconstrained system with is therefore written in an uncontrollable canonical form. The only way to control is through the collisions, i.e., by using as an input. The basic idea in [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF] and [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF] is to study the controllability properties of a partial impact Poincaré map with state and input (see, e.g., [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF]Lemma 3]). Reachable sets and accessibility are defined accordingly. The Poincaré section is chosen as , . This controllability problem may be stated as follows [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF].

Problem 1: Given with , and , find , , and , such that and .

In other words, one looks for a sequence of impacts to join the initial and end points. The locations, velocities, and number of impacts, are unknown of the problem. Denote , , , . Let us denote as an input vector , and . As shown in [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF] when the vector fields , and in [START_REF] Çamlibel | On linear passive complementarity systems[END_REF] possess some linearity, finding a solution to problem 1 is equivalent to finding a solution to the constrained equation [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF] where the matrices , , and depend nonlinearly on the initial state and on . If ( 21) possesses a solution , then is reachable from . The example of an impacting pair (modeling dynamic backlash [START_REF] Jimenez | Analysis of PD and nonlinear control of mechanical systems with dynamic backlash[END_REF]) is treated in [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF], and some general results are given which make use of the particular structure of the constrained equation in [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF] and of CP constructed from the Karush-Kuhn-Tucker conditions [START_REF] Murty | Linear complementarity, linear and nonlinear programming[END_REF]. Comparing the constrained equation in [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF] with the linear system of equations that characterizes the controllability of discrete-time invariant linear systems [START_REF] Sontag | Mathematical Control Theory[END_REF]Sec. 3.2], shows how nonlinear jugglers are.

OP 9: Investigate the structure of the reachable sets. How may one use tools from linear and nonlinear programming to get a general criterion on accessibility of jugglers? Can local controllability criteria be set up, and if so are they always relevant? Indeed in certain cases global criteria may be needed.

It is particularly interesting to note that jugglers may not be locally accessible, but only globally accessible [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF], [START_REF] Lynch | Recurrence, controllability, and stabilization of juggling[END_REF]. In other words, problem 1 may not possess any solution for and all intermediate locations and velocities lying in a neighborhood of . This is the case of a planar juggler ( , ) whose reachable sets have been fully characterized in [START_REF] Lynch | Recurrence, controllability, and stabilization of juggling[END_REF].

OP 10: Characterize the controllability of (20) from . The major ingredients are: the controllability of the -dynamics, the controllability of the -dynamics in the sense of problem 1, the invertibility of the collision mapping. Use the aforementioned framework to study the influence of various characteristic physical of systemical constants (restitution coefficient, Darboux-Keller shock dynamics [12, Sec. 4.2.5], [START_REF] Stronge | Impact Mechanics[END_REF], relative degrees between , , , etc.) on the controllability.

D. Affine Complementarity Systems 1) General Considerations:

Let us consider the ACS in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF]. It is crucial to note that the slack variable is not an exogenous disturbance, so even when a matching condition between and is satisfied, one may not compensate for directly with . The switches between the modes play a crucial role. If is a function, instantaneous switches at arbitrary time are impossible in general. In other words, in most CS the event times are not directly controllable. OP 11: Is it possible to include distributional inputs in , whose action would be similar to a discrete control by taking instantaneously the state from one mode to another one? As we shall see in Section V-D-2, in some cases of ACS, bounded discontinuous inputs can act as discrete controls and help invert some nonsmooth characteristics. The application of distributional inputs (e.g., Dirac measures and derivatives) to a CS, certainly requires much care. Until now we have only considered state jumps which are ruled by a reinitialization mapping. Applying distributional inputs to ACS [even if solutions are AC for all bounded ] certainly is a delicate matter in general since they become complex MDE (see Section IV). The case where both reinitialization mapping and distributional input coexist, is by far not yet well understood. Notice that such questions may be fundamental when dealing with optimal control problems. Remark 6: Consider example 8. The dynamics in [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF] can be rewritten as a first-order sweeping process [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF] (

The right-hand side of the inclusion in ( 22) is noncompact and depends on both and , whereas in [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF] it is compact and depends on the state only, and in (4) it is noncompact and depends on the state only. Which are the consequences of these properties on controllability? It is important to remark here that the nature of (continuous, piecewise continuous, RCLBV function) influences much the nature of in the sweeping process. If is Lipschitz, then is Lipschitz. If is RCLBV then is RCLBV [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF]. Therefore, controllability may be more difficult to study, since modifying the input space implies modifying the solutions as well (especially introducing jumps).

OP 12: The controllability of the ACS in ( 13) with and AC solutions may be attacked as follows. Assume that is full-column rank and let , . Then, one obtains [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] where and is the solution of the LCP:

, and

How may one use the CP in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] and the reduced-order dynamics in [START_REF] Cortès | Geometric, Control, and Numerical Aspects of Nonholonomic Systems[END_REF] (that may be a mixed or a simple LCP in ) for controllability purpose? Can one formulate similar conditions as ( 23) and ( 24) from a time-discretization of ( 14) and then deduce some properties of the continuous-time system by studying the limit of the discrete-time solutions [START_REF] Mabrouk | A unified variational model for the dynamics of perfect unilateral constraints[END_REF], [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF]? What is the influence of the convergence (strong, weak, weak [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF], [START_REF] Stewart | Rigid-body dynamics with friction and impacts[END_REF], [START_REF] Schatzman | A class of nonlinear differential equations of second order in time[END_REF]) on such a study? Is it possible to formulate some verification or feasibility criterion [START_REF] Bemporad | Observability and controllability of piecewise affine and hybrid systems[END_REF] that would allow the designer to test numerically some approximate controllability properties from a time-discretization? It is known that mere LCPs may be quite simple or almost untractable problems [START_REF] Billups | Complementarity problems[END_REF], depending on the properties of the LCP matrix. This is expected to impose severe restrictions on the data in ( 23) and ( 24) to get solutions of OP 12.

OP 13: Since and can be seen as an exogenous and a state (and ) dependent inputs, respectively, what is the role of the pairs , and in the controllability of ( 13)?

The study of simple first-order examples as in (2), with the input acting both in and in the vector field , might constitute a good starting point.

OP 14: Let us consider an ACS in (13) with , , , , . Then from ( 14) its DI formalism reads . Mimicking linear invariant systems [START_REF] Sontag | Mathematical Control Theory[END_REF], its conjugate (or adjoint) system may be defined as , , where is the conjugate of . How is controllability of the system related to observability of the system?

2) Inversion of Piecewise Linear Characteristics: The inversion of the operator PL , PL , whose graph is as in Fig. 1 is certainly an important property for the controllability of the overall system. The problem here is to find a such that PL for some desired signal . The deadzone of Fig. 3(b) with is invertible on : let us denote its output as and let us consider , . Then for all . Actually, one sees that the graph of PL: is exactly the inverse graph [86, p. 219] of the deadzone graph. The trick is that such a allows the system to jump between modes and , avoiding mode : it acts as a discrete control. This is closely linked to having in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF], see [START_REF] Bemporad | On hybrid systems and closed-loop MPC systems[END_REF]. One sees that the complementarity formalism clearly shows why some nonlinearities are invertible:

directly acts inside the complementarity conditions and allows one to impose switches of .

OP 15: Determine the conditions on the subsystem in Fig. 1 and/or the set of admissible inputs such that invertibility of PL implies controllability of the overall system.

In conclusion, the controllability of CS as in ( 1) is a difficult problem because they possess no discrete control, and steering trajectories with the input involves an interplay between the continuous dynamics and complementarity conditions.

VI. STABILIZATION AND FEEDBACK CONTROL

Consider for instance an ACS , . As shown in [75, ex. 3.2], the stability of the system does not imply that of this ACS [which is an evolution variational inequality (EVI)]. Stabilization of CS is therefore a subject that requires full attention. The characterization of invariant sets is a prerequisite to stabilization. The first section provides some insights on fixed points calculation for CS. Then we discuss stability, dissipativity, and optimal control. Finally the tracking control of two classes of CS (Lagrangian systems and jugglers) is briefly reviewed.

A. Existence of Fixed Points

Let us consider the system in (4) with . Then the point is a fixed point of this MDI if and only if the generalized equation (see Appendix B) [START_REF] Curnier | Unilateral contact: Mechanical modeling[END_REF] holds. The function is the potential energy of the system and , where denotes Coriolis and centripetal torques. We also used the fact that in the impact law, since . Let us notice that the condition [START_REF] Curnier | Unilateral contact: Mechanical modeling[END_REF] does not imply at all the uniqueness of the fixed point (for instance systems with clearance in which there is an infinity of fixed points [START_REF] Jimenez | Analysis of PD and nonlinear control of mechanical systems with dynamic backlash[END_REF]).

Example 10: Let us consider the simple example [START_REF] Dam | The contact problem for linear continuous-time dynamical systems: A geometric approach[END_REF] Obviously is the unique fixed point and one can check that condition [START_REF] Curnier | Unilateral contact: Mechanical modeling[END_REF] [START_REF] De Schutter | The linear dynamic complementarity problem is a special case of the extended linear complementarity problem[END_REF]. The results in [START_REF] Goeleven | Solvability theory and projection method for a class of singular variational inequalities: Elastostatic unilateral contact applications[END_REF] may also be used to study and solve (25) (if then this ELCP has one solution from [36, Th. 2.2]).

Generally, the fixed points of CS are the solutions of generalized equations .

B. Dissipative Systems 1) Lagrange-Dirichlet Theorem: It is crucial to develop a stability analysis framework for systems as in [START_REF] Abadie | Dynamic simulation of rigid bodies: Modeling of frictional contact[END_REF]. For Lagrangian mechanical systems as in (3), the mathematical results of [START_REF] Stewart | Rigid-body dynamics with friction and impacts[END_REF] and [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF] provide a nice ground. In summary, they imply (under some conditions which are not recalled here) the following. a) , (hence any quadratic function of is RCLBV as well [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF]). b) Therefore, the distributional derivative can be decomposed as the sum of three terms: an atomic measure , a Lebesgue integrable function , and a nonatomic measure singular with respect to the Lebesgue measure . c) The set of impact times is countable, and there exist possible left-accumulations . These very specific features are at the core of the developments in [START_REF] Brogliato | Nonsmooth Mechanics[END_REF], [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF], [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF], [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF], and [START_REF] Bourgeot | Tracking control of complementarity Lagrangian systems[END_REF]. Item b) is crucial, since it means that despite the time derivative of a Lyapunov function is not a function, it is a measure. Consequently, characterizing the sign of makes perfect sense, and if the measure , the function is decreasing [START_REF] Dieudonné | Eléments d'Analyse[END_REF] (in other words, RCLBV functions are the primitive of their distributional derivative). Such conclusions would not have been possible if had been proved to be measurable only. Item c) secures that the set is an admissible event times set [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF]Def. 3.3.4]. The measure in (5) can be chosen as . We note that without Coulomb friction and if all data are piecewise analytic, then [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF]. It is noteworthy that the fact that velocities are RCLBV does not preclude some very complicated phenomena like accumulations or accumulations of impacts (concretely, let and be two sequences of impacts with limits and , respectively, and let tend toward so that both accumulations are merged into a single one). So far, it has not been proved that such phenomena do not occur in mechanical systems with unilateral constraints and nonpurely elastic collisions.

Lemma 2: Consider a mechanical system as in [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF] where consists of Coriolis, centripetal and conservative generalized forces with a smooth potential , . Then, if has a strict minimum at , the equilibrium point satisfying [START_REF] Curnier | Unilateral contact: Mechanical modeling[END_REF] is stable in the sense of Lyapunov. Items a) and b) are in force here. The proof [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF] is based on the use of the nonsmooth Lyapunov candidate function [START_REF] Dankowicz | Exploiting discontinuities for stabilization of recurrent motions[END_REF] and on basic nonsmooth and convex analysis. Since for all , it follows that RCLBV and one has along trajectories of (5) [START_REF] Deimling | Multivalued differential equations[END_REF] where and Moreau's rule for differentiation of quadratic function of RCLBV functions [START_REF] Monteiro-Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction[END_REF] has been used. The advantage of using the MDI formalism, is that it might be a powerful tool in studying systems such that the measure is more complex [e.g., stochastic friction models inherently containing nonzero measures as in item b), or such that the support of is not contained in ]. It is of primary importance to note that the mapping in ( 5) is monotone, since and are polar cones. Lemma 2 brings an answer to whether or not the system Collision mapping [START_REF] Dieudonné | Eléments d'Analyse[END_REF] where , , , , are constant, has bounded solutions for all . The reader may check its application on simple systems like one degree of freedom jugglers [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF], impacting pair modeling backlash [START_REF] Jimenez | Analysis of PD and nonlinear control of mechanical systems with dynamic backlash[END_REF], with collocated PD feedback controllers.

OP 16: Extend Lemma 2 to systems with unilateral constraints and Coulomb friction, with nonconvex and . Extend it to all mechanical systems subject to set valued force laws that derive from a superpotential.

It is likely that solving OP 16 requires first modeling investigations [START_REF] Frémond | Non-Smooth Thermomechanics[END_REF], [START_REF] Ch | Set Valued Force Laws[END_REF].

2) Absolute Stability: Let us now turn our attention to a class of ACS. The absolute stability problem (see, e.g., [START_REF] Lozano | Dissipative Systems Analysis and Control. Theory and Applications[END_REF]) consists of studying the stability of a the negative feedback interconnection of (a positive real transfer function ) with a sector static nonlinearity with slope in . Usually, is required to be piecewise continuous in and locally Lipschitz in . It is of interest to extend this result to the case where the feedback nonlinearity is a maximal monotone multivalued mapping.

A multivalued mapping is monotone if for any couples and in its graph. Monotone mappings correspond to completely nondecreasing curves in . Maximality of multivalued monotone operators refers to graph inclusion (i.e., the gaps are filled in). The relay function defined as , if , is not maximal. However, the graphs in Fig. 3(e), (f), and(h) are maximal monotone.

The considered state-space equations are [START_REF] Dimnet | Collisions involving solids and fluids[END_REF] where a.e. is almost everywhere because solutions are expected to be AC. The link with CS is clear (choose, e.g., , ). The system in ( 16) fits within [START_REF] Dimnet | Collisions involving solids and fluids[END_REF] with and , as well as in (2) with , see [START_REF] Brogliato | A controllability criterion for linear juggling mechanical systems[END_REF]. The fixed points of (30) can be characterized as in Section VI-A. Let us assume that is a strictly positive real transfer function [START_REF] Lozano | Dissipative Systems Analysis and Control. Theory and Applications[END_REF], that is lsc and convex, so that is a maximal monotone multivalued mapping [START_REF] Rockafellar | Convex Analysis[END_REF], and that the graph of contains .

Lemma 3 [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF]: The solutions of (30) satisfying are AC and uniqueness holds. The fixed point of the system in ( 30) is unique and is globally asymptotically stable.

The stability proof relies on a quadratic Lyapunov function with the solution of the Kalman-Yakubovic-Popov lemma conditions [START_REF] Lozano | Dissipative Systems Analysis and Control. Theory and Applications[END_REF]. Related results can be found in [START_REF] Çamlibel | On linear passive complementarity systems[END_REF] (passive LCS) and in [START_REF] Nagurney | Projected Dynamical Systems and Variational Inequalities With Applications[END_REF]Ch. 3] (evolution VI). Passive LCS have the property that the transfer function is positive real. Passifiability by pole shifting is an interesting result for stabilization by feedback [START_REF] Çamlibel | On linear passive complementarity systems[END_REF]. The well-posedness proof of Lemma 3 requires some manipulations and the fixed point is the solution of the generalized equation . It seems that Brézis' theorem on global existence and uniqueness of solutions [START_REF] Brezis | Opérateurs Maximaux Monotones[END_REF] to DI , a maximal monotone operator, is central in the study of interconnections as in [START_REF] Dimnet | Collisions involving solids and fluids[END_REF]. The PR condition on can be relaxed when [START_REF] Yu | On wellposedness of relay systems[END_REF], at the price of restricting the set of solutions to piecewise differentiable functions. Let us notice that (30) is a particular DI. Many extensions exist, e.g., considering

, where is a monotone operator. When and the identity, (30) corresponds to the DI studied in [START_REF] Kenmochi | A class of nonlinear elliptic-parabolic equations with time-dependent constraints[END_REF]. Therefore, infinite dimensional generalizations of lemma 3 should be possible.

OP 17: A next step is also to extend Lemmas 2 and 3 toward ACS as in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF]. Then, extend toward dissipative nonlinear dynamics , . Actually, Lemma 3 readily extends to nonlinear systems of the form , , , , with storage function . The operator is strictly passive with dissipation function [START_REF] Lozano | Dissipative Systems Analysis and Control. Theory and Applications[END_REF]. Remark 7: The following is an issue [START_REF] Brogliato | Nonsmooth Mechanics[END_REF], [START_REF] Çamlibel | On linear passive complementarity systems[END_REF]: Since the supply rate may be meaningless in the sense of Schwarz' distributions at times , how to extend the dissipation equality (integrated or infinitesimal forms) to CS? Possible paths have been pointed out in [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]Sec. 1.2.2].

Finally, a natural extension of lemmas 2 and 3 is: OP 18: Characterize classes of CS such that the Krakovskii-LaSalle invariance lemma applies.

The invariance lemma relies on properties of the positive limit sets (compactness, invariance) which in turn are a consequence of properties of solutions like uniqueness, continuous dependence in the initial data, as well as properties of the Lyapunov function itself. By far, not all CS meet all these requirements.

C. Optimal Control

Let us focus on mechanical systems as in (3), to which items a), b), and c) apply. As pointed out in [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]Sec. 3.5.4] the following problem remains unsolved.

OP 19: Find that is critical for some action integral , under the dynamics , , . How do the number of impacts and the restitution influence ? In which space should one consider ? Then generalize to higher order systems.

Applications are in the optimal control of all mechanical systems subject to impacts and unilateral constraints. It is note-worthy that the optimal control of MDE (see, e.g., [START_REF] Schmaedeke | Optimal control theory for nonlinear vector differential equations containing mesures[END_REF]), does not solve OP 19. The optimal control of a time invariant linear system with virtual state constraints yields optimality conditions [START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF] (31) for some and matrices , , . Notice the striking similarity between ( 31) and ( 6), though both systems represent quite different problems.

A prerequisite to a better understanding of optimal control for the aforementioned classes of systems, is a better understanding of the dynamics of CS. The optimal control of a simple ACS has been solved in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]Sec. 5.3] and extensions toward more general ACS may be an objective. Applications are in optimal control of electrical networks with diodes and/or MOS transistors.

D. Tracking Control for Lagrangian Systems

This problem has been studied in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF], [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF], and [START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF], for systems as in (3), with a single constraint, and the identity matrix. In [START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF], planar systems inside a disc are considered, with elastic impacts. PD inputs are shown to asymptotically stabilize particular impacting trajectories. The objective in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF] and [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF] is to design a controller that assures stable tracking of some reference trajectory , relying on a suitable stability framework that encompasses both unconstrained, persistently constrained, and impacting orbits. Features a), b), and c) enumerated in the introduction of Section VI-B hold, and the goal is to extend lemma 2 to the case of tracking control. There are four main features in this extension.

i) Impacts have to be included in any stability analysis, and the designer should take advantage that impacts imply a loss of kinetic energy in the stabilization. ii) There are two antagonist facts: robustness of the stabilization process on requires that a "bouncing-ball"like dynamics be realized in closed loop [notice that in [START_REF] Dam | The contact problem for linear continuous-time dynamical systems: A geometric approach[END_REF] needs not to be known to get stabilization], but asymptotic stability implies that velocity jumps vanish asymptotically. iii) Requiring that for all [see item b)] is too stringent in most cases, especially during transition phases including rebounds. iv) The underlying nature of the evolution process and the trajectories has to be taken into account. Item i) leads to the choice of a Lyapunov function which is as close as possible to the process total energy. The function is a suitable one, . Item ii) implies a specific transition phase in order to stabilize the system on . The dynamics should be similar to [START_REF] Dam | The contact problem for linear continuous-time dynamical systems: A geometric approach[END_REF] during the transient, whereas a tangent approach should be asymptotically guaranteed. Attaining the constraint tangentially, and without incorporating impacts in the stability analysis, cannot work neither theoretically nor in practice. The main source of difficulty for stability lies in the coupling between normal and tangential directions to , i.e., the generally cannot be transformed in a block-diagonal matrix via a suitable generalized coordinate change. When the dynamics is decoupled, the control problem is much easier [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]Sec. 8.6.3]. Item iii) means that it is often more convenient to think of the dynamics during the transition phase, in terms of an impact Poincaré map with Poincaré section as in problem 1. Item c) secures that such a discrete-time system can be constructed. Then stability may be characterized as the Lyapunov stability of with Lyapunov function the restriction of to . Item iv) means that the dynamics in ( 26) and [START_REF] Ferris | Engineering and Economic Applications of Complementarity Problems[END_REF] are of different natures. In [START_REF] Dam | The contact problem for linear continuous-time dynamical systems: A geometric approach[END_REF], the invariant set is the solution of the generalized equation ( 25) and is, therefore, characterized by all the ingredients of the dynamics (including the impact law). In [START_REF] Ferris | Engineering and Economic Applications of Complementarity Problems[END_REF], the invariance is simply that of the continuous dynamics, since . The stability framework and the controller proposed in [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF] and [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF] take these peculiar features into account. The proposed controller is based on the Paden and Panja controller [57, Sec. 6.2.5] for unconstrained systems. In particular the transition phase uses a signal as depicted in Fig. 5, with a closed-loop dynamics similar to [START_REF] Ferris | Engineering and Economic Applications of Complementarity Problems[END_REF] in the normal direction to , to cope with item ii). The design of during the stabilization phase on is a crucial step. The use of the (fixed parameter) Slotine and Li controller might be of interest too in this context [START_REF] Bourgeot | Tracking control of complementarity Lagrangian systems[END_REF], since it possesses the advantage over the Paden and Panja's one of assuring , , on free motion phases. However, the function is less close to the mechanical energy for this controller. In fact, Moreau's MDI in [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF] has not yet been fully exploited in this setting.

The tracking problem for fully actuated complementarity Lagrangian systems is not a simple extension of the unconstrained case. The design of a suitable desired trajectory during transition phases, is a crucial step.

The aforementioned results apply to the case , i.e., a unique constraint with in (3) continuously differentiable. Let . OP 20: Study control strategies that stabilize the system on the codimension surface , with any initial conditions. In particular, the transition phase controller should be robust with respect to all types of uncertainties that may modify some dynamical properties. Then extend the tracking controllers of [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF] and [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF] which concern (3), to the case .

The impact with the surface is called a multiple impact. Let us recall that Lyapunov stability is equivalent to some continuity of solutions with respect to initial data, on an infinite time interval. Multiple impacts may destroy this property, see [START_REF] Heemels | Linear complementarity systems[END_REF], [START_REF] Schatzman | A class of nonlinear differential equations of second order in time[END_REF], [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]. This depends a lot on the domain geometry, the mass matrix and the impact law. Therefore uncertainties on these physical data may drastically modify the trajectories which collide in the neighborhood of . Once OP 20 has been solved, proceed to the following.

OP 21: Extend the results in [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF] and [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF] to the case of joint flexibilities.

Both OP 20 and 21 have important practical consequences. Indeed in most mechanical systems . Concerning flexibilities, it is expected that the effect of vibrations in the structure due to the impacts, may be modeled this way. The consequences on stabilization may be crucial. The control of biped robots in a walking motion seems quite particular and its link with OP 20 needs to be clarified. Indeed the basic stability of a walking biped requires that at each instant the state and the input satisfy inequalities outside impacts and at impacts [START_REF] Hurmuzlu | Modeling, stability and control of biped robots-A general framework[END_REF], for some matrices and vectors , and . Designing such a feedback control is an open problem. The extension toward higher order CS as in ( 2) is challenging as well.

Control of Jugglers: Quite interesting stabilization results have been obtained in [START_REF] Lynch | Recurrence, controllability, and stabilization of juggling[END_REF] for a planar juggler, using ideas of control recurrence. A framework for the control design of mechanical systems as in [START_REF] Çamlibel | On linear passive complementarity systems[END_REF] has been proposed in [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF], and the control of an impacting pair modeling dynamic backlash has been studied in [START_REF] Jimenez | Analysis of PD and nonlinear control of mechanical systems with dynamic backlash[END_REF]. A feedback stabilization technique using the linearization of ( 19) is used in [START_REF] Dankowicz | Exploiting discontinuities for stabilization of recurrent motions[END_REF]. An interesting problem is in the control of buildings subject to earthquake excitations, modeled with the rocking block approach [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]; where should the actuators be placed so as to avoid overturning, and minimize the angular motion magnitude? This is thought to be a tough problem (including multiple impact and friction).

OP 22: Characterize classes of tasks in terms of the closed-loop invariant sets to be stabilized, and derive a general stability and controller design framework.

VII. CONCLUSION

The class of nonlinear nonsmooth controlled dynamical systems considered in this paper (namely CS) is quite interesting, since it still offers many challenges to various scientific communities (including systems and control) and at the same time finds many applications. Such systems may be recast in the class of hybrid dynamical systems, but can also be seen as measure differential inclusions. The adopted formalism depends on the objective of the study. The goal of this paper is to expose the difficulties and main problems associated to such complex dynamics. Several notions which are peculiar to complementarity dynamical systems are introduced, such as sets of admissible states and inputs, normal controllability, viability of controls. One interesting question is whether it is possible to classify complementarity dynamical systems into subclasses with specific control properties, or not. Until now, the studies have focused on linear CS [START_REF] Çamlibel | On linear passive complementarity systems[END_REF], [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF], [START_REF] Heemels | The rational complementarity problem[END_REF]- [START_REF] Heemels | Linear complementarity systems[END_REF], [START_REF]The complementarity-slackness class of hybrid systems[END_REF], nonlinear CS [START_REF] Van Der Schaft | Complementarity modeling of hybrid systems[END_REF] and mechanical systems [START_REF] Brogliato | On the control of finite dimensional mechanical systems with unilateral constraints[END_REF]- [START_REF] Brogliato | On tracking control of a class of complementarity-slackness mechanical systems[END_REF], [START_REF] Brogliato | On the control of complementarityslackness juggling mechanical systems[END_REF], [START_REF] Huang | Time optimal control for a robotic contour following problem[END_REF], [START_REF] Lin | Yoyo dynamics: Sequence of collisions captured by a restitution effect[END_REF], [START_REF] Lynch | Recurrence, controllability, and stabilization of juggling[END_REF], [START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF], [START_REF] Menini | State estimation of (otherwise unobservable) linear mechanical systems through the use of nonsmooth impacts: The case of two mating gears[END_REF], [START_REF] Niculescu | Force measurement time-delays and contact instability phenomena[END_REF], [START_REF] Pfeifer | [END_REF]. Complementarity dynamical systems do represent a very specific class of hybrid systems for which many problems heavily rely on the use of complementarity problems and convex analysis. This peculiar structure is very useful since it allows one to deeply investigate their properties, and at the same time is not restrictive in terms of potential applications which are numerous. The possible relationships with other types of nonsmooth evolution problems are pointed out. Some open problems are proposed; clearly many others exist and will hopefully be given solutions. Finally, it is worth noting that the numerical simulation of complementarity systems is also a hard problem and that most available software packages do not allow to treat correctly neither complementarity problems nor finite accumulation of events [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF]. the th component of the vector . Many other types of complementarity problems exist [START_REF] De Schutter | The linear dynamic complementarity problem is a special case of the extended linear complementarity problem[END_REF] which cannot be described here for the sake of briefness of the presentation. A generalized equation is an equation of the form , where is a multivalued mapping [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications[END_REF] [i.e., and does not imply ; the mappings whose graphs are in Fig. 3(d), (e), (f), (g) and (h) are multivalued]. In particular the complementarity condition can equivalently be rewritten as the generalized equation , which also shows the link between complementarity and convex analysis [86, p. 226]. This will also be useful to transform complementarity systems into various sorts of differential inclusions.

Therefore, one sees that there are strong links between generalized equations, convex analysis, and complementarity problems.

Fig. 1 .

 1 Fig. 1. Systems with a piecewise linear interconnection.

Fig. 2 .

 2 Fig. 2. Painlevé system.

Fig. 3 .

 3 Fig. 3. Graphs of the piecewise-linear relations.

Fig. 4 .

 4 Fig. 4. Controllability in the normal cone.

Fig. 5 .

 5 Fig. 5. Transition phase trajectory.

  are fundamental tools for the analysis of complementarity systems, and we will use most of them. Let denote a closed convex set.

							Its in-
	dicator function is defined as		if		and
		if	. The subdifferential of a lower
	semicontinuous (lsc) convex function		at	is denoted
	as	and is the set of subgradients, i.e., vectors sat-
	isfying	for all . The subdifferen-
	tial replaces the derivative, for nonsmooth convex functions.
	If	for	,	for		,	,
	then	is convex lsc and		. The subdif-
	ferential of the indicator of	(which is convex if and only
	if	is convex) is given by		if		,
		if	,		if		, where
		is the boundary of , and				,
		is the outwards normal cone to	at		. The
	inwards tangent cone	is the polar cone to		and
	is defined as		,			. Both
	the normal and tangent cones are convex and are defined at
	points of the boundary

  12, Remark 8.12]. But this is not true in general [87, Sec. 1.2.5].

		Piecewise Smooth Systems (PSS): PSS may be defined as
		if	,	, each	has a nonempty
	interior,	is either a zero-measure set included in	or
	in	(e.g.,	and	), or (e.g.,	and	). The
		are smooth vector fields, is an index (positive integer).
	Mimicking OP 2 and OP 3, let us state the following.
		OP 4: What CS can be represented equivalently as PSS, and
	vice versa?

Some of these open problems may be very challenging, and some others more trivial. It is rather difficult to classify them at this stage. Other research directions can be found in[START_REF] Heemels | Linear complementarity systems. A study in hybrid dynamics[END_REF] and[START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF].

ACKNOWLEDGMENT

The author would like to thank M. Monteiro-Marques, M. Heemels, M. Mabrouk, and M. di Bernardo for some fruitful discussions on parts of this paper and preliminary versions.

APPENDIX A MEASURE DIFFERENTIAL INCLUSIONS

The classical equation of dynamics is an equality of functions, like

. Imagine now that the velocity has jumps at times . Then, the left-and right-hand sides are measures instead of mere functions, and one may rewrite such dynamics as an equality of measures , where is an atomic measure with atoms at . In this case, writing the dynamics as is meaningless because at a jump time has no density with respect to the Lebesgue measure (it has a density with respect to the Dirac measure which in turn is singular to ). Now if the right-hand side is a multivalued function , one may write the dynamics as . The concept of MDI merges both equality of measures and inclusion and has been coined by Moreau [START_REF]Liaisons Unilatérales Sans Frottement et Chocs Inélastiques[END_REF] and Schatzman [START_REF] Schatzman | A class of nonlinear differential equations of second order in time[END_REF]. This is an evolution problem of the form , where is a positive measure and is a cone, is Lebesgue integrable, and the inclusion holds almost everywhere. The notation generalizes that of a derivative, for measures. For instance if and if jumps at , then . The fact that can be any positive measure comes from the fact that is a cone. If is merely convex this no longer holds. If is discontinuous and is continuous at , , one may choose and write the dynamics as

. At one has for some whereas . Outside impacts and . Finally the functions are evaluated at their right limits because the solutions of Moreau's MDI are right continuous.

APPENDIX B COMPLEMENTARY PROBLEMS (CONTINUED)

Solving the quadratic program , , subject to , is equivalent [74, Sec. 9.3.1] to solving a LCP with , . This has important practical consequences, e.g., for numerical calculations [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF], [START_REF] Çamlibel | Complementarity methods in the analysis of piecewise linear dynamical systems[END_REF], [START_REF]Some numerical methods in multibody dynamics: Application to granular materials[END_REF]. An ELCP is a complementarity problem that can be written as , , , for some matrices , , vectors , , and subsets of ; denotes