Yuri Mikhlin

Sergei Mitrokhin

NONLINEAR NORMAL VIBRATION MODES AND THEIR APPLICATIONS IN SOME APPLIED PROBLEMS

Keywords: Nonlinear normal modes, absorber, suspension

Nonlinear normal vibration modes (NNMs) are a generalization of the normal vibrations in the linear systems. In conception of NNMs by Kauderer-Rosenberg all position coordinates can be defined from any one of them. In conception of NNMs by Lyapunov-Shaw-Pierre all phase coordinates can be defined from two selected ones. Curvilinear trajectories of NNMs in a configuration space, or in a phase space, can be obtained as power series.

The NNMs theory is used to study vibrations of some linear structure connected with the single-DOF nonlinear absorber. An essentially nonlinear oscillator, a snap-through truss with three equilibrium positions, and a vibro-impact oscillator are chosen as absorbers. Construction and stability analysis of the NNMs are presented. If the localized mode is stable, and the non-localized vibration mode is unstable, the vibration energy is concentrated in the absorber.

Free damped oscillations of the double tracked road vehicle with a nonlinear response of the suspension can be considered by the NNMs theory too. The 7-DOF nonlinear model is used to analyze the suspension dynamics with smooth characteristics. The quarter-car model is considered for a case of the nonsmooth characteristic of the shock absorber.

Introduction

Nonlinear normal vibrations modes (NNMs) are a generalization of the normal vibrations of the linear systems. In the normal mode, a finite-dimensional system behaves like a single DOF conservative one, and all position coordinates can be analytically parametrized by any one of them. This conception of NNMs was proposed by Kauderer and Rosenberg.

Rosenberg [START_REF] Rosenberg | The normal modes of nonlinear n-degrees-of-freedom systems[END_REF][START_REF] Rosenberg | Nonlinear vibrations of systems with many degrees of freedom[END_REF] defined NNMs as 'vibrations in unison' and introduced a broad class of essentially nonlinear conservative systems allowing for NNMs with rectilinear trajectories in a configuration space.

In general, the NNMs trajectories are curvilinear instead of straight lines in linear systems. The power series method was proposed in [START_REF] Manevich | Periodic solutions close to rectilinear normal vibration modes[END_REF][START_REF] Manevich | The Method of Normal Oscillation for Essentially Nonlinear Systems[END_REF] to construct the curvilinear trajectories of NNMs. Shaw and Pierre reformulated the concept of NNMs for a general class of nonlinear discrete oscillators [Shaw andPierre, 1991, 1993]. The analysis is based on the computation of invariant manifolds on which the NNM oscillations take place. All phase coordinates can be well defined from a pair of phase coordinates. This idea was first proposed by Lyapunov [START_REF] Lyapunov | The General Problem of the Stability of Motion[END_REF]. In [START_REF] Mikhlin | Matching of local expansions in the theory of nonlinear vibrations[END_REF] Padé approximations are used for an analysis of NNMs with large amplitudes.

Rauscher's ideas and the power-series method for trajectories in a configurational space are used in the construction of NNMs in nonautonomous and selfexcited systems, close to conservative ones [START_REF] Mikhlin | Resonance modes of nearconservative nonlinear systems[END_REF][START_REF] Mikhlin | Normal vibrations in near-conservative self-exited and viscoelastic nonlinear systems[END_REF]. In such systems some additional potentiality conditions must be used. It means that a loss of energy on the average over the period of the periodic solutions under consideration, is absent.

Basic results on NNMs are presented in the book by Vakakis et al. [START_REF] Vakakis | Normal Modes and Localization in Nonlinear Systems[END_REF] which describes quantitative and qualitative analyses of NNMs, including localized modes, an analysis of stability, and an investigation of NNMs in distributed systems.

Nonlinear normal modes in a system with an essentially nonlinear absorber

Numerous publications contain an analysis of different devices for the vibration absorption of mechanical systems. The general theory of the linear/nonlinear absorbers is presented in [START_REF] Kolovski | Nonlinear theory of vibroabsorber systems[END_REF][START_REF] Frolov | Vibrations in engineering[END_REF].

Oscillations of the two-DOF system which contains the essentially nonlinear absorber can be studied by the NNMs approach. Both the non-localized, and the localized vibration modes are possible here. The localized NNM is appropriate for the absorption, when the main linear system and the absorber have small and large amplitudes, respectively. If the localized mode is stable, and the non-localized vibration mode is unstable, the vibration energy is concentrated in the absorber. A system is considered where the nonlinear single-DOF absorber connected with a fixed point by cubically nonlinear spring (Fig. 1). Here a mass of the absorber m is essentially smaller than that of the linear subsystem M . . This is the non-localized vibration mode. The classical procedure of the small parameter method permits to present the NNM trajectory as power series with respect to ε . The localized vibration mode can be analized if the next time transformation is introduced:

τ ε = t
. One has in this case the following zero approximation with respect to ε : 0 0 = y . The first approximation solution can be obtained in the form of power series by x . Test numerical calculations show a good accuracy of the analytical solutions. A harmonic approximation of the non-localized vibration mode permits to reduce a problem of the NNM stability to a single variational equation in the form of the standard Mathieu equation parameters • • δ ε , (these parameters are defined by the system (1) parameters). The vibration amplitudes are not limited here. The Fig. 2 shows a position of point, corresponding to the NNM, in a plane of the Mathieu equation parameters (for some fixed values of the parameter γ and some other parameters). Direction of the vibration amplitudes increase is showed by the arrows. The non-localized vibration mode is situated in the region of instability (this region is shaded), but for not very big vibration amplitudes the solution is near the boundary of the region. A problem of the localized vibration mode stability can be reduced to the Mathieu equation too. It be obtained that the localized mode is stable for almost all values of the system parameters [START_REF] Mikhlin | Dynamical interaction of an elastic system and essentially nonlinear absorber[END_REF]. Other approach of the NNMs stability analysis is based on the so-called algebraization by Ince. In this case a new independent variable associated with the solution under consideration is chosen [START_REF] Ince | Ordinary Differential Equations[END_REF]. Then the variational equation is converted to the equation with singular points. This approach was used earlier in the NNMs theory [START_REF] Mikhlin | An application of the Ince algebraization to the stability of nonlinear normal vibration modes[END_REF]. This algebraization gives us more exact results than those obtained by the reduction of the stability problem to the Mathieu equation [START_REF] Mikhlin | Dynamical interaction of an elastic system and essentially nonlinear absorber[END_REF]. One introduces the following transformation of the independent variable: x t → . One has after some transformations the variational equation with singular points. It can be demonstrated that in this case solutions corresponding to boundaries of the stability/instability regions are the following:

… + + + + = 3 3 2 2 1 0 1 x a x a x a a v ,
( )

… + + + + ⋅ - = 3 3 2 2 1 0 2 2 0 2 x b x b x b b x X v .
Substituting the series to the variational equation and matching respective powers of x , we can write recurrent systems of linear algebraic equations to determine coefficients of the expansions. These systems have nontrivial solutions if their determinants are equal to zero. One obtains from here the stability/instability regions boundaries. It generally corresponds to results obtained for the Mathieu equation. But for small vibration amplitude the nonlocalized vibration mode is situated in the domain of stability near its boundary and gets into the instability domain if the vibration amplitude increases. Additional narrow instability regions are obtained for the localized vibration mode in a case when the stiffness coefficient γ is small.

Snap-through truss as a vibration absorber

Possibility of the elastic oscillations absorption by means of the snap-through truss is considered here. The nonlinear absorber with three equilibrium positions (snap-through truss) is attached to the linear oscillator. By assumption, the truss is shallow and its γ m с М mass and stiffness are significantly smaller then the corresponding parameters of the linear system.

; ; ≺≺ ε γ ε = γ µ ε = µ . The new variable is introduced too: ) 1 /( ) k 1 ( u 1 u γ + - γ + =
. Retaining the linear, quadratic and cubic terms by 1 u and w we can write the equations of motion as

; 0 2 w 2 2 2 w 1 u 3 1 u ) 1 ( 1 u = ρ γ ε - ρ γ ε - γ ε + + , 0 3 w 2 2 1 wu 2 w 2 w = β γ + ρ γ - α γ - µ (2) where . 3 1 3 1 2 ; 2 1 1 2 ; 1 κ + ρ = β - κ + ρ = α γ + κ + γ = ρ
To study the periodic motions of the system (2) with large amplitudes we determine here the NNM in the form u 1 =u 1 (w) [Avramov and Mikhlin, 2004]. The solution is presented as power series, which is then substituted into the corresponding equation and to the boundary conditions to obtain the coefficients of the series. Numerical calculations show a good accuracy of the analytical solution. The snap-through truss has significant amplitudes of oscillations and the linear oscillator has small amplitudes. If such motions are stable, the vibration absorption is guaranteed.

The snap-through periodic motion stability

A small curvature of the obtained NNMs is used here to analyze their stability [Avramov and Mikhlin, 2004]. Let us introduce new variables (ξ,η). The ξ axis is directed along the rectilinear approximation of the NNM trajectory and the orthogonal variation η(t) defines the orbital stability of NNMs. One considers a stability of the periodic motions with large amplitudes taking into account that

) ( O u 1 ε =
and introducing the following representation: w=w 0 +O(ε). We study variations η(t) of periodic motions 1 u : η + = 1 1 u u . One obtains after some transformations the following variational equation:

, 0 ) 2 0 w 3 1 ( = η ρ γ ε - γ ε + + η (3) where ). k ; 4 H 4 1 p ( cn H 4 1 1 2 0 w + ατ + + β α =
Here k is the elliptic integral modulus; H is the oscillator total energy. Substituting the Fourier-series expansion of cn 2 (t,k 0 ) into (3), we can rewrite it as the Hill's equation which is analyzed by the multiple scales method. Resonances of the orders s

(s=1,2,3…) are considered: , 0 2 * s s εσ + ω = Ω
where σ s is the detuning parameter. Since the snapthrough truss is shallow, the next notations can be introduced:

.

1 * ; 1 с * с 1 ≺≺ ε ε = -
After some analysis of the corresponding modulation equations, which are not presented here, we can obtain the stable/unstable region boundaries on the system parametric plane ) * W , с ( as

), ( O ) 2 * ( O 2 * W 1 c 2 c * 1 c 2 * W s p ) 0 k ( сK 2 ε + ε + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ε - = π (4) where ); 2 * ( O 2 * W 1 c 2 c * 2 1 2 0 k ε + ε + = max W * W = ; ). 2 * ( O 2 * W 2 1 c 2 c 2 1 K * 2 1 K ) 0 k ( K ε + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ε + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =
Here K(k 0 ) is the complete elliptic integral of the first kind. Fig. 4 shows curves (OA 1 ),(OA 2 ),(OA 3 ) in the plane ) * W , с ( , which are in correspondence with the equation ( 4). The boundaries of the stable/unstable regions are shown qualitatively as curves (B 1 C 1 D 1 ), (B 2 C 2 D 2 ), (B 3 C 3 D 3 ). If ϕ is small, the unstable oscillations regions have the order O(ε). If value ϕ is increased, the width and number of the unstable regions are decreased. We can choose such values of ϕ, that is the periodic motions are always stable.

Figure 4: The instability regions of the snap-through motion.

Remark

Note that a similar analysis of localized and nonlocalized NNMs is realized for the two-DOF system containing a vibro-impact oscillator as absorber. Besides, the forced resonances are investigated in such systems with three types of absorbers, namely, an essentially nonlinear oscillator with a single equilibrium position [START_REF] Mikhlin | Dynamical interaction of an elastic system and essentially nonlinear absorber[END_REF]; a snap-through truss with three equilibrium positions [Avramov and Mikhlin, 2004]; and a vibroimpact oscillator.

Principal model of the vehicle suspension nonlinear dynamics.

7-DOF model of the vertical and axial vehicle dynamics is considered here for a case of independent-solid axle suspension to predict the vehicle body and wheel states. It is possible to consider, by using this model, all principal vehicle motions. The correctness of this model is substantiated by comparison with some experiments [START_REF] Wong | The Theory of Ground Vehicles[END_REF][START_REF] Kim | Improving the vehicle performance with active suspension using road-sensing algorithm[END_REF][START_REF] Pilipchuk | Vehicle state estimation using dynamic 7 degree-of-freedom observer[END_REF]. It is known that nonlinear effects in the suspension dynamics are important if corresponding displacements are in the order of 0.05-0.1 m, or larger. Here the smooth nonlinear spring characteristics in front and rear suspensions are taken into account. NNMs and corresponding amplitudefrequency relations are obtained for the nonlinear 7-DOF system.

In order to describe the double-tracked road vehicle dynamics, the 7-DOF mathematical model is used (Fig. 5). It considers the heave, roll and pitch motions of the car body. Here z is the vertical displacement, α is the pitch angle, β is the roll angle, i x is the vertical displacement of i-th suspended mass which are equivalent to the wheel, 1 d , 2 d are the front and the rear track widths and 1 l , 2 l are the front and the rear wheel bases. In this model tires are presented as elastic elements with linear characteristics. The suspension is characterized by nonlinear elastic characteristics of the front and rear springs, and linear damping characteristics. Typical elastic characteristics are shown in the Fig. 6.

… … + + + + + = = + + + + + = = v b uv b u b v b u b v u Y y v a uv a u a v a u a v u X x i i i i i i i i i i i i i i (5)
To calculate coefficients of the series (5) one has a system of algebraic equations. One of the NNMs obtained here is shown in the Fig. 7, where the coordinate z is chosen as the independent variable u . If the NNM in the form ( 5) is obtained, the series are substituted to equations of motion, and functions u = u(t) and v = v(t) can be obtained too. As a result, seven NNMs were determined. Numerical calculations show a good accuracy of the obtained analytical results. To construct the NNMs skeletons the harmonic linearization method together with a continuation procedure were used. In a case of large initial displacements only the lowfrequency vibration modes of the car body are stable, and other vibration modes tend to the low-frequency modes.

β α z X Z Y f 1 (x) f 1 (x) k 2 k 2 k 1 k 1 m 2 m 2 m 1 m 1 M c t c t c t c t l 1 l 2 d 1 d 2 x 1 x 2 x 3 x 4 f 2 (x f 2 (x)

Nonlinear normal modes and the transients (shock absorber, non-smooth characteristic)

To investigate the suspension dynamics taking into account a non-smooth characteristic of the shock absorber, the quarter-car model is considered (Fig. 8).

⎩ ⎨ ⎧ = + - + - + = - + - + . 0 ) ( ) ( , 0 ) ( ) ( 2 1 2 1 2 2 2 1 2 1 1 x c x x d x x f x m x x d x x f x M t (6) where ) (x f is a stiffness function, ) (x d is a piecewise damping function of the suspension, namely, ⎩ ⎨ ⎧ ≥ - < - = . 0 , ) ( , 0 , ) ( ) ( 2 1 2 2 1 1 x x x d x x x d x d (7)
An analysis of NNMs is made for a case of nonsymmetric piecewise linear damping characteristic. Both NNMs were obtained by using the method which is described below. Motions on places, corresponding to NNMs, are shown in Fig. 9. After each gap of the piecewise linear damping characteristic, the short transient from one place to another one can be observed. It is shown that methods of the NNMs theory permit to describe a behavior of systems containing nonlinear dynamical absorbers. The nonlinear dynamics of the double tracked road vehicle with a nonlinear response of the suspension can be analyzed by using the NNMs approach both for a case of the smooth nonlinear characteristic of suspension, and for a case of the non-smooth characteristic of the shock absorber.

Fig. 1 .

 1 Fig.1. The system under consideration One has the following equations of motion: ( ) ( )

Fig. 2 :

 2 Fig. 2: Region of the non-localized mode instability.

Fig. 3 .

 3 Fig.3. The system with the snap-through truss Let ) W , U ( be the generalized coordinates; L is a length of the spring; ϕ is the angle, corresponding to the equilibrium position; κ is a spring stiffness of the truss; κ 1 is a stiffness of the main elastic system. The dimensionless variables L / W w ; L / U u = = and

Fig. 5 .

 5 Fig. 5. Mathematical model of a double-tracked road vehicle under consideration One has seven generalized coordinates to describe vibrations of the model.

  a) The angle α versus u and v b) The displacement x 1 versus u and v c) The angular velocity α versus u and v d) The velocity 1 x versus u and v Fig. 7. The first NNM (independent variable z)

Fig. 8 .

 8 Fig. 8. The quarter-car model Equations of motion for the quarter-car model are

  Fig. 9. First NNM. Independent variables x 1 and the corresponding velocity (piecewise linear system)

Fig. 10 .

 10 Fig. 10. The piecewise cubic shock absorption characteristic in the suspension

Acknowledgements

This work was supported in part by the grant Ф25.1/042 of the Foundation of the fundamental researches by Ukraine.


On existence, uniqueness, and convergence, of optimal control problems governed by parabolic variational inequalities

Mahdi Boukrouche ⋆ and Domingo A. Tarzia ⋆⋆ Mahdi Boukrouche, Lyon University UJM F-42023, CNRS UMR 5208, ICJ, France. Domingo A. Tarzia, CONICET and Austral University, Rosario, Argentina.

Abstract. I) We consider a system governed by a free boundary problem with Tresca condition on a part of the boundary of a material domain with a source term g through a parabolic variational inequality of the second kind. We prove the existence and uniqueness results to a family of distributed optimal control problems over g for each parameter h > 0, associated to the Newton law (Robin boundary condition), and of another distributed optimal control problem associated to a Dirichlet boundary condition. We generalize for parabolic variational inequalities of the second kind the Mignot's inequality obtained for elliptic variational inequalities (Mignot, J. Funct. Anal., 22 (1976), 130-185), and we obtain the strictly convexity of a quadratic cost functional through the regularization method for the non-differentiable term in the parabolic variational inequality for each parameter h. We also prove, when h → +∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition. II) Moreover, if we consider a parabolic obstacle problem as a system governed by a parabolic variational inequalities of the first kind then we can also obtain the same results of Part I for the existence, uniqueness and convergence for the corresponding distributed optimal control problems. III) If we consider, in the problem given in Part I, a flux on a part of the boundary of a material domain as a control variable (Neumann boundary optimal control problem) for a system governed by a parabolic variational inequality of second kind then we can also obtain the existence and uniqueness results for Neumann boundary optimal control problems for each parameter h > 0, but in this case the convergence when h → +∞ is still an open problem.

Keywords: Parabolic variational inequalities, convex combination of solutions, regularization method, optimal control problems, strict convexity of cost functional.

Introduction

The goal of this paper is to show the existence and uniqueness results to a family of distributed (see Sections 2 and 3) or Neumann boundary (see Section 4) optimal control problems for each parameter h > 0, associated to the Newton law (Robin boundary condition on a part of the boundary of the material domain), and of another distributed optimal control problem associated to a Dirichlet boundary condition. The system of these optimal control problems are governed by free boundary problems (with Tresca boundary condition (see Sections 2 and 4) or of an obstacle type problem (see Section 3) through a parabolic variational inequalities of the first (see Section 3) or second (see Sections 2 and 4) kind [START_REF] Barbu | Optimal control of variational inequalities[END_REF], [START_REF] Boukrouche | Convergence of distributed optimal controls for second kind parabolic variational inequalities[END_REF]. An optimal control problem for elliptic variational inequality of the second kind is given in [START_REF] De | Optimal control of a class of variational inequalities of the second kind[END_REF]. In order to prove the existence and uniqueness results we generalize for parabolic variational inequalities of the second kind the Mignot's inequality obtained for elliptic variational inequalities [START_REF] Mignot | Contrôle dans les inéquations variationelles elliptiques[END_REF], and then we obtain the strictly convexity of a quadratic cost functional through the regularization method for the non-differentiable term for each parameter h > 0.

We also prove, when h → +∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition.

We obtain these convergence without using the adjoint state which is a great advantage with respect to the proof given previously for optimal control problems governed by elliptic and parabolic variational equalities [START_REF] Ben Belgacem | Singular perturbation for the Dirichlet boundary control of elliptic problems[END_REF], [START_REF] Gariboldi | Convergence of distributed optimal controls on the internal energy in mixed elliptic problems when the heat transfer coefficient goes to infinity[END_REF], [START_REF] Gariboldi | Convergence of boundary optimal controls problems with restrictions in mixed elliptic Stefan-like problems[END_REF], [START_REF] Menaldi | A distributed parabolic control with mixed boundary conditions[END_REF].

These convergence when h → +∞ are valid for the optimal control problems given in Sections 2 and 3, and it is still an open problem for the Neumann boundary optimal control problem given in Section 4.

2 Distributed optimal control problems governed by parabolic variational inequality of second kind

Let Ω a bounded open set in R N with smooth boundary ∂Ω = Γ 1 ∪ Γ 2 such that Γ 1 ∩ Γ 2 = ∅, and meas(Γ 1 ) > 0. We set V = H 1 (Ω), V 0 = {v ∈ V : v | Γ 1 = 0}, H = L 2 (Ω), H = L 2 (0, T ; H), V = L 2 (0, T ; V ), and the closed convex set K b = {v ∈ V : v | Γ 1 = b}. Let given b ∈ L 2 (0, T ; H 1/2 (Γ 1 )), b > 0, g ∈ H, g ≥ 0, q ∈ L 2 ((0, T ) × Γ 2 ), q > 0, u b ∈ K b . (1)
We consider the following variational problems [START_REF] Boukrouche | Convergence of distributed optimal controls for second kind parabolic variational inequalities[END_REF] Problem 1. Let given g, q, b and u b as in [START_REF] Amassad | Optimal control of an elastic contact problem involving Tresca friction law[END_REF].

Find u = u g ∈ C(0, T, H) ∩ L 2 (0, T ; K b ) with u ∈ H, such that u(0) = u b
, and solution of the parabolic variational inequality of second kind:

< u, v -u > + a(u, v -u) + Φ(v) -Φ(u) ≥ (g, v -u), ∀v ∈ K b , t ∈ (0, T ).
Problem 2. Let given g, q, b and u b as in [START_REF] Amassad | Optimal control of an elastic contact problem involving Tresca friction law[END_REF]. For all h > 0, find u = u hg in C(0, T, H) ∩ V with u ∈ H, such that u(0) = u b , and solution of the parabolic variational inequality of second kind

< u, v -u > + a h (u, v -u) + Φ(v) -Φ(u) ≥ (g, v -u) +h Γ1 b(v -u)ds, ∀v ∈ V, t ∈ (0, T ).
Where u = u t , <, > denotes the duality brackets between V ′ and V , a is a symmetric, continuous and coercive bilinear form over V 0 , and Φ is given by

Φ(v) = Γ2 q|v|ds, (2)
and

a(u, v) = Ω ∇u∇vdx, a h (u, v) = a(u, v) + h Γ1 uvds, (g, v) = Ω gvdx.
Moreover from [START_REF] Kinderlehrer | An introduction to variational inequalities and their applications[END_REF], [START_REF] Tabacman | Sufficient and/or necessary condition for the heat transfer coefficient on Γ1 and the heat flux on Γ2 to obtain a steady-state two-phase Stefan problem[END_REF], [START_REF] Tarzia | Una familia de problemas que converge hacia el caso estacionario del problema de Stefan a dos fases[END_REF] we have that:

∃λ 1 > 0 such that λ h v 2 V ≤ a h (v, v) ∀v ∈ V, with λ h = λ 1 min{1 , h}
that is, a h is also a bilinear continuous, symmetric and coercive form on V . We remark that on

Γ 1 × (0, T ), Problem 1 is with the Dirichlet condition u | Γ 1 = b, while Problem 2 is with the Robin's condition -∇u • n = h(u -b),
where n is the exterior unit vector normal to Γ . The functional Φ comes from the Tresca condition on Γ 2 [START_REF] Amassad | Optimal control of an elastic contact problem involving Tresca friction law[END_REF], [START_REF] Boukrouche | On a non-isothermal, non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions[END_REF]. The existence and uniqueness of the solution to each of the above Problem 1 and Problem 2, is well known see for example [START_REF] Brézis | Problèmes unilatéraux[END_REF], [START_REF] Chipot | Elements of nonlinear Analysis[END_REF], [START_REF] Duvaut | Les inéquations en Mécanique et en Physique[END_REF]. Therefore, it allows us to consider g → u g as a function from H to C(0, T, H) ∩ V.

Let M > 0 be a constant and H + = {g ∈ H : g ≥ 0}. We consider the following distributed optimal control problems defined by:

Find g op ∈ H + such that J(g op ) = min g∈H+ J(g), (3)
Find

g op h ∈ H + such that J(g op h ) = min g∈H+ J h (g), (4)
where the cost functional J : H → R and J h : H → R such that [START_REF]Lions Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF] (see also [START_REF] Kesavan | Low-cost control problems on perforated and nonperforated domains[END_REF], [START_REF] Kesavan | Optimal control on perforated domains[END_REF], [START_REF] Tröltzsch | Optimal control of partial differential equations: Theory, methods and applications[END_REF])

J(g) = 1 2 u g 2 H + M 2 g 2 H , and
J h (g) = 1 2 u hg 2 H + M 2 g 2 H , (5)
being here u g , u hg the unique solutions of the parabolic variational Problem 1, and Problem 2 respectively, and corresponding to the control g in H. In order to prove the strict convexity of the cost functional J and J h , we generalize for parabolic variational inequalities a main property [START_REF] Mignot | Contrôle dans les inéquations variationelles elliptiques[END_REF] that : For any two control g i ∈ H, i = 1 or i = 2, we have

u µg1+(1-µ)g2 ≤ µu g1 + (1 -µ)u g2 , ∀µ ∈ [0, 1], u h(µg1+(1-µ)g2) ≤ µu hg1 + (1 -µ)u hg2 , ∀µ ∈ [0, 1],
by using a regularization method for the non-differentiable functional Φ (see [START_REF] Boukrouche | Convergence of distributed optimal controls for second kind parabolic variational inequalities[END_REF]).

Then we prove the following Theorem 1. [START_REF] Boukrouche | Convergence of distributed optimal controls for second kind parabolic variational inequalities[END_REF] Let u hgop h , g op h and u gop , g op be the states and the optimal controls defined in Problem 1 and Problem 2 respectively. Then, we obtain the following asymptotic behavior:

lim h→+∞ u hgop h -u gop V = 0, (6)
lim h→+∞ g op h -g op H = 0. ( 7
)
3 Distributed optimal control problems governed by parabolic variational inequality of first kind

We will examine in this section, some distributed optimal control problems, for which the strong formulation can be linked to a free boundary problems of complementarity type (Obstacle problems [START_REF] Rodrigues | Obstacle problems in mathematical physics[END_REF]), given for example by the following conditions:

u ≥ 0, u( u -∆u -g) = 0, u -∆u -g ≥ 0 in Ω, (8)
u = b ≥ 0 on Γ 1 , - ∂u ∂n = f on Γ 2 , and u(0) = u b (9)
and

u ≥ 0, u( u -∆u -g) = 0, u -∆u -g ≥ 0 in Ω, (10)
- ∂u ∂n = h(u -b) on Γ 1 , - ∂u ∂n = f on Γ 2 , and u(0) = u b ( 11
)
where Ω is a multidimentional regular domain whose boundary is

∂Ω = Γ 1 ∪ Γ 2 with Γ 1 ∩ Γ 2 = ∅. Let consider the convex set K b as in Section 2. It is classical that, for a given positive b ∈ L 2 (0, T ; H 1 2 (Γ 1 )), f ∈ L 2 (0, T ; L 2 (Γ 2 )
), and g ∈ H, the variational formulations of Problems (8)-( 9) and ( 10)- [START_REF] Gariboldi | Convergence of distributed optimal controls on the internal energy in mixed elliptic problems when the heat transfer coefficient goes to infinity[END_REF] are respectively given by the following parabolic variational problems:

Problem 3. Let given g, b and u b as in (1) and f ∈ L 2 (0, T ; L 2 (Γ 2 )). Find u = u g ∈ C(0, T, H) ∩ L 2 (0, T ; K b ) with u ∈ H, such that u(0) = u b , and < u, v -u > +a(u, u -v) ≥ (g, v -u) - Γ2 f (v -u)ds, ∀v ∈ K b , ∀t ∈ (0, T ).
Problem 4. Find u = u hg ∈ C(0, T, H) ∩ V with u ∈ H, such that u(0) = u b , and

< u, v -u > +a h (u, u -v) ≥ (g, v -u) + h Γ1 b(v -u)ds - Γ2 f (v -u)ds, ∀v ∈ V, ∀t ∈ (0, T ).
where a and a h are as in Section 2. Then the existence and uniqueness of the solution to Problem 3 and Problem 4, is also well known see for example [START_REF] Brézis | Problèmes unilatéraux[END_REF], [START_REF] Chipot | Elements of nonlinear Analysis[END_REF], [START_REF] Duvaut | Les inéquations en Mécanique et en Physique[END_REF]. Then it allows us to consider g → u g as a function from H to C(0, T, H) ∩ V. Let M > 0 be a constant. We consider the same family of distributed optimal control problems ( 3)-( 4) and we obtain the same results of the previous Theorem 1.

Theorem 2. Let g, b, u b as in [START_REF] Amassad | Optimal control of an elastic contact problem involving Tresca friction law[END_REF] and f ≤ 0 in Γ 2 × (0, T ), we can obtain the same results as in Section 2, for the corresponding distributed optimal control problems ( 3)-( 4) when g ≥ 0 is the control variable.

Neumann boundary optimal control problem governed by parabolic variational inequalities of second kind

We assume in this section that the boundary of a multidimensional regular domain Ω is decomposed in three parts ∂Ω = Γ 1 ∪ Γ 2 ∪ Γ 3 with meas(Γ 1 ) > 0 and meas(Γ 3 ) > 0.

We consider a Neumann boundary optimal control problem whose system is governed by a free boundary problem with Tresca conditions on a portion Γ 2 of the boundary, with the flux f on Γ 3 as the control variable, given by: Problem 5.

u -∆u = g in Ω × (0, T ), ∂u ∂n < q ⇒ u = 0, on Γ 2 × (0, T ), ∂u ∂n = q ⇒ ∃k > 0 : u = -k ∂u ∂n , on Γ 2 × (0, T ), u = b on Γ 1 × (0, T ), - ∂u ∂n = f on Γ 3 × (0, T ),
with the initial condition

u(0) = u b on Ω,
and the compatibility condition on Γ 1 × (0, T )

u b = b on Γ 1 × (0, T ),
where q > 0 is the Tresca friction coefficient on Γ 2 ([1], [START_REF] Boukrouche | On a non-isothermal, non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions[END_REF], [START_REF] Duvaut | Les inéquations en Mécanique et en Physique[END_REF]). We define the space F = L 2 (0, T ; L 2 (Γ 3 )).

The variational formulation of Problem 5 leads to the following parabolic variational problem: Problem 6. Let given g, q, b and u b as in (1) and f ∈ F, f ≤ 0. Find u = u f in C(0, T, H) ∩ L 2 (0, T ; K b ) with u ∈ H, such that u(0) = u b , and for t ∈ (0, T )

< u, v -u > +a(u, u -v) + Φ(v) -Φ(u) ≥ (g, v -u) - Γ3 f (v -u)ds, ∀v ∈ K b .
where a and Φ are defined as in Section 2.

We consider also the following problem where we change, in Problem 5, only the Dirichlet condition on Γ 1 × (0, T ) by the Newton law or a Robin boundary condition.

Problem 7. u -∆u = g in Ω × (0, T ), ∂u ∂n < q ⇒ u = 0, on Γ 2 × (0, T ), ∂u ∂n = q ⇒ ∃k > 0 : u = -k ∂u ∂n , on Γ 2 × (0, T ), - ∂u ∂n = h(u -b) on Γ 1 × (0, T ), - ∂u ∂n = f on Γ 3 × (0, T ),
with the initial condition

u(0) = u b on Ω,
and the condition of compatibility on Γ 1 × (0, T )

u b = b on Γ 1 × (0, T ).
The variational formulation of the problem (7) leads to the the following parabolic variational problem Problem 8. Let given g, q, b, u b and f as in Problem 6. For all h > 0, find u = u hf ∈ C(0, T, H) ∩ V with u ∈ H, such that u(0) = u b , and for t ∈ (0, T )

< u, v -u > +a h (u, u -v) + Φ(v) -Φ(u) ≥ (g, v -u) - Γ3 f (v -u)ds +h Γ1 b(v -u)ds, ∀v ∈ V,
where a h and Φ are defined as in Section 2.

Neumann boundary optimal control problems

Let M > 0 be a constant and we define the space F -= {f ∈ F : f ≤ 0}. We consider the new following Neumannn boundary optimal control problems defined by: Problem 9. Find the optimal control f op ∈ F -such that

J(f op ) = min f ∈F- J(f ) (12)
where the cost functional J : F → R + 0 is given by

J(f ) = 1 2 u f 2 H + M 2 f 2 F (M > 0) ( 13
)
and u f is the unique solution of the Problem 6.

Problem 10. Find the optimal control f op h ∈ F -such that

J(f op h ) = min f ∈F- J h (f ) (14)
where the cost functional J h : F → R + 0 is given by

J h (f ) = 1 2 u hf 2 H + M 2 f 2 F (M > 0, h > 0) (15)
and u hf is the unique solution of Problem 8.

Theorem 3. Under the assumptions given in Problem 6, we have the following properties:

a) The cost functional J is strictly convex on F -, b) There exists a unique optimal f op ∈ F -solution of the new Neumannn boundary optimal control Problem 9.

Proof. We give some sketch of the proof. i) We generalize for parabolic variational inequalities of the second kind the estimates obtained for convex combination of solutions for elliptic variational inequalities [START_REF] Boukrouche | On a convex combination of solutions to elliptic variational inequalities[END_REF] that is, the estimate between u 4 (µ) = u µf1+(1-µ)f2 , and u 3 (µ) = µu f1 + (1µ)u f2 , for any two element f 1 and f 2 in F.

ii) The main difficulty, to prove this result comes from the fact that the functional Φ is not differentiable. To overcome this difficulty, we use the regularization method and consider for ε > 0 the following approach of Φ defined by:

Φ ε (v) = Γ2 q ε 2 + |v| 2 ds, ∀v ∈ V, (16)
which is Gateaux differentiable, with

Φ ′ ε (w) , v = Γ2 qwv ε 2 + |w| 2 ds ∀(w, v) ∈ V 2 .
We define u ε as the unique solution of the corresponding parabolic variational inequality for all ε > 0. We obtain that for all µ ∈ [0, 1] we have u ε 4 (µ) ≤ u ε 3 (µ) for all ε > 0.

iii) When ε → 0 we have that:

u ε i → u i strongly in V ∩ L ∞ (0, T ; H) for i = 1, 2, 3, 4, (17)
for all µ ∈ [0, 1] and therefore we get:

0 ≤ u 4 (µ) ≤ u 3 (µ) in Ω × [0, T ], ∀µ ∈ [0, 1]. ( 18
)
iv) For all µ ∈]0, 1[, and for all f 1 , f 2 in F, and by using f 3 (µ) = µf 1 + (1µ)f 2 we obtain that:

µJ(f 1 ) + (1 -µ)J(f 2 ) -J(f 3 (µ)) = 1 2 u 3 (µ) 2 H -u 4 (µ) 2 H + 1 2 µ(1 -µ) u f1 -u f2 2 H + M 2 µ(1 -µ) f 1 -f 2 2 F . ( 19
)
Then J is strictly convex functional on F -and therefore there exists a unique optimal f op ∈ F -solution of the new Neumannn boundary optimal control Problem 9.

Theorem 4. Under the assumptions given in Problem 6, we have the following properties:

a) The cost functional J h are strictly convex on F -, for all h > 0, b) There exists a unique optimal f op h ∈ F -solution of the new Neumannn boundary optimal control Problem 10, for all h > 0.

Proof. We follow a similar method to the one developed in Theorem 3 for all h > 0.

Open problem

The convergence of the new Neumann boundary optimal control Problem 10 to the new Neumann boundary optimal control Problem 9 when h → ∞ is an open problem.

Acknowledgements: The autors would like to thank very much the unknown referee for helpful comments which allowed to improve the paper. This paper was partially sponsored by the Institut Camille Jordan ST-Etienne University for first author and the project PICTO Austral # 73 from ANPCyT and Grant AFOSR FA9550-10-1-0023 for the second author.


Ekaterina Kostina

Olga Kostyukova

Werner Schmidt

New Necessary Conditions for Optimal Control Problems in Discontinuous Dynamic Systems

Keywords: Optimal control problem, maximum principle, discontinuous dynamic system

In the paper we derive new necessary optimality conditions for optimal control of differential equations systems with discontinuous right hand side. The main attention is paid to a situation when an optimal trajectory slides on the discontinuity surface. The new conditions, derived in the paper, are essential and do not follow from any known necessary conditions for such systems.

Introduction

Optimal control of systems of differential equations with discontinuous right hand side are widely used to describe numerous applications in natural sciences and engineering, where, e.g., there is a necessity to model dynamics with different scales or even with jumps. Such models appear e.g. in economics, mechanics (e.g. optimal control of mechanical systems with Coulomb friction), chemical processes, electrical and radio engineering, aerodynamics, automatic control theory, and theory of hybrid systems [START_REF] Bock | Optimal Control of Velocity Profiles for Minimization of Energy Consumption in the New York Subway System[END_REF][START_REF] Branicky | Introduction to Hybrid Systems[END_REF][START_REF] Filippov | Differential Equations with Discontinuous Right-Hand Sides[END_REF][START_REF] Kirches | Time-Optimal Control of Automobile Test Drives with Gear Shifts[END_REF][START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF].

There is a row of papers devoted to the numerical solution of such optimal control problems, see [START_REF] Stewart | Optimal Control of Systems with Discontinuous Differential Equations[END_REF] and references therein. However, due to the complexity of these optimal control problems there are few papers devoted to their theoretical studies.

The aim of this paper is to present new necessary optimality conditions for optimal control of differential equations systems with discontinuous right hand side. It is assumed that the discontinuity of the function, which describes the dynamic system, appears at some surface defined in the state space. This surface is called the discontinuity or switching surface. Most papers on the topic consider the situation when an optimal trajectory crosses the surface [START_REF] Ashchepkov | Optimal Control of Discontinuous Systems[END_REF][START_REF] Boltyanskii | The Maximum Principle for Systems with Discontinuous Right-Hand Side[END_REF][START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF]. In this case the optimality conditions are a slight modification of the maximum principle of Pontryagin. However, the more interesting and practically non-studied case is the situation when an optimal trajectory slides on the surface. There are only very few papers [START_REF] Ashchepkov | Optimal Control of Discontinuous Systems[END_REF][START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF][START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF][START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF], which deal with this case, however, the maximum conditions presented there are weaker than the conditions presented in this paper.

Dynamic systems with discontinuous right hand side may be presented as differential inclusions. Necessary optimality conditions in optimal control problems for differential inclusions have been intensively studied, see e.g. [START_REF] Donchev | Discrete Approximations, Relaxation, and Optimization of One-Sided Lipschitzian Differential Inclusions in Hilbert Spaces[END_REF][START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation, II: Applications[END_REF][START_REF] Vinter | Optimal Control[END_REF] and references therein. However, the results obtained there are based on the assumption that the differential inclusion is Lipschitz continuous or possesses a modified one-sided Lipschitz property, which is not the case for the differential inclusion describing the dynamic system of this paper.

Problem Statement and Assumptions

Let us consider the following optimal control problem with discontinuous right hand side:

min u(•) f 0 (x(t * )), ẋ = f -(x, u) if S(x(t)) < 0, f + (x, u) if S(x(t)) > 0, (1)
x(t * ) = x 0 , h(x(t * )) = 0, |u(t)| ≤ 1, t ∈ T = [t * , t * ].
Here f ± (x, u) = f ± (x) + b(x)u, x = x(t) ∈ R n denotes the state n-vector, u = u(t) ∈ R is a scalar control, S(x) := d T xγ = 0 is a surface of discontinuity (switching surface), the functions

f 0 (x) ∈ R, f ± (x), b(x) ∈ R n , h(x) ∈ R s * are
given sufficiently smooth scalar and vector functions, d, x 0 are given vectors, γ, t * , t * are given numbers, s * = dim(h) is the dimension of the vector-function h = h(x).

In the general case, even for a fixed control u(•) = (u(t), t ∈ T ) and a given initial value x(t * ) = x 0 , the system (1) may not have a classical solution, since the system is not defined at the switching surface. Therefore, we redefine a solution of the system (1) at the switching surface following Filippov [START_REF] Filippov | Differential Equations with Discontinuous Right-Hand Sides[END_REF]. Then the problem may be reformulated as min

u(•),α(•) f 0 (x(t * )), ẋ =    f -(x, u), if d T x(t) < γ, f + (x, u), if d T x(t) > γ, F (x, u, α), if d T x(t) = γ, (2)
x(t * ) = x 0 , h(x(t * )) = 0, |u(t)| ≤ 1, 0 ≤ α(t) ≤ 1, t ∈ T = [t * , t * ]. where F (x, u, α) := α f -(x, u) + (1 -α) f + (x, u) = f + (x) + a(x)α + b(x)u, a(x) := f -(x) -f + (x). In this problem the control is u(t), α(t), t ∈ T.
Using classical optimal control theory [START_REF] Lee | Foundations of Optimal Control Theory[END_REF], we may conclude that if the problem ( 2) is feasible, it has an optimal solution in the class of measurable functions u(t), α(t), t ∈ T.

Let u * (•) = (u * (t), t ∈ T ), α * (•) = (α * (t), t ∈ T ), x * (•) = (x * (t), t ∈ T ) be an optimal control and the corresponding trajectory of the system (2). Here and in what follows, we formally suppose that

α * (t) = 1 if d T x * (t) < γ, α * (t) = 0 if d T x * (t) > γ, 0 ≤ α * (t) ≤ 1 if d T x * (t) = γ, t ∈ T.
For the formulation and the proof of the maximum principle we need several assumptions.

Assumption 1

The functions u * (•) = (u * (t), t ∈ T ), α * (•) = (α * (t), t ∈ T ) are piecewise continuous and piecewise smooth. We denote T s = {t ∈ T : d T x * (t) = γ}. In the general case this set contains points, which correspond to crossing the discontinuity surface by a trajectory, and segments, which correspond to the case when the trajectory lies on the surface. As it was mentioned before, the case when the trajectory crosses the discontinuity surface is well-studied. The aim of this paper is to study the case when the trajectory may lie on the discontinuity surface. For this purpose we need the following assumption.

Assumption 2 The set T s consists of a finite number of segments

[τ k , τ k ], k = 1, ..., p, t * < τ 1 < τ 1 < τ 2 < τ 2 < ... < τ p < τ p < t * ,
and the following inequalities hold,

d T ẋ * (τ k -0) = 0, d T ẋ * (τ k + 0) = 0, k = 1, ..., p. (3)
Here and in what follows, z( t -0) and z( t + 0) for a given function z(t), t ∈ T, are defined by z( t + 0) := lim t→ t,t≥ t z(t), z( t -0) := lim t→ t,t≤ t z(t). We denote

T 0 s = {t ∈ T s : α * (t) = 0}, T 1 s = {t ∈ T s : α * (t) = 1}, T * s = T s \ (T 0 s ∪ T 1 s ).
Assumption 3 The "active" set T 0 s ∪ T 1 s does not contain isolated points and the following relations hold true for an optimal control:

∃ ǫ 0 > 0, |u * (t)| ≤ 1 -ǫ 0 , t ∈ T 0 s ∪ T 1 s , |d T b(x * (t))| ≥ ǫ 0 , t ∈ T 0 s ∪ T 1 s ; |d T a(x * (t))| ≥ ǫ 0 , t ∈ T * s , rank ∂h(x * (t * )) ∂x = s * .

Necessary Optimality Conditions

The main goal of this section is to formulate and prove new necessary optimality conditions in the form of the maximum principle. For this purpose we will need some auxiliary results.

Without loss of generality, we will suppose for simplicity that the following relations are satisfied

d T x * (t) > γ, t ∈ [t * , τ 1 ), d T x * (t) = γ, t ∈ [τ 1 , τ 1 ], d T x * (t) < γ, t ∈ (τ 1 , t * ], (4) α * (t) = 1, |u * (t)| < 1, t ∈ (τ 1 , τ 0 ), 0 < α * (t) < 1, t ∈ (τ 0 , τ 1 ).
Therefore, for the case under consideration we have

p = 1, T s = [τ 1 , τ 1 ], T + α = (τ 1 , τ 0 ), T 0 α = ∅, T * α = (τ 0 , τ 1 ), T + = [t * , τ 1 ), T -= (τ 1 , t * ].
We introduce the set of parameters µ = (y 0 , y, λ 1 , λ 1 ), [START_REF] Branicky | Introduction to Hybrid Systems[END_REF] where y 0 ≥ 0, λ 1 , λ 1 are scalars, y ∈ R s * , and denote by ψ(t|µ), t ∈ T, a solution of the system 6)

ψT (t) = -ψ T (t) ∂ f + (x * (t), u * (t)) ∂x , t ∈ [t * , τ 1 ); ψT (t) = -ψ T (t) ∂ f -(x * (t), u * (t)) ∂x , t ∈ [τ 1 , t * ]; ψT (t) = -ψ T (t) ∂F (x * (t), u * (t), α * (t)) ∂x -q * 1 (t)d T , t ∈ [τ 1 , τ 0 ), ψT (t) = -ψ T (t) ∂F (x * (t), u * (t), α * (t)) ∂x -q * 2 (t)d T , t ∈ [τ 0 , τ 1 ), (
ψ(t * ) = -y 0 ∂f 0 (x * (t * )) ∂x - ∂h T (x * (t * )) ∂x y, ψ(τ 1 -0) = ψ(τ 1 + 0) + dλ 1 , ψ(τ 1 -0) = ψ(τ 1 + 0) + dλ 1 .
Further we pick up any m ∈ N, m ≥ 2, and consider a set of points t1 , t2 , ..., t2m , such that

τ 1 = t1 < t2 < ... < t2m-1 = τ 0 < t2m = τ 1 . (7)
This set of points satisfies the following lemma:

Lemma 1. Let u * (•), α * (•),
x * (•) be an optimal control and the trajectory in the problem [START_REF] Ashchepkov | Optimal Control of Discontinuous Systems[END_REF], for which Assumptions 1-3 are fulfilled. For any choice of the points [START_REF] Dubovitskii | Criterion of Existence of a Substantive Maximum Principle in a State Constraint Problem[END_REF] there exists a vector of parameters µ, µ = 1, (5), such that along the corresponding solution ψ(t) = ψ(t|µ), t ∈ T, of the system (6) the following relations hold true,

ψ T (t)b(x * (t))u * (t) = max |u|≤1 ψ T (t)b(x * (t))u, a.e. t ∈ T ; (8) ψ T (t)a(x * (t))α * (t) = max 0≤α≤1 ψ T (t)a(x * (t))α, (9)
a.e. t ∈ [ t2i-1 , t2i ], i = 1, 2, ..., m, ψ T (t)q * 1 (t) ≥ 0, a.e. t ∈ [ t2i , t2i+1 ], i = 1, 2, ..., m -1. ( 10
)
The proof of the lemma can be found in [START_REF] Kostina | New Maximum Principle for Optimal Control Problems in Discontinuous Dynamic Systems: Proof and Examples[END_REF] and its main idea is sketched here. For a fixed m ≥ 2 we consider a set of parameters θ = (t 1 , t 2 , ..., t 2m ) and formulate the optimal control problem of a hybrid system:

min u(•),α(•),θ f 0 (x(t * )), ẋ = f + (x, u), d T x(t) ≥ γ, t ∈ [t * , t 1 [ ẋ = F (x, u, α), d T x(t) = γ, t ∈ [t 2i-1 , t 2i [, ẋ = f -(x, u), d T x(t) ≤ γ, t ∈ [t 2i , t 2i+1 [, i = 1, ..., m, (11)
x(t * ) = x 0 , h(x(t * )) = 0,

t * = t 0 ≤ t 1 ≤ ... ≤ t 2m ≤ t 2m+1 = t * , |u(t)| ≤ 1, t ∈ T ; 0 ≤ α(t) ≤ 1, t ∈ m i=1 [t 2i-1 , t 2i ].
Let us note that in the problem [START_REF] Kostina | New Maximum Principle for Optimal Control Problems in Discontinuous Dynamic Systems: Proof and Examples[END_REF], the decision variables are the control u(•), α(•) and a vector θ.

With the notations

z i (τ ) = x(t i-1 + τ (t i -t i-1 )), i = 1, 2, ..., 2m + 1, v i (τ ) = u(t i-1 + τ (t i -t i-1 )), i = 1, 2, ..., 2m + 1, ( 12
)
β i (τ ) = α(t i-1 + τ (t i -t i-1 )), i = 2, 4, ..., 2m,
we form the extended state vector

Z(τ ) = (z i (τ ), i = 1, ..., 2m + 1; t i (τ ), i = 1, ..., 2m) ∈ R n×(2m+1)+2m , ( 13
) τ ∈ [0, 1],
and the extended control vector

V (τ ) = (v i (τ ), i = 1, ..., 2m + 1, β i (τ ), i = 2, 4, ..., 2m) ∈ R 3m+1 , (14)
τ ∈ [0, 1].

Using the introduced notations we may rewrite the problem [START_REF] Kostina | New Maximum Principle for Optimal Control Problems in Discontinuous Dynamic Systems: Proof and Examples[END_REF] as follows, min

V (•) f 0 (z 2m+1 (1)), Ż(τ ) = F (Z(τ ), V (τ )), Φ(Z(0), Z(1)) = 0, (15)
G T 1 Z(τ ) ≥ γ, G T 2i Z(τ ) = γ, G T 2i+1 Z(τ ) ≤ γ, i = 1, 2, ..., m; |v i (τ )| ≤ 1, i = 1, ..., 2m + 1, 0 ≤ β i (τ ) ≤ 1, i = 2, 4, ..., 2m, t i (τ ) ≤ t i+1 (τ ), i = 0, ..., 2m, τ ∈ [0, 1]. ( 16
)
Here V (τ ) is the control vector ( 14), Z(τ ) is the state vector ( 13),

F T (Z, V ) = (t 1 -t 0 ) f + (z 1 , v 1 ), (t 2i -t 2i-1 )F (z 2i , v 2i , β 2i ), (t 2i+1 -t 2i ) f -(z 2i+1 , v 2i+1 ), i = 1, ..., m, 0, ..., 0 2m , Φ(Z(0), Z(1)) =     z 1 (0) -x 0 z i (1) -z i+1 (0), i = 1, ..., 2m h(z 2m+1 (1)) d T z i (1) -γ, i = 1, ..., 2m     , G T i = (O T , ..., O T , i-1 d T , O T , ..., O T , 2m+1-i 0, ..., 0 2m
), i = 1, ..., 2m + 1, O ∈ R n is a vector of zeros, t 0 (τ ) ≡ t0 = t * , t 2m+1 (τ ) ≡ t2m+1 = t * .
Now consider the set of points t1 , t2 , ..., t2m satisfying [START_REF] Dubovitskii | Criterion of Existence of a Substantive Maximum Principle in a State Constraint Problem[END_REF] and denote by Z * (τ ), V * (τ ), τ ∈ [0, 1], the functions ( 12) -( 14), constructed using this set, the optimal control u * (•), α * (•) and the trajectory x * (•) of the problem (2). Since the control u * (•), α * (•) and the trajectory x * (•) are optimal in the problem (2), it is obvious that V * (τ ), Z * (τ ), τ ∈ [0, 1], are an optimal control and the corresponding trajectory of the problem (15), [START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF]. By the assumptions (see [START_REF] Bock | Optimal Control of Velocity Profiles for Minimization of Energy Consumption in the New York Subway System[END_REF] and ( 4)) we have

G T 1 Z * (τ ) > γ, τ ∈ [0, 1), G T 1 Z * (1) = γ, G T 1 Ż * (1 -0) = 0, G T 2m+1 Z * (τ ) < γ, τ ∈ (0, 1], G T 2m+1 Z * (0) = γ, G T 2m+1 Ż * (+0) = 0.
Hence, V * (τ ), Z * (τ ), τ ∈ [0, 1], is also a strong local extremal in the problem that results from the problem ( 15), ( 16) by removing the state constraints

G T 1 Z(τ ) ≥ γ, G T 2m+1 Z(τ ) ≤ γ, τ ∈ [0, 1]
, and the constraints [START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF], namely in the problem min

V (•) f 0 (z 2m+1 (1)), Ż(τ ) = F (Z(τ ), V (τ )), Φ(Z(0), Z(1)) = 0, (17)
G T i Z(τ ) ≤ γ, i = 3, 5, ..., 2m -1; G T i Z(τ ) = γ, i = 2, 4, ..., 2m; |v i (τ )| ≤ 1, i = 1, ..., 2m + 1, 0 ≤ β i (τ ) ≤ 1, i = 2, 4, ..., 2m; τ ∈ [0, 1],
with the control V (τ ) (see ( 14)) and the state vector Z(τ ) (see [START_REF] Lee | Foundations of Optimal Control Theory[END_REF]).

The problem ( 17) is an optimal control problem with inequality and equality state constraints and boundary constraints Φ(Z(0), Z(1)) = 0. Due to Assumptions 1-3 and the specific structure of this problem, regularity conditions (see [START_REF] Arutyunov | Optimality Conditions: Abnormal And Degenerate Problems[END_REF]) are satisfied for the control V * (τ ) and the trajectory Z * (τ ), τ ∈ [0, 1]. Thus we can apply Theorems 4.1 and 12.1 from [START_REF] Arutyunov | Optimality Conditions: Abnormal And Degenerate Problems[END_REF] and results from [START_REF] Dubovitskii | Criterion of Existence of a Substantive Maximum Principle in a State Constraint Problem[END_REF], according to which the control V * (τ ) and the trajectory Z * (τ ), τ ∈ [0, 1], satisfy certain relations. Analyzing these relations and taking into account the structure of the vectors V * (τ ) and Z * (τ ) allow us to get the assertions of the lemma. ⋄

Let us note that it follows from Lemma 1 that the continuity of the function ψ(t|µ), t ∈ (τ 1 , τ 1 ), the relations ( 9), [START_REF] Kirches | Time-Optimal Control of Automobile Test Drives with Gear Shifts[END_REF] and the assumption α * (t) = 1, t ∈ (τ 1 , τ 0 ), imply the inequalities

ψ T ( ti |µ)a(x * ( ti )) ≥ 0, ψ T ( ti |µ)q * 1 ( ti ) ≥ 0, i = 2, 3, ..., 2m -1, (18)
for each point set t1 , t2 , ..., t2m satisfying [START_REF] Dubovitskii | Criterion of Existence of a Substantive Maximum Principle in a State Constraint Problem[END_REF].

Now we are ready to formulate and prove new necessary optimality conditions for problem (2) in the form of the maximum principle.

Theorem 1. Let u * (•), α * (•), x * (•) be an optimal control and the corresponding trajectory of the problem [START_REF] Ashchepkov | Optimal Control of Discontinuous Systems[END_REF], which satisfy Assumptions 1-3. Then there exist numbers λ k , λ k , k = 1, ..., p, y 0 ≥ 0, and a vector y ∈ R s * , not all trivial, p k=1

(|λ k | + |λ k |) + y 0 + ||y|| > 0,
such that along a solution of the adjoint system

ψT (t) =                      -ψ T (t) ∂(f ± (x * (t))+b(x * (t))u * (t)) ∂x , t ∈ {t ∈ T : ±(d T x * (t) -γ) > 0}, -ψ T (t) ∂F (x * (t),u * (t),α * (t)) ∂x -q * 1 (t)d T , t ∈ T 0 s ∪ T 1 s , -ψ T (t) ∂F (x * (t),u * (t),α * (t)) ∂x -q * 2 (t)d T , t ∈ T * s , ψ(t * ) = -y 0 ∂f 0 (x * (t * )) ∂x - ∂h T (x * (t * )) ∂x y, ψ(τ k -0) = ψ(τ k + 0) + dλ k , ψ(τ k -0) = ψ(τ k + 0) + dλ k , k = 1, ..., p,
the following relations hold true:

ψ T (t)b(x * (t))u * (t) = max |u|≤1 ψ T (t)b(x * (t))u, a.e. t ∈ T ; ψ T (t)a(x * (t))α * (t) = max 0≤α≤1 ψ T (t)a(x * (t))α, a.e. t ∈ T s ; ( 19
)
ψ T (t -0) ẋ * (t -0) = ψ T (t + 0) ẋ * (t + 0), t = τ k , t = τ k , k = 1, ..., p, ψ T (t)q * 1 (t) ≤ 0, t ∈ int T 0 s , ψ T (t)q * 1 (t) ≥ 0, t ∈ int T 1 s . (20)
Here q * i (t) := q i (x * (t), u * (t), α * (t)); i = 1, 2;

q 1 (x, u, α) := ∂F (x, u, α) ∂x b(x) - ∂b(x) ∂x F (x, u, α) /d T b(x), q 2 (x, u, α) := ∂F (x, u, α) ∂x a(x) - ∂a(x) ∂x F (x, u, α) /d T a(x).
Proof. Again we will suppose for simplicity that relations (4) are satisfied. For an arbitrary m ∈ N, m ≥ 2, we consider the set of points

t (m) i = τ 1 + (i -1) (τ 0 -τ 1 ) 2m -2 , i = 1, ..., 2m -1, t (m) 2m = τ 1 . ( 21
)
For any m ≥ 2 the set (21) satisfies relations [START_REF] Dubovitskii | Criterion of Existence of a Substantive Maximum Principle in a State Constraint Problem[END_REF]. Hence Lemma 1 and the relation [START_REF] Stewart | Optimal Control of Systems with Discontinuous Differential Equations[END_REF] imply that for a set (21) there exists a vector

µ(m) = (y 0 (m), y(m), λ 1 (m), λ 1 (m)), ||µ(m)|| = 1,
such that the following relations hold true,

ψ T (t|µ(m))b(x * (t))u * (t) = max |u|≤1 ψ T (t|µ(m))b(x * (t))u, a.e. t ∈ T, ψ T (t|µ(m))a(x * (t))α * (t) = max 0≤α≤1 ψ T (t|µ(m))a(x * (t))α (22) a.e. t ∈ [τ 0 , τ 1 ], ψ T (t -0|µ(m)) ẋ * (t -0) = ψ T (t + 0|µ(m)) ẋ * (t + 0), t = τ 1 , t = τ 1 ; ψ T (t (m) i |µ(m))a(x * (t (m) i )) ≥ 0, ψ T (t (m) i |µ(m))q * 1 (t (m) i ) ≥ 0, (23)
i(m) = i(m|t) ∈ {2, 3, ..., 2m -1}, m = 2, 3, 4, ..., such that t (m) i(m
) → t as m → ∞. By construction (see (23)), ψ T (t (m) i(m) |µ(m))a(x * (t (m) i(m) )) ≥ 0, ψ T (t (m) i(m) |µ(m))q * 1 (t (m) i(m) ) ≥ 0, (24)
m = 2, 3, 4, ....

For m → ∞ in the last inequalities we get

ψ T (t|µ * )a(x * (t)) ≥ 0, ψ T (t|µ * )q * 1 (t) ≥ 0, t ∈ [τ 1 , τ 0 ]. (25)
Similarly, for m → ∞ in (22) we obtain

ψ T (t|µ * )b(x * (t))u * (t) = max |u|≤1 ψ T (t|µ * )b(x * (t))u, a.e. t ∈ T ; ψ T (t|µ * )a(x * (t))α * (t) = max 0≤α≤1 ψ T (t|µ * )a(x * (t))α a.e. t ∈ [τ 0 , τ 1 ]; (26)
ψ T (t -0|µ * ) ẋ * (t -0) = ψ T (t + 0|µ * ) ẋ * (t + 0), t = τ 1 , t = τ 1 .
The relations (25) and ( 26) are nothing but the assertions of Theorem 1 for the considered structure of the solution of the problem (2) (see the assumption ( 4)).

Analogously we may prove the Theorem for other types of solution structure. ⋄

Discussion of the New Necessary Optimality Conditions

Because of the problem complexity, there are only few papers, e.g. [START_REF] Ashchepkov | Optimal Control of Discontinuous Systems[END_REF][START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF][START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF][START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF], where necessary optimality conditions for optimal control problems with discontinuous dynamics are presented. The theorem formulated in this paper contains new crucial conditions [START_REF] Vinter | Optimal Control[END_REF] and [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF].

Let us briefly discuss differences between our maximum principle and maximum principles known from the literature.

The necessary conditions in [START_REF] Ashchepkov | Optimal Control of Discontinuous Systems[END_REF] are proved under the very strong assumption T 0 s = T 1 s = ∅, which is not needed in our theorem. As in [START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF], the optimality conditions derived here are based on the so-called direct approach (see [START_REF] Hartl | A Survey of the Maximum Principles for Optimal Control Problems with State Constraints[END_REF]). However, the necessary conditions in [START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF] contain the conditions [START_REF] Vinter | Optimal Control[END_REF] but not the condition [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF].

The optimality conditions in [START_REF] Ashchepkov | Optimal Control of Discontinuous Systems[END_REF][START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF] are formulated based on an so-called indirect approach (see [START_REF] Hartl | A Survey of the Maximum Principles for Optimal Control Problems with State Constraints[END_REF]), which a priori leads to weaker results compared with the direct approach.

In [START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF], conditions [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF], [START_REF] Vinter | Optimal Control[END_REF] are not considered. However, the conditions ( 19), [START_REF] Vinter | Optimal Control[END_REF] are essential and they are not a consequence of any other conditions mentioned in [START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF], [START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF].

The maximum principle in [START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF] is weaker than the Theorem 1. Indeed, one can easily show that for the problems of the form (2) the maximum principle from [START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF] is satisfied trivially for any feasible control. Furthermore, one can construct examples (one of them is presented below) where a non-optimal control satisfies the conditions of maximum principle from [START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF], but not the conditions of Theorem 1.

To finish the discussion of the new necessary conditions, we would like to stress the importance of the conditions [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF].

Let us consider an optimal control problem with state constraints in the form

min f 0 (x(t * )), ẋ = f -(x) + b(x)u, x(t * ) = x 0 , h(x(t * )) = 0, d T x(t) ≤ γ, |u(t)| ≤ 1, t ∈ T = [t * , t * ].
Suppose that this problem has an optimal control u * (t), t ∈ T, and the corresponding trajectory x * (t), t ∈ T, such that mes T s > 0, T s := {t ∈ T : d T x * (t) = γ}. Then the control u * (t), t ∈ T, and the function α * (t) = 1, t ∈ T , are feasible in the original problem (2) for any function f + (x) and satisfy the necessary optimality conditions from [START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF], [START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF]. However, one can easily construct functions f + (x) (and corresponding optimal control problems (2)), which together with the functions u * (t), α * (t) = 1, t ∈ T, violate the maximum condition [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF]. Hence, the control u * (t), α * (t), t ∈ T, is not optimal in (2) according to Theorem 1.

Illustrative examples

In order to demonstrate differences of our maximum principle from maximum principles known from the literature we have constructed several examples. The aim of the examples is to show that the necessary conditions presented in this paper are stronger than known necessary conditions. Namely, we want to show that the new conditions ( 20) and ( 19) may be violated for controls that satisfy other necessary optimality conditions known from the literature, and hence our maximum principle guarantees that such controls are not optimal. Furthermore, we want to show that the new conditions are essential and does not follow from other known optimality conditions.

Example 1. Consider the optimal control problem depending on a parameter c,

P(c) : min x 1 (t * ) -2.5x 2 (t * ), x(t * ) = x 0 , x 3 (t * ) = 1, |u(t)| ≤ 1, 0 ≤ α(t) ≤ 1, t ∈ [t * , t * ], ẋ1 = x 2 , ẋ2 = x 3 + 5, ẋ3 = u + 1/2,    if x 3 < 0, ẋ1 = x 2 + c, ẋ2 = x 3 , ẋ3 = u,    if x 3 > 0, ẋ1 = x 2 + c(1 -α), ẋ2 = x 3 + 5α, ẋ3 = u + 1/2α,    if x 3 = 0, with x T 0 = (19/32, -37/16, -3/4), t * = -0.5, t * = 2
and the control u * (•), α * (•):

u * (t) =    1, t ∈ [-0.5, 0], -0.5, t ∈ [0, 1], 1, t ∈ [1, 2], α * (t) =    1, t ∈ [-0.5, 0], 1, t ∈ [0, 1], 0, t ∈ [1, 2]. For the control u * (•), α * (•) we have T s = [0, 1], T 1 s = [0, 1], T 0 s = ∅, T ⋆ s = ∅.
If we choose c = c 0 = 0, then in the problem P (c 0 ) the control u * (•), α * (•) is feasible, locally optimal and satisfies all necessary optimality conditions from Theorem 1.

If we choose c = c * = -5.5, then in the problem P (c * ) the control u * (•), α * (•) is feasible and not optimal but it satisfies all necessary conditions from [START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF][START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF], and it satisfies all necessary conditions from Theorem 1 except for the condition [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF]. Hence, according to [START_REF] Kugushev | Necessary Optimality Conditions for Systems Described by Equations with Discontinuous Right-Hand Side[END_REF][START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF], this control may be locally optimal in the problem P (c * ). On the other hand, following Theorem 1, it cannot be locally optimal in the problem P (c * ). For the control u * (•), α * (•) we have T s = [0, 1], T 1 s = [0, 1], T 0 s = T * s = ∅. The control u * (•), α * (•) satisfies all necessary optimality conditions from [START_REF] Oberle | Numerical Computation of a Singular-State Subarc in an Economic Optimal Control Model[END_REF], satisfies all necessary optimality conditions from Theorem 1, except for the condition [START_REF] Vinter | Optimal Control[END_REF], and is not locally optimal.

Note that results from [START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF] can not be applied to this example because Assumption 3) from [START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF] (namely the condition d T ( f -(x, u * (t))-f + (x, u * (t))) > 0, ∀ x ∈ S 0 (t), t ∈ T s ) is not satisfied.

i = 2 , 3 ,

 23 ..., 2m -1. Consider the sequence of the vectors µ(m), m = 2, 3, .... Since ||µ(m)|| = 1, m = 2, 3, ..., there exists a converging subsequence. Without loss of generality, we assume that the sequence µ(m), m = 2, 3, ... converges itself, µ * = lim m→∞ µ(m). Obviously, ||µ * || = 1. It follows from (21) that for any point t ∈ [τ 1 , τ 0 ] there exists a sequence of indices

Example 2 .

 2 Consider the optimal control problem min 2x 1 (t * ) -2x 2 (t * ),x(t * ) = x 0 , x 3 (t * ) = -1, |u(t)| ≤ 1, 0 ≤ α(t) ≤ 1, t ∈ [t * = -0.5, t * = 2], ẋ1 = x 2 , ẋ2 = x 3 , ẋ3 = u, -23/48, -1/8, 1/2), and the control u * (•), α * (•) u * (t) = -1, α * (t) = 0; t ∈ [-0.5, 0], u * (t) = 0, α * (t) = 1; t ∈ [0, 1], u * (t) = -1, α * (t) = 1; t ∈ [1, 2].

Example 3 .

 3 Consider the following problem, P* : min c T x(t * ),x(t * ) = x 0 , d T x(t * ) = 1, |u(t)| ≤ 1, 0 ≤ α(t) ≤ 1, t ∈ [t * , t * ], ẋ = Ax + bu + g -, if d T x < 0, ẋ = Ax + bu + g + , if d T x > 0, ẋ = Ax + bu + αg -+ (1α)g + , if d T x = 0, with x ∈ R n , n = 4, t * = -0.5, t * = 2,

The vector x 0 is uniquely defined by the condition that the trajectory x(t), t ∈ [-0.5, 0], of the system ẋ = Ax + b + g -, x(-0.5) = x 0 , should satisfy the equality x(0

and the corresponding trajectory

For the control (28) we have

We can show that the control (28) is not locally optimal in the problem P * , satisfies all assumptions and all necessary optimality conditions from [START_REF] Morozov | Optimal Control of Sliding Modes of Discontinuous Dynamical Systems[END_REF] and satisfies all conditions from Theorem 1 except for the condition [START_REF] Vinter | Optimal Control[END_REF].

Conclusions

We have presented a new maximum principle for optimal control problems in discontinuous systems, which takes into account a situation when a solution of the dynamic system lies on the switching surface. We have shown that the new maximum principle is stronger than known optimality conditions and contains new conditions which are essential and do not follow from other known optimality conditions.


Sonja Steffensen
email: steffensen@mathc.rwth-aachen.de

Michael Herty
email: herty@mathc.rwth-aachen.de

⋆⋆ Lorenzo Pareschi
email: lorenzo.pareschi@unife.it

Numerical Methods for the Optimal Control of Scalar Conservation Laws

Keywords: IMEX schemes, optimal control, conservation laws, Runge-Kutta methods

We are interested in a class of numerical schemes for the optimization of nonlinear hyperbolic partial differential equations. We present continuous and discretized relaxation schemes for scalar, oneconservation laws. We present numerical results on tracking type problems with nonsmooth desired states and convergence results for higherorder spatial and temporal discretization schemes.

Introduction

We consider an optimal control problem for scalar conservation laws of the type minimize u0 J(u(T ), u 0 ) subject to u t + f (u) x = 0, u(0, x) = u 0 (x),

Here, J and f are assumed to be smooth and possibly nonlinear functions. The initial value u 0 acts as control to the problem. It can be observed that the wave interactions that occur in the solution u in the case of a nonlinear flux function f pose the serious analytical challenges. Recently, the differentiability of J with respect to u 0 could be proven in the sense of shift-differentiability. We refer to [6, 9-11, 28-30, 4, 32] for more details.

Here, a class of numerical methods applied to the optimal control problem (1) is studied. We only consider the case of smooth initial data and smooth solutions u and refer to [START_REF] Banda | Adjoint imex-based schemes for control problems governed by hyperbolic conservation laws[END_REF] for more details. For a numerical analysis including shock waves and in the case of the Lax-Friedrichs scheme we refer to [START_REF] Giles | Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: Adjoint approximations and extensions[END_REF][START_REF] Castro | An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks[END_REF] and the references therein.

Relaxation Method

As motiviation for a numerical scheme we follow the ideas of Jin and Xin [START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF]. Therein, a linear approximation (2) of the nonlinear hyperbolic equation

∂ t u + ∂ x f (u) = 0
has been discussed. For initial conditions u(x, 0) = u 0 the approximation is

∂ t u + ∂ x v = 0 , u(x, 0) = u 0 , ∂ t v + a 2 ∂ x u = 1 ǫ (f (u) -v) , v(x, 0) = f (u 0 ) (2)
where ε > 0 is the relaxation rate and a is a given constant satisfying the subcharacteristic condition max u |f ′ (u)| ≤ a. For ε being small, the solution u of (2) satisfies [START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF]). Applying the relaxation to the optimal control problem (1), we obtain

∂ t u + ∂ x f (u) = ε∂ x ((a 2 -f (u) 2 )∂ x u) (cf.
min u0 J(u(•, T ), u 0 ) subject to      u t + v x = 0, v t + a 2 u x = 1 ǫ (f (u) -v) , u(0, x) = u 0 , v(0, x) = f (u 0 ) (3)
The corresponding adjoint equations for (3) are given by (cf. [?])

-p t -a 2 q x = q ǫ f ′ (u), p(T, x) = p T (x), -q t -p x = - q ǫ , q(T, x) = q T (x).
For more information on the relaxation system, its limiting scheme for ǫ = 0, further numerical analysis and extensions we refer to [START_REF] Aregba-Driollet | Natalini Discrete kinetic schemes for multidimensional systems of conservation laws[END_REF][START_REF] Areba-Driollet | Convergence of relaxation schemes for conservation laws[END_REF][START_REF] Banda | Higher-order relaxation schemes for hyperbolic systems of conservation laws[END_REF][START_REF] Gottlieb | Strong stability preserving high-order time discretization methods[END_REF][START_REF] Ascher | Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations[END_REF][START_REF] Boscarino | Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit[END_REF][START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF]25,[START_REF] Pareschi | Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations Recent Trends in Numerical Analysis[END_REF] and the references therein. Also, the computations are valid provided that all appearing functions are at least once differentiable. This is in general not the case for conservation laws.

IMEX-Runge-Kutta Discretization

Numerical discretization of the relaxation system using higher order temporal discretizations combined with higher order spatial discretization has been investigated in several recent publications as for example [START_REF] Jin | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF][START_REF] Pareschi | Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations Recent Trends in Numerical Analysis[END_REF]. We apply so called implicit-explicit Runge-Kutta methods [START_REF] Pareschi | Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation[END_REF][START_REF] Pareschi | Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations Recent Trends in Numerical Analysis[END_REF][START_REF] Ascher | Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations[END_REF] as temporal discretization (IMEX RK). Here, the expliciti integration is used for the linear hyperbolic transport part and an implicit method is applied to the the stiff source term. Implicit-explicit Runge-Kutta method have been studied in the context of control problems for example in [START_REF] Banda | Adjoint imex-based schemes for control problems governed by hyperbolic conservation laws[END_REF][START_REF] Herty | Time discretizations for numerical optimization of hyperbolic problems[END_REF]. Define

y = (u, v) T , g(y) = (v, a 2 u) T and r(y) := (0, -(v -f (u))) T then (2) becomes y t + g(y) x = 1 ε r(y), and
y(0, x) = (u 0 , f (u 0 )) T (x)
Applying a suitable discretization D x of the spatial derivative yields the semidiscrete state equations

y ′ = -D x g(y) + 1 ǫ r(y), y(0) = y 0 . ( 4
)
Remark 1. Spatial discretizations for the linear transport part are well-known. The simplest possible is a first-order Upwind method:

∂ ∂t y j = - 1 ∆x 0 1 a 2 0 (y j+1/2 -y j-1/2 ) + 1 ǫ r(y j ) ,
where y j+1/2 is obtained by applying the first-order upwind method to characteristic variables v ± au. Higher order MUSCL schemes, WENO schemes or central schemes have also been studied in this context.

The resulting semi-discrete optimal control problem is then given by: minimize j(y(T ), y 0 ) subject to y ′ = -D x g(y)

+ 1 ǫ r(y), y(0) = y 0 . t ∈ [0, T ] (5)
In the context of relaxation schemes the semi-discrete problem is seen as a time-integration problem with stiff source which is discretized by an IMEX RK methods. For the numerical discretization we therefore consider the previous problem as an optimal control problem involving ordinary differential equations. Literature concerning the numerical analysis of Runge-Kutta methods for the optimality system of ( 5) have been studied in [START_REF] Hager | Runge-Kutta methods in optimal control and the transformed adjoint system[END_REF][START_REF] Bonnans | Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control[END_REF][START_REF] Lang | Methods in optimal control Preprint[END_REF]. In [START_REF] Bonnans | Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control[END_REF][START_REF] Hager | Runge-Kutta methods in optimal control and the transformed adjoint system[END_REF] partitioned Runge-Kutta methods for the optimality system are obtained using the discretize-then-optimize approach. The derived partitioned Runge-Kutta methods have been analysed with regard to symplecticity and order of convergence. In [START_REF] Herty | Time discretizations for numerical optimization of hyperbolic problems[END_REF], Herty and Schleper, moreover, analysed the associated adjoint imex Runge-Kutta method that one obtains if an explicit method is applied to D x g(y) and a (diagonally) implicit method to 1 ǫ r(y). In the following, we will analyse general partitioned Runge-Kutta methods using IMEX RK methds. More details can be found in [START_REF] Herty | Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems[END_REF]. Therein, the following IMEX Runge-Kutta discretization of ( 4) is studied.

Y (i) n = y n + h i-1 j=1 ãij D x g(Y (j) n ) + h i j=1 a ij 1 ǫ r(Y (j) n ) i = 1, .., s y n+1 = y n + h s i=1 ωi D x g(Y (i) n ) + h s i=1 ω i 1 ǫ r(Y (i) n ), n = 0, 1, 2, . (6)
A nonlinear variable transformation and two intermediate states

K(i) n and K (i) n
give the equivalent system

K(i) n = D x g y n + h s j=1 ãij K(j) n + h s j=1 a ij K (j) n i = 1, .., s K (i) n = 1 ǫ r y n + h s j=1 ãij K(j) n + h s j=1 a ij K (j) n i = 1, .., s y n+1 = y n + h s i=1 ωi K(i) n + h s i=1 ω i K (i) n , n = 0, 1, 2, . (7)
The associated optimality systems for the two previous optimization problems then coincide and we refer to [START_REF] Herty | Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems[END_REF] for mor details. It is proven that the adjoint schemes are equivalent to

P (i) = p n -h s j=1 αij g ′ (Y (j) n ) T Dx P (j) -h s j=1 α ij 1 ǫ r ′ (Y (j) n ) T P (j) i = 1, .., s P (i) = p n -h s j=1 βij g ′ (Y (j) n ) T Dx P (j) -h s j=1 β ij 1 ǫ r ′ (Y (j) n ) T P (j) i = 1, .., s (8)
p n+1 = p n -h s i=1 ωi g ′ (Y (i) n ) T Dx P (i) -h s i=1 ω i 1 ǫ r ′ (Y (i) n ) T P (i) n = 0, 1, .., N -1
Here, the coefficients of the Runge-Kutta method αij , α ij , βij and β ij are given by

αij := ωj - ωj ωi ãji , α ij := ω j - ω j ωi ãji , βij := ωj - ωj ω i a ji , β ij := ω j - ω j ω i a ji .

Properties of Discrete IMEX-RK Optimality System

For the resulting scheme ( 6),( 8) order conditions can be stated [START_REF] Herty | Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems[END_REF]. To this end we add a suitable equation for p to the previous system.

pn+1 = pn -h s i=1 ωi f y (Y (i) n ) T P (i) -h s i=1 ω i g y (Y (i) n ) T P (i) . (9)
The full method therefore is a standard additive Runge-Kutta scheme for Theorem 1. Consider the Runge-Kutta scheme ( 6),( 8),( 9). This scheme is of

y ′ = -D x g(y) + 1 ǫ r(y) p′ = g ′ (y) T D x p + 1 ǫ r ′ (y) T p p ′ = g ′ (y) T D x p + 1 ǫ r ′ (y) T p

If we define

-First-Order : if (SRK1) is of first order -Second-Order : if (SRK1) is of second order -Third-Order : if (SRK1
) is of third order and either

s i=1 ω i γ 2 i = 1 3 , s i=1 ω i γ2 i = 1 3 , s i=1 ω i γ i γi = 1 3 , are satisfied or if s i=1 ω i a ij γ i = 1 6 , s i=1 ω i ãij γi = 1 6
and if

s i=1 ω i a ij γi = 1 6 or s i=1 ω i ãij γ i = 1 6
are satisfied.

Note that the system (6) and ( 8) is not completely coupled, since the forward scheme ( 6) is solved independently of the adjoint scheme [START_REF] Boscarino | Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit[END_REF]. General order conditions can be found e.g. in [START_REF] Kennedy | Additive Runge-Kutta schemes for convectiondiffusion-reaction equations[END_REF]. The proof of Theorem 1 and together with more details are discussed in [START_REF] Herty | Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems[END_REF].

3 Numerical Results

Scalar Example

As a simple example, we use a tracking type functional J(u) together with Burgers' equation

u t + u 2 2 x = 0,
and the desired state u d at final time T = 2.0, that belongs to the initial condition u d (0, x) = 1 2 + sin(x) and we start the optimization with the initial data u s (0, x) ≡ 0.5. Moreover, the spatial interval is given by x ∈ [0, 2π], As discretization of the objective functional, we use

J(u(•, T ), u 0 , u d ) = ∆x 2 K i=1 u i -u d i 2 .
Moreover, the discrete gradient of the reduced cost functional is given by ∇ u0,i J = p 0,i + (Df (u 0 ) T q 0 ) i .

In order to solve the optimal control problem, we apply a steepest descent method (with respect to the reduced cost functional) with fixed stepsize 0 < α < 1, i.e. we set u k+1 0 = u k 0 + α∇ u0,i J. As stopping criterion for the optimization process we test | J(u 0 , u d )| < tol where tol = 1E -2 denotes a predefined stopping tolerance. We observe grid independence in the case where u and u 0 are differentiable in space and time. As first-order scheme, we test the Implicit-Explicit Euler scheme

u * i = u n i v * i = v n i -∆t ǫ (v * i -f (u * i )) u n+1 i = u * i -∆t D x v * i v n+1 i = v * i -∆t a 2 D x v * i
for the forward, as well as for the backward

q * i = q n+1 i -∆tD * x p n+1 i p * i = p n+1 i -∆t a 2 D * x q n+1 i q n i = q * i -∆t ǫ q n i p n i = p * i + ∆t ǫ q n i f ′ (u n i )
The spatial gridsize is chosen to be N x = 300, whereas the time discretization is done according to the CFL condition with constant c CF L = 0.5.

Summary

We briefly discussed a class of numerical methods applied to an optimal control problem for scalar, hyperbolic partial differential equations. Order conditions for the temporal numerical discretization in the case of differentiable functions have been stated. Future work includes the analysis of additional properties of the derived numerical discretizations as for example strong stability and asymptotic preservation properties.

On the other hand, using (6) the associated discrete adjoint equations are

ζ (i) n = h ωi f y (Y (i) n ) + ω i g y (Y (i) n ) T p n+1 + s j=1 ãji f y (Y (i) n ) T ζ (j)

a

  ji g y (Y (i) n ) T ζ (j) i = 1, .., s p n = p n+1 + s i=1 ζ (i) n , i = 1, .., N -1, p N = j ′ (y N )which can be transformed into the scheme (8) using the variable transformation P (i) n := p n+1 +

Acknowledgments.

This work has been supported by DFG HE5386/7-1, HE5386/8-1 and by DAAD 50727872, 50756459 and 54365630. We also acknowledge the support of Ateneo Italo-Tedesco (AIT) under the Vigoni project 2010-2012 'Adjoint implicit--explicit methods for the numerical solution to optimization problems'.

Appendix.

The discrete adjoint equations that correspond to the discrete optimization problem associated with [START_REF] Bonnans | Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control[END_REF] are

Moreover, the variable transformation that is needed to obtain ( 8) is given by

h ω i (i = 1, .., s; n = 0, .., N -1) .