Emilio Turco

Francesco Dell'isola

Antonio Cazzani

Nicola Luigi Rizzi

Hencky-type discrete

Keywords: Pantographic structures, Micro-mechanical model, Second gradient continuum, Nonlinear problems

come

Introduction

In this paper, we introduce ab initio a discrete model for planar pantographic structures and we correspondingly formulate, using MATLAB, an effective computation tool which overcomes some technical difficulties confronted in [START_REF] Dell'isola | Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response[END_REF][START_REF] Dell'isola | Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution[END_REF] via heavier finite elements-based computational schemes, whose formulation starts from second gradient continuum models. The developed computation tool, in particular, is better suited for determining the deformed shapes of pantographic sheets when: (1) large extensions are activated locally, (2) shear angles approach the value π 2 and (3) locking or "hourglass deformation effects" could arise due to the high heterogeneity of the material properties.

To explicit its effectiveness, we compare the newly developed computational tool with the one previously adopted using the FEM implemented in COMSOL Multiphysics (see [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF]) based on a second gradient continuum model. The presented comparison considers some typical deformation problems including: (1) standard extension bias test; (2) shear/extension tests; (3) bending/extension tests; and (4) deformations induced by displacements imposed to single fibers (extraction tests).

The discrete model which we consider here includes both pairwise and triple elastic interactions between material particles and generalizes Hencky discrete model for the elastica [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF][START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF][START_REF] Challamel | Discrete and non-local elastica[END_REF][START_REF] Wang | Hencky bar-chain model for buckling and vibration of beams with elastic end restraints[END_REF][START_REF] Challamel | Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams[END_REF][START_REF] Madeo | Wave propagation in pantographic 2D lattices with internal discontinuities[END_REF].

The mechanical systems which could be described by the introduced discrete model include those considered and studied in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF][START_REF] Dell'isola | Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution[END_REF][START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF][START_REF] Greco | In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions[END_REF][START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model[END_REF]. Indeed, (1) the pivots interconnecting (see Fig. 1 for a 3D printing) the two arrays of beams are to be identified with the material particles which we introduce here; [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF] the fact that these pivots may store elastic energy may be efficiently modeled introducing suitable "shearing" springs (that is the rotational springs called r 3 in the following Figs. 3,4); [START_REF] Dell'isola | Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response[END_REF] the bending phenomena occurring between the two closest pivots and the corresponding stored elastic energy are modeled by the rotational springs called r 1 and r 2 in the following Figs. 3 and4; and (4) the extensional phenomena occurring between two closer pivots are modeled by the extensional springs called r 0 .

However, we believe that the potential scope of the presented model is wider: it very likely may include the most relevant deformation phenomena in the woven fabrics described in [START_REF] Boisse | The bias-extension test for the analysis of in-plane shear properties of textile composite reinforcements and prepregs: a review[END_REF][START_REF] Barbagallo | Bias extension test on an unbalanced woven composite reinforcement: experiments and modeling via a second-gradient continuum approach[END_REF][START_REF] Cao | Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results[END_REF][START_REF] Harrison | Shear characterisation of viscous woven textile composites: a comparison between picture frame and bias extension experiments[END_REF][START_REF] Harrison | Characterising the shear-tension coupling and wrinkling behaviour of woven engineering fabrics[END_REF][START_REF] Harrison | Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh[END_REF][START_REF] D'agostino | Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements[END_REF][START_REF] Caggegi | Experimental global analysis of the efficiency of carbon fiber anchors applied over cfrp strengthened bricks[END_REF]. Indeed, the fibers constituting these fabrics intersect one with the others very similarly than in pantographic structures, and in their deformation, one can still distinguish extension, bending and shear: even if the hypothesis of elastic behavior in woven fabrics is applicable in a smaller range of deformation states, the model presented here seems capable of catching some relevant aspects of its phenomenology. Similarly, some biological living tissues which are characterized by fibers embedded in a soft matrix can be described with the proposed model (see e.g., [START_REF] Grillo | Remodelling in statistically oriented fibre-reinforced materials and biological tissues[END_REF][START_REF] Federico | Nonlinear elasticity of biological tissues with statistical fibre orientation[END_REF][START_REF] Grillo | Growth, mass transfer, and remodeling in fiber-reinforced, multi-constituent materials[END_REF][START_REF] Federico | A transversely isotropic, transversely homogeneous microstructural-statistical model of articular cartilage[END_REF][START_REF] Federico | Elasticity and permeability of porous fibre-reinforced materials under large deformations[END_REF]).

In conclusion, we are sure that the synthetic discrete description given in this paper may supply some insight in all those phenomena where chains of forces may be activated by directional strain fields. A careful comparison with available experimental evidence will be the subject of some following papers.

After the concise statement concerning the original results, we complete this introduction by framing the presented results in the more general context of the mechanics of metamaterials and their continuum modeling. The dichotomy between discrete and continuum modeling will be shortly treated in a digression preceding the synthetic description of the structure of the paper.

Metamaterials and their modeling with generalized continua

The reader will find an increasing literature dealing with the class of materials which has been collectively gathered under the unifying concept of metamaterials. In this context, we have found particularly interesting the works by [START_REF] Engheta | Metamaterials: Physics and Engineering Explorations[END_REF][START_REF] Zouhdi | Metamaterials and Plasmonics: Fundamentals, Modelling, Applications[END_REF], where metamaterials are described as those materials which, while not existing in Nature, are conceived by means of a theoretical abstraction and then engineered to have "exotic" properties.

Metamaterials are obtained by the "architectured" assemblage of multiple individual "microscopic" elements constituted by standard materials, usually arranged in (quasi-)periodic substructures [START_REF] Del Vescovo | Dynamic problems for metamaterials: review of existing models and ideas for further research[END_REF]. It results that the macroscopic physical properties of metamaterials depend in a critical way on the topology of their microstructures and the nature of the interaction among constituting elements, while they are less sensitive to the standard properties of the micro-material constituting their microstructural elements. The macroscopic physical properties tailored in the design of metamaterial may be, for instance, optical, electromagnetic, thermal, mechanical and any combination (and coupling) of many of them. For instance, the particular shape, geometry, size, orientation and arrangement of the microstructural elements affect, in some particular metamaterials, the propagation of waves in a not-often-observed manner, e.g., not allowing the propagation in some band gaps (see, e.g., [START_REF] Misra | Granular micromechanics based micromorphic model predicts frequency band gaps[END_REF][START_REF] Auriault | Long wavelength inner-resonance cut-off frequencies in elastic composite materials[END_REF][START_REF] Boutin | Acoustics of porous media with inner resonators[END_REF][START_REF] Placidi | Euromech 563 Cisterna di Latina 17-21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions[END_REF]) even if such a property is not present in the material constituting the microstructural elements (see also [START_REF] Placidi | Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials[END_REF][START_REF] Rosi | Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids[END_REF][START_REF] Berezovski | Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations[END_REF][START_REF] Abd-Alla | Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity[END_REF] for others unusual wave propagation phenomena).

Mechanical metamaterials.

In the literature, more recently, a particular class of metamaterials has attracted the attention of mechanicians: the so-called mechanical metamaterials. The properties for which these architectured materials show unexpected or exotic features are purely mechanical. The newly synthesized metamaterial studied in [START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF] and considered here belongs to this class.

To give a hint of the possible performance of newly designed metamaterials, we list here some whose applications seem closer to those which we have in mind (for a more detailed discussion the reader is referred to [START_REF] Dell'isola | Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response[END_REF][START_REF] Carcaterra | Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials[END_REF][START_REF] Madeo | Modeling and designing micro-and nano-structured metamaterials: towards the application of exotic behaviors[END_REF]): (1) it has been synthesized a (micro-)composite medium showing a negative effective bulk modulus, or a negative effective mass density, or both properties; (2) it is possible to design and fabricate metamaterials with negative Poisson's ratio; (3) it is possible to tailor materials showing selective damping effects using special material microstructures (see e.g., [START_REF] Enakoutsa | A model for elastic flexoelectric materials including strain gradient effects[END_REF][START_REF] Giorgio | Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications[END_REF]).

The main problem in the theory of metamaterials is actually an inverse problem: given a continuum/discrete model (i.e., a set of kinematical parameters, either finite or infinite dimensional, and a Lagrangian action functional), one has to find if there actually exist mechanical systems which, at a specific length scale, behave as it is prescribed by the chosen (continuum or discrete) model.

The problem actually consists in finding the microscopic properties and structure of such mechanical systems to be able to construct (or how sometimes it is said with a word derived from Greek: synthesize) them, after having solved the corresponding technological problems.

The metamaterial conceived in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF][START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF][START_REF] Madeo | Wave propagation in pantographic 2D lattices with internal discontinuities[END_REF][START_REF] Cuomo | Simplified analysis of a generalized bias test for fabrics with two families of inextensible fibres[END_REF] has been designed to be governed, when considering macroscopic behavior, as a second gradient continuum and to have an enhanced extensional toughness: it has been actually synthesized exploiting the novel possibilities given by 3D printing technologies. This new class of bidimensional metamaterials, which have been also called extensible pantographic sheets, has been modeled using bidimensional continua which generalize those introduced in [START_REF] Steigmann | Equilibrium of elastic nets[END_REF][START_REF] Pipkin | Some developments in the theory of inextensible networks[END_REF][START_REF] Pipkin | Plane traction problems for inextensible networks[END_REF]: indeed they are conceived to account for the elastic energy in the extensional deformation and geodesic bending of constituting fibers simultaneously.

1.1.2. Generalized continuum models. Continuum mechanics is a powerful mean for the design of mechanical structures: indeed starting from the first half of nineteenth century this design has been based on the application of predictive mathematical models. Starting from the pioneering works by Navier and Piola (see the discussion in the following subsection), it was established that continuum models could be deduced from discrete microscopic ones: the underlying idea is that materials can be ultimately modeled at a micro-level as finite-dimensional Lagrangian systems and that their effective properties can be all obtained via a suitable homogenization procedure. However, the working assumptions put forward by Cauchy were later assumed to be of universal validity. The works of Piola on generalized continua (see [START_REF] Dell'isola | Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola's peridynamics and generalized continuum theories[END_REF]) and what has been later rediscovered and called peridynamics (see [START_REF] Dell'isola | At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola[END_REF]) remained for a long time nearly unknown. The limiting assumptions accepted by Cauchy include the absence of length scale phenomena and of high contrast in microscopic properties. On the other hand, the particular class of microstructured mechanical systems which present high contrast1 in microscopic properties once homogenized has been shown to produce generalized continua in [START_REF] Pideri | A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium[END_REF][START_REF] Camar-Eddine | Determination of the closure of the set of elasticity functionals[END_REF][START_REF] Reis | Construction of micropolar continua from the asymptotic homogenization of beam lattices[END_REF][START_REF] Goda | Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization[END_REF][START_REF] Cecchi | Heterogeneous elastic solids: a mixed homogenization-rigidification technique[END_REF][START_REF] Misra | Granular micromechanics model of anisotropic elasticity derived from Gibbs potential[END_REF][START_REF] Misra | Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics[END_REF][START_REF] Misra | Micro-poromechanics model of fluid-saturated chemically active fibrous media[END_REF].

Actually, the continuum model introduced by Cauchy is very accurate for a large class of phenomena but cannot be applied to all materials in every physical condition (and the contrary would be really surprising). When introducing generalized continua, the true conceptual frame settled by Cauchy and Navier is to be drastically modified. The concept of stress becomes secondary and the main role is played by deformation measures together with action and dissipation functionals (see [START_REF] Dell'isola | Variational Principles are a Powerful Tool also for Formulating Field Theories[END_REF] or e.g., [START_REF] Federico | An energetic approach to the analysis of anisotropic hyperelastic materials[END_REF][START_REF] Nadler | A model for frictional slip in woven fabrics[END_REF][START_REF] Scerrato | A simple non-linear model for internal friction in modified concrete[END_REF][START_REF] Scerrato | A micro-structural model for dissipation phenomena in the concrete[END_REF]). The Euler-Lagrange equation obtained in this more encompassing modeling process cannot be anymore regarded to coincide with the balance of force unless one generalizes the concept of force. This can be done by introducing generalized actions as the dual quantities in the work of the gradients of displacements (see e.g., [START_REF] Germain | The method of virtual power in continuum mechanics. Part 2: microstructure[END_REF][START_REF] Mindlin | Second gradient of strain and surface-tension in linear elasticity[END_REF][START_REF] Dell'isola | How contact interactions may depend on the shape of cauchy cuts in n-th gradient continua: approach á la D'Alembert[END_REF][START_REF] Carcaterra | Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials[END_REF] or [START_REF] Altenbach | On generalized Cosserat-type theories of plates and shells: a short review and bibliography[END_REF][START_REF] Altenbach | On the linear theory of micropolar plates[END_REF][START_REF] Gabriele | A 1D higher gradient model derived from Koiter's shell theory[END_REF] for 2D or 1D generalized continua characterized by a specific microstructure).

All these difficulties are easily surmounted when discrete models are usable: what we prove is possible in the present paper. Moreover, a further consideration is needed here: at the end a continuum model, for getting predictions, needs to be discretized. In other words, in any case a discrete model is used, whose structure is rather out of the control of the modeler as it is mediated by many complex steps. In the case of pantographic structures, they can be shortly listed as follows: (1) to find the right class of generalized continua which is able to capture their behavior (and it seems that such class must be at least the class of second gradient Piola continua, if not the class of micro-stretch continua); (2) to find the correct FEM for getting algorithms converging to the physically meaningful shapes of introduced continua; and (3) to establish the correct identification of constitutive parameters for getting physically founded predictions.

Here, we control totally the modeling process by introducing a convincing discrete model, we formulate a Lagrangian equilibrium condition, and we find with a robust code under our complete control, since it has been developed by ourselves, to determine all equilibrium configurations. The comparison with the previously obtained results found in [START_REF] Dell'isola | Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response[END_REF][START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF] is really comforting: there is a total correspondence of the predictions. When one of the two models fails to converge, it is always the one obtained via the mediation of generalized continuum theories.

A historical perspective

The controversy about the prevalence of continuum models on discrete models (or vice versa, depending on the tastes and the schools) for mechanical (or more generally physical) systems is very old. Probably, it started with the antithesis between Epicurean atomism and Heraclitean continuum approach: the more recent dispute between Mach and Boltzmann is an example of how bitter this contraposition may become. In this paper, we do not try a historical analysis of this contraposition and we do not dare to take position on the aforementioned controversy. Actually, we believe that one has to assume a pragmatic view on the question. In general, a mathematical model to be used for describing physical systems has to have the following obvious features: ( 1) it has to be formulated based on a clear minimal set of postulates which should be self-consistent and ( 2) it has to be predictive when applied to describe physical phenomena.

In order to get predictions out of a set of postulates, some exercises of mathematics need to be solved: the obtained solutions, when engineering applications are needed, must involve the performance of specific computations, which allow for the exact quantitative verification of considered models and for the precise design of novel devices.

Before the advent of modern computing machines, the most powerful computing tool has been given by mathematical analysis and its ancillary disciplines, i.e., the theories of analytical, elementary or special functions. In fact, once a model was postulated on the basis of a variational principle (see, e.g., [START_REF] Dell'isola | Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola's peridynamics and generalized continuum theories[END_REF][START_REF] Dell'isola | At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola[END_REF][START_REF] Dell'isola | The complete works of Gabrio Piola: volume I-Commented English translation[END_REF]) then the corresponding Euler-Lagrange conditions were found in order to be able to apply for their solutions all the methods which are made available by mathematical analysis. We refer, for instance, to the approximate methods of solutions by series, by polynomial approximations, separation of variables, or to the so-called analytical or semi-analytical methods. Instead discrete mathematics, and in particular the theory of discrete difference equations, could not produce calculation methods based on closed form solutions, as it is done, instead, very effectively by infinitesimal analysis. As a consequence, also those scientists who, being followers of Epicurean school, believed in the atomistic and discrete fundamental nature of matter did study continuum models. Piola [START_REF] Dell'isola | The complete works of Gabrio Piola: volume I-Commented English translation[END_REF], Navier [START_REF] Navier | Mémoire sur les lois de l'équilibre et du mouvement des corps solides élastiques[END_REF], and later Boltzmann [START_REF] Cercignani | Ludwig Boltzmann: The Man who Trusted Atoms[END_REF] (just to name a few of them) started their postulation starting with a discrete finite-dimensional model where Lagrangian coordinates consist of N -tuples of material particles coordinates and then, via suitable homogenization methods, deduced the governing equations of the continuum approximating one.

Therefore, without the modern powerful tools of automatic computation, also the Epicureans needed to deduce continuum models to get applicable equations from which, via suitable analytical and simple numerical computations, they could get quantitative predictions.

A paradoxical situation was then created. Continuum models, which had been first created as auxiliary tools needed for performing calculations and obtained via a deductive process, were regarded subsequently as constituting the most fundamental ones, simply because they were considered the most effective way for describing macroscopic phenomena (see [START_REF] Truesdell | The Non-Linear Field Theories of Mechanics[END_REF]). The basic postulates in these continuum models have been identified with a list of balance laws (see [START_REF] Dell'isola | Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola's peridynamics and generalized continuum theories[END_REF]) to which a (presumed) firmer physical meaning has been (more or less arbitrarily) attributed. Subsequently, discrete models were introduced as an ancillary tool for getting effective calculation algorithms: variational principles for continuum models were therefore deduced as "theorems" (and it did not seem to many an oxymoron to talk about the theorem (sic!) of the principle of virtual work ) to be used for introducing the continuum model discretization. Actually, as the historical circumstances were changed, discrete models had become the most suitable ones for supplying quantitative predictions, as von Neumann machines are able to calculate the solution of algebraic equations having millions of unknowns. This shift on the role given, in the passage from a cultural paradigm (in the sense of Kuhn [START_REF] Kuhn | The Structure of Scientific Revolutions[END_REF]) to another paradigm, to the same concept is not new in the history of science. One concept may be considered as "fundamental" by a school and "ancillary" in another one and vice versa. We postpone a careful study of these cultural phenomena to further investigations as it seems that it plays a major role also in mechanical sciences. Simply we want to recall here (see, e.g., [START_REF] Russo | Forgotten Revolution: How Science was Born in 300 BC and Why it Had to be Reborn[END_REF]) that in Hellenistic science a predictive model describing the motion of planets seems to have been formulated. The main tool for calculating these motions was some mechanisms (like Antikythera Machine see [START_REF] De Solla | Gears from the Greek. The Antikythera mechanism: a calendar computer from ca. 80 BC[END_REF]) whose functioning mechanism was epicyclic gears: Ptolemy, being a practical astronomer, instead of teaching the underlying astronomic theory decided to describe simply the theory needed for constructing the relative calculation ancillary tools. As a consequence, during Middle Age, the astronomers believed to the existence of several heavens around the Earth, the closer ones embedding the "moving planets" which were supported by moving structures and the most distant one being constituted by a fixed vault on which also fixed stars were blocked. The cultural shift was again of the same nature: those concepts, which had been conceived as secondary ones needed for supplying calculating tools, did eventually become the most fundamental ones, assumed as the basis of the whole theoretical speculation.

Structure of the paper

In Sect. 2, the Lagrangian finite-dimensional model proposed and studied in this paper is introduced. The spirit of Piola, when he describes the molecular mechanical system which he places at the foundation of his presentation, is faithfully followed. A lattice of positions is defined which characterizes the reference configuration of all particles constituting the system. Lagrangian mechanics formalism is then assumed: a deformation energy is associated with the set of interaction springs introduced and the problem of its minimization is presented.

In Sect. 3, a solution strategy of the aforementioned minimization problem is presented, and all the details needed for its eventual reproduction are given.

In Sect. 4, some numerical solutions are shown for some equilibrium problems related to engineering applications. In particular there are presented solutions which are relevant to standard extension bias test with the correspondent reactive force distributions and resultants induced by imposed displacements on constraints.

The reader will remark that in the dedicated literature the determination of reactive forces for bias test is considered somehow difficult. Moreover, some equilibrium shapes are shown in the presence of displacements imposed on single fibers and in the test of extraction of a fiber.

In Sect. 5, the performance of the presented numerical model is compared with those shown in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF]. The code which is presented here needed an ad hoc programming activity while previously the code was obtained by using a standard FE integrator (COMSOL). As expected, the convergence capacities (to solution having a correct physical interpretation) of the newly presented model are improved remarkably. Of course, with FE codes one can improve the results related to higher gradient continuum theories using newly developed numerical tools with an intrinsic high continuity as done, e.g., in [START_REF] Bilotta | Performance of a high-continuity finite element in three-dimensional elasticity[END_REF][START_REF] Cazzani | An analytical assessment of finite elements and isogeometric analysis of the whole spectrum of Timoshenko beams[END_REF][START_REF] Cazzani | An analytical assessment of finite elements and isogeometric analysis of the whole spectrum of Timoshenko beams[END_REF][START_REF] Cazzani | Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches[END_REF][START_REF] Cazzani | Constitutive models for strongly curved beams in the frame of isogeometric analysis[END_REF][START_REF] Greco | An implicit G 1 multi patch B-spline interpolation for Kirchhoff-Love space rod[END_REF][START_REF] Cuomo | A variational model based on isogeometric interpolation for the analysis of cracked bodies[END_REF][START_REF] Greco | Consistent tangent operator for an exact Kirchhoff rod model[END_REF][START_REF] Greco | An isogeometric implicit G1 mixed finite element for Kirchhoff space rods[END_REF]; however, herein we want to explore the potential of the proposed discrete model whose features are based on the microstructure properties of the system under study and compare its performances with already tested and verified models.

In conclusions, the future immediate research perspectives are listed and some major possible modeling improvements are delineated.

Hencky-Piola-type model for planar pantographic sheets

As discussed in Introduction, among the many modeling possibilities, we have chosen, in this paper, to introduce a finite-dimensional Lagrangian model for describing pantographic planar sheets. This model is Piola type as we consider a finite number of material particles occupying, in a reference configuration, the nodes of a rectangular square cell lattice having length ε (see Fig. 2, where it was assumed that n f = 10 square cells constitute the shorter side of the rectangular lattice).

The lattice is, roughly speaking, composed by two arrays of fibers which are oriented at an angle π/4 and -π/4 with respect to x 1 axis: we will call array 1 that at the angle π/4 and array 2 the other one.

Each of the introduced material particles models the pivots which in the 3D printed specimen interconnect the two considered arrays of beams, see Fig. 1.

This model is Lagrangian because we introduce the reference position of the generic ith material particle via the position P i and we consider as a Lagrangian kinematical parameter for each of them the actual position, denoted by p i .

r 0 r 0 r 0 r 0 r 1 r 2 r 3 x 1 , u 1 x 2 , u 2 i 1 j 1 = j 2 k 1 i 2 k 2 I II III IV ε Fig. 3. Discrete mechanical model
This model is also a Hencky type one because, as done in his pioneering work (see [START_REF] Hencky | Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette[END_REF][START_REF] Challamel | Discrete and non-local elastica[END_REF][START_REF] Wang | Hencky bar-chain model for buckling and vibration of beams with elastic end restraints[END_REF][START_REF] Challamel | Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams[END_REF]) for elasticae, we model the elastic interactions among the particles by means of rotational and extensional springs such as those represented in Fig. 3, so that we assume to need pairwise and triple particles interactions.

Our conjecture is that it is formulated, in this way, a discrete mathematical model which is able to supply an accurate prediction of the structural response in the regime of (very) large displacements.

The modeling procedure is to be completed by defining the deformation energy of each spring element. The problem of determining equilibrium configurations is successively solved by imposing the stationarity of total potential energy following the algorithm described in Sect. 3.

Our Ansatz concerning deformation energies can be summarized as follows: Deformation energy for axial springs Each axial spring stores elastically an energy which depends quadratically on its variation of length:

w 0 = 1 2 r 0 ( p j -p i -ε) 2 , ( 1
)
where p i and p j are the actual position, see Figs. 3 and4, of the nodes connected by the considered extensional spring whose rigidity is denoted r 0 while ε is the initial unstressed length of the axial spring.

Deformation energy for bending springs Three consecutive particles along array 1 or 2 interact via a rotational spring whose stored energy depends on the angle formed by the two consecutive segments connecting, in the actual configurations, the particles' positions:

w 1,2 = r 1,2 (cos γ 1,2 + 1) , (2)
where

cos γ 1,2 = p j1,2 -p i1,2 2 + p k1,2 -p j1,2 2 -p k1,2 -p i1,2 2 2 p j1,2 -p i1,2 p k1,2 -p j1,2 , (3)
and p i1,2 , p j1,2 and p k1,2 are the ordered actual position of the three aligned nodes along array 1 or 2 and r 1 , see Figs. 3 and4, and r 2 denote the rigidities of the corresponding rotational springs. The deformation energies for springs at the nodes where displacement is imposed need the specification of suitable boundary conditions. In particular, if the short sides of the lattice are constrained to be fixed on two rigid bodies and the bending springs are connected with material segments of these bodies then the previous formulae obviously supply the needed relations when setting (if the constrained node j has not the subsequent node k in the interior of the lattice)

x 1 , u 1 x 2 ,
p k1,2 = P k1,2 + R k1,2 ,
or (if the constrained node j has not the preceding node i in the interior of the lattice)

p i1,2 = P i1,2 + R i1,2 ,
where R k1,2 or R i1,2 represent the corresponded rigid displacement. Deformation energy for shear springs Via the interconnecting pivots, the two arrays of fibers interact elastically: if the actual angle between the two arrays is γ 3 we assume that the stored energy depends quadratically on the difference γ 3 -π

2 w 3 = 1 2 r 3 γ 3 - π 2 2 , (4)
where (see Figs. 3,4)

cos γ 3 = p j1 -p k 2 2 + p k1 -p j1 2 -p k1 -p k2 2 2 p j1 -p k 2 p k1 -p j1 , (5)
if p j1 and p k1 , p k 2 are the actual position of the nodes relative to the rotational spring of rigidity r 3 connecting array 1 with array 2. Remark that we assume to have one of such shear springs also in quadrants II, III and IV, see Fig. 3.

The strain energy defined for each kind of introduced springs supplies a Hencky-type model for the pantographic sheet and allows us to easily define the total strain energy by simply summing each contribution.

In order to have a complete solution of the considered equilibrium problem, displacements and forces or couples exerted by each spring, a step-by-step procedure was implemented to reconstruct the complete equilibrium path of the pantographic sheet, as it will be described in detail in the following Sect. 3.

The solution algorithm

We collect the nodal displacements in the lattice in the vector d in order to write the total energy of the pantographic structure in the form:

W (d) = e w 0 + w 1 + w 2 + w 3 -L ext , (6)
where e ranges on all the spring, extensional, bending and shear and L ext is the work of the external loads and all quantities on the RHS depend on the variable d.

The equilibrium problem which we want to consider is a mixed one: we assume that the displacements of some particles are imposed and that some external conservative forces are applied to the remaining particles. Let us therefore decompose d into the pair composed by two vectors: the imposed displacements u a and the free displacements u. For notational aims, we will reorder d to get the decomposition

d = (u, u a ) .
Because of our assumption, we have that L ext depends only on u.

The nonlinear system of equilibrium equations is obtained by imposing that the first variation of W vanishes:

s(u) -p(u) = 0, (7)
where p(u) is the vector which collects the Lagrangian components of external forces (which may be assumed to be dead loads, for instance, so that p becomes independent of u) and s(u) is the vector of the internal forces (called also, in the context of structural mechanics, structural reaction), as defined by:

s(u) = dW du , p(u) = dL e du . ( 8
)
The tangent stiffness matrix is defined as the derivative of the structural reaction s(u) with respect to the displacements vector u, in formulae:

K T (u) = ds(u) du = d 2 W du 2 , ( 9
)
If the external load potential depends on a parameter μ and the imposed displacements depend on a parameter λ (as in the case of quasi-static loading and imposed displacements), then Eq. ( 7) becomes: r(u, λ, μ) := s(u, u a (λ))p(u, μ) = 0.

An equilibrium manifold is defined as a surface u(λ, μ) such that r(u(λ,μ), λ,μ) = 0. [START_REF] Greco | In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions[END_REF] For example, in the case of an external dead load, we will have that p is independent of u, so that if a linear dependence on μ is assumed, we can set:

p(μ) = p 0 + μp, ( 11
)
Moreover, when the equilibrium equation ( 7) can be linearized in the neighborhood of the solution u 0 relative to the dead load p 0 then, as s(u 0 )p 0 = 0 and setting (uu 0 ) =: Δu we have

0 = s(u (μ)) -p 0 -μp s(u 0 ) -p 0 + K T (u 0 )Δu-μp = K T (u 0 )Δu-μp, from which we deduce Δu =μ (K T (u 0 )) -1 p, ( 12
)
which will be of use in the sequel. The solution of the nonlinear equilibrium system of Eq. ( 10) can be found by means of an incremental iterative procedure based on the Newton-Raphson scheme.

As we will limit ourselves to the case of equilibrium paths depending only on imposed displacements, we introduce here only the parameter λ. Starting from an estimated point of the equilibrium path (λ j , u j ) verifying the condition r(u j , λ j ) ≤ η, [START_REF] Barbagallo | Bias extension test on an unbalanced woven composite reinforcement: experiments and modeling via a second-gradient continuum approach[END_REF] i.e., with a pair being an η-approximate solution of the equilibrium condition [START_REF] Wang | Hencky bar-chain model for buckling and vibration of beams with elastic end restraints[END_REF], the iterative scheme, once the step Δλ is fixed, is obtained by constructing the η-approximate solution (u j+1 =: u j + Δu j , λ j+1 := λ j + Δλ) by using a sub-iteration scheme for calculating Δu j .

This sub-iteration is specified by the following steps 1. We calculate the so-called residual nodal forces relative to the initial tentative value Δu j,0 u j , λ j = 0 r u j , λ j + Δλ =: p j,0 .

Once calculated the hth approximation

h l=0 Δu j,l u j , λ j of the increment Δu j , we calculate the corresponding residual

r u j + h l=0
Δu j,l u j , λ j , λ j + Δλ =: p j,h .

If the inequality

p j,h > η,
is verified we set, on the basis of ( 12) and assuming μ = -1 (this is equivalent to trying to find the solution by "imposing" fictitious external dead loads being opposed to the calculated residuals)

Δu j,h+1 = - ∂r(u j + h l=0 Δu j,l u j , λ j , λ j + Δλ) ∂u -1 p j,h ,
and continue the sub-iteration. 4. If the inequality p j,h ≤ η, is verified, then we set Δu j := h l=0 Δu j,l u j , λ j . and we stop the iteration.

Some numerical results

The geometrical parameters considered to model the pantographic structure are:

= 70 mm, ε = √ 2 2 n f ,
where n f is the number of fibers which intercept the smaller side of the lattice, e.g., in Fig. 2 n f = 10 and consequently ε ≈ 4.95 mm. The mechanical parameters of the lattice are the rigidities of the springs for which the following values are used:

r 0 = 134 N mm, r 1 = r 2 = 19.2 N mm, r 3 = k3 ε 2 = 3.90 4 N mm.

Standard extension bias test

This test is characterized by assuming, see again Fig. 2:

u 1 (0, x 2 ) = u 2 (0, x 2 ) = u 2 (3 , x 2 ) = 0, u 1 (3 , x 2 ) = 0.8 = 56 mm.
In Fig. 5, the deformation of the lattice, compared with its reference position (in gray), along with the color bar which indicates the level of the internal forces on the extensional springs for the case n f = 10 is reported. For the same test, Fig. 6a shows the global reaction of constraints on the side x 1 = 3 , more precisely the x 1 component in blue and the x 2 component in red, as a function of the parameter λ ∈ (0, 1) which guides the imposed displacements from 0 to u 1 (3 , x 2 ). Figure 6b shows the discrete density r(x 2 ) of the global reaction on x 1 = 0 when λ = 1 (the x 1 component in blue and the x 2 component in red), and Fig. 6c shows the axial E a (λ), bending E b (λ) and shear E s (λ) energies.

Bending-extension test

In this case, the imposed displacements are:

u 1 (0, x 2 ) = u 2 (0, x 2 ) = u 2 (3 , x 2 ) = 0, u 1 (3 , x 2 )) = 80 - 5 7
x 2 mm, which produce, for the case n f = 10, the deformation plot reported in Fig. 13 along with the color bar of the internal forces on extensional springs. Figure 14, as for the previous numerical test, reports the global reaction R(λ), its density r(x 2 ) and the strain energies subdivided in the extensional, bending and shear parts.

Figures 15 and16 show, for the same test, the results obtained from a second gradient model confirming the reliability of the discrete model results.

Shear-extension test

Shear-extension test is defined by the following displacements: 17 along with the color bar of the internal forces on extensional springs. Analogously, Fig. 18 reports the global reaction R(λ), its density r(x 2 ) and the strain energies subdivided in the extensional, bending and shear parts.

u 1 (0, x 2 ) = u 2 (0, x 2 ) = 0,

Extraction test

The last numerical test is characterized by the following imposed displacements:

u 1 (0, 0) = u 2 (0, 0) = u 1 (0, ) = u 2 (0, ) = 0.0 u 1 (3 , 0) = u 2 (3 , 0) = 0.8 = 56 mm, u 1 (3 , ) = u 2 ( 3
, ) = 0.8 = 56 mm, which produce, for the case n f = 10, the deformation plot reported in Fig. 19 along with the color bar of the internal forces on extensional springs. Finally, Fig. 20, reports the global reaction R(λ), its density r(x 2 ) and the strain energies subdivided in the extensional, bending and shear parts.

Conclusions and future perspectives

This paper is intended to supply the designing tool for an extensive campaign of measurements to be performed on pantographic structures, in planar deformations. The features of the code which is elaborated and described here are promising:

1. It is robust: its applications to fiber extraction test show that it can be applied in limit cases; the reader will remark that in this case first gradient continuum mechanics cannot be applied and that higher gradient continua require more complex FEM tools; 2. The constitutive parameters characterizing the model are only five: two extensional, two bending moduli to which one has to add a single shear modulus; 3. It allows for a rather intuitive micro-macro identification procedure, so that it is possible to easily identify the aforementioned moduli in terms of the material properties constituting the pantographic sheet and in terms of the geometric properties of the beam lattice constituting them.

The results presented here open interesting perspectives in different contexts:

1. First of all, they supply a strong motivation to the study of several complex mathematical problems, especially in homogenization theory; indeed the methods presented in [START_REF] Alibert | Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof[END_REF] need to be applied to the present context to prove that the correct homogenized model is indeed the second gradient model presented in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF]; this results will be of use when, in perspective, the pantographic structure will be constituted by nanofibers, so that at the smaller scale the considered system will have several millions of degrees of freedom, making the presented discrete approach not applicable; 2. Secondly, we have obtained an efficient predictive and designing tool, when the pantographic microstructure has not a too small characteristic length scale; it has been proven that with several thousands material particles the code is producing reliable results with a very limited computing time; as a consequence, we plan to use it to direct the 3D printing of many specimen and use the presented code for predicting their behavior; 3. Finally, we have the motivation to optimize the presented code and generalize it to the study of pantographic structures which are more general than those considered here; in particular, we will generalize it to 2D pantographic structures moving in the space, or to more general 3D pantographic materials.

Some future developments concern contact problem between fibers, see, for instance, [START_REF] Andreaus | Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle[END_REF][START_REF] Andreaus | Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation[END_REF][START_REF] Cuomo | Complementary energy approach to contact problems based on consistent augmented Lagrangian formulation[END_REF]; although stability problem is not explicitly considered in this paper, the extension to these cases could be based on the following papers [START_REF] D'annibale | On the failure of the 'similar piezoelectric control' in preventing loss of stability by nonconservative positional forces[END_REF][START_REF] D'annibale | Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces[END_REF][START_REF] Piccardo | Some research perspectives in galloping phenomena: critical conditions and post-critical behavior[END_REF][START_REF] Luongo | Postcritical behavior of cables undergoing two simultaneous galloping modes[END_REF][START_REF] Rizzi | Initial postbuckling behavior of thin-walled frames under mode interaction[END_REF][START_REF] Aminpour | A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis[END_REF] while pantographic structures with initially curved beams can take advantage of the theory of the symmetry group treated in [START_REF] Eremeyev | Material symmetry group and constitutive equations of micropolar anisotropic elastic solids[END_REF][START_REF] Eremeyev | Material symmetry group of the non-linear polar-elastic continuum[END_REF] A final remark is needed: we do not manage to take a clear position in the "atomistic" (or discrete) versus "continuum" (or infinite dimensional) eternal contraposition, also when giving up to discuss about the ultimate nature of matter and willing to limit one-self to the more utilitarian perspective of the choice of the most effective calculation tool. Indeed, while in the presented context discrete models simply seem more suitable, we have presented in the previous point (1) a perspective in which the synthetic approach given by continuum higher gradient models may be more fruitful. Actually, when the set of material particles is very large, also rotational and extensional spring correspondingly increase in number, and the intermediate step represented by the introduction of a continuum model may be a clever tool for capturing in a synthetic way "global" behavior of large subgroups of particles and springs.

Fig. 1 . 4 x 1 , u 1 x 2 , u 2 Fig. 2 .

 1411222 Fig. 1. Pantographic structure

Fig. 4 .

 4 Fig. 4. Kinematics of extensional, bending and shear springs

Fig. 5 .

 5 Fig. 5. Bias test for n f = 10: reference configuration (gray), deformation and color bar of the internal forces on extensional springs (color figure online)

Figure 7 Fig. 6 .Fig. 7 .+ x 1 Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .

 76718910111213 Fig. 6. Bias test for n f = 10: global reaction of constraints R(λ) (a), its density r(x 2 ) when λ = 1 (b) and strain energies (c). a x 1 component in blue and x 2 component in red. b x 1 component in blue and x 2 component in red. c axial, Ea (blue), bending, E b (red), and shear, Es (green), strain energy versus λ (color figure online)

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Bending-extension test for n f = 10: global reaction of constraints R(λ) versus λ = ū/ūmax (a) and its density r(x 2 ) when λ = 1 (b) and strain energies (c). a x 1 component in blue and x 2 component in red. b x 1 component in blue and x 2 component in red. c axial, Ea (blue), bending, E b (red), and shear, Es (green), strain energy versus λ = ū/ūmax (color figure online)

Fig. 16 .Fig. 17 .

 1617 Fig. 16. Bending-extension test by using a second gradient model: global reaction of constraints R(λ) (a), its density r(x 2 ) when λ = 1 (b) and strain energies (c). a x 1 component in blue and x 2 component in red. b x 1 component in blue and x 2 component in red. c axial, Ea (blue), bending, E b (red), and shear, Es (green), strain energy versus λ = ū/ūmax (color figure online)

Fig. 18 .Fig. 19 .

 1819 Fig. 18. Shear-extension test: global reaction of constraints R(λ) versus λ = ū/ūmax (a), its density r(x 2 ) when λ = 1 (b) and strain energies for n f = 10 (c). a x 1 component in blue and x 2 component in red. b x 1 component in blue and x 2 component in red. c axial, Ea (blue), bending, E b (red), and shear, Es (green), strain energy versus λ = ū/ūmax (color figure online)

Fig. 20 .

 20 Fig. 20. Extraction test for n f = 10: global reaction of constraints R(λ) versus λ = ū/ūmax (a), its density r(x 2 ) when λ = 1 (b) and strain energies (c). a x 1 component in blue and x 2 component in red, b x 1 component in blue and x 2 component in red, c axial, Ea (blue), bending, E b (red), and shear, Es (green), strain energy versus λ = ū/ūmax (color figure online)

These systems are defined as quasiperiodical systems having some of the physical properties diverging when the size of the representative elementary volume tends to zero, while simultaneously some others properties are vanishing in this limit. Pantographic structures verify this definition, see[START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF].


O Zakary

M Rachik

The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems

Keywords: discrete-time, linear system, 𝜖-capacity, 𝜖-tolerable, disturbances, perturbed initial state, admissible set. I

The -capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems

O Zakary, M Rachik

Introduction

Generally, in practice of control systems, we have to shed light on the existence of disturbances, due to the principle of action and counter-action which exists between a system and its environment. There has a lot of work on this topic [START_REF] Rakovic | Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances[END_REF][START_REF] Bouyaghroumni | Admissible disturbance sets for discrete perturbed systems[END_REF][START_REF] Cheng | High Gain Disturbance Observer-Based Control for Nonlinear Affine Systems[END_REF], it is the case of systems with perturbed parameters [START_REF] Kunisch | A review of some recent results on the output least squares formulation of parameter estimation problems[END_REF]. For instance, we can cite the works of Dahleh et al., in Reference [START_REF] Dahleh | Rejection of persistent bounded disturbances: Nonlinear controllers[END_REF] and Wang et al., in Reference [START_REF] Wang | Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances[END_REF], where the authors studied respectively the rejection of persistent bounded disturbances, and the exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances. In the mathematical modeling, disturbances lead to the emergence of undesired parameters. Moreover, these parameters could decrease the performance of the system, thus in the worst case destabilize it. In this paper, we are concerned with disturbances that infect the initial state for discrete-time linear system, such disturbances are said to be 𝜖-tolerable if the associated output is insensitive to their effects, i.e. the effects of these disturbances are relatively tolerable. In [START_REF] Namerikawa | Disturbance and Initial State Uncertainty Attenuation Control for Magnetic Bearings[END_REF], an application of 𝐻 ∞ control attenuating initial-state uncertainties to the magnetic bearing where the authors examine the 𝐻 ∞ DIA Disturbance and Initial state uncertainty Attenuation) control problem for a class of linear time-invariant systems. For a given scalar 𝜖 > 0, we will define the 𝜖-capacity of a gain matrix 𝐾 a set of all initial state that satisfies a certain inequalities with respect to 𝜖. We will show that for a given 𝜖 > 0, if the 𝜖-capacity of some gain matrix 𝐾 verifies some conditions then the system will be insensitive to the effects of all disturbances when the control law u i = Kx i is used, i.e. all disturbances will be 𝜖-tolerable. In this paper, the 𝜖-capacity of some gain matrix 𝐾, that makes the system insensitive to the effects of all disturbances, will be called an admissible 𝜖capacity. We are interested with the investigation of the set of all gain matrix those have an admissible 𝜖capacity. More precisely, we consider the following example. A linear, controlled, discrete-time system described by

x i+1 = Ax i + Bu i x 0 = ατ 0 + β ∈ ℝ n (1)
where β = β j e j ∈ ℝ n n j=1

and α ∈ ℝ are unknown and unavoidable disturbances that infect the initial state τ 0 ∈ ℝ n , (𝑒 𝑖 ) i , i = 1, … , n, is the canonical basis of ℝ n . The associated output function is:

y i = Cx i ∈ ℝ p
where A, B and C are real matrices of appropriate dimension. The control law is

u i = Kx i ∈ ℝ m
We will propose a technique to determine among these controls law which makes the system insensitive the effects of these disturbances. A disturbance (α, β) ∈ ℝ n × ℝ is said to be 𝜖-tolerable if:

∂y i ∂α ≤ ϵ and ∂y i ∂β j ≤ ϵ ∀j = 1, … , n; ∀i ≥ 0 (2)
The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems DOI: 10.9790/5728-113XXXXX www.iosrjournals.org Some of the control objectives are to stabilize the system and to maintain its output trajectory within the domain of constraints. The constraints may be summarized by a single set inclusion

y i ∈ Y α, β, ϵ = y i ∈ ℝ p / ∂y i ∂α ≤ ϵ and ∂y i ∂β j ≤ ϵ ∀j = 1, … , n; ∀i ≥ 0 (3)
If these constraints are violated for any 𝑖 ≥ 0, serious damage may happen. We say that a gain matrix K have an admissible 𝜖-capacity Θ 𝜖 (K), if the output of the system never exceed the specified constraints [START_REF] Cheng | High Gain Disturbance Observer-Based Control for Nonlinear Affine Systems[END_REF]. With ( 1) and

(3), it is desired to determine the set Φ of all gain matrices K those have an admissible 𝜖-capacity, to be explicit:

Φ = K ∈ ℝ m×n / 𝜕𝑦 𝑖 𝜕𝛼 ≤ 𝜖 𝑎𝑛𝑑 𝜕𝑦 𝑖 𝜕𝛽 𝑗 ≤ 𝜖 ∀𝑗 = 1, … , 𝑛; ∀𝑖 ≥ 0
The outline of the rest of the manuscript is: Section 2, the goal of this paper is stated and the required assumptions are discussed, and the main theoretical results are developed in Section 3, while the algorithmic determination for the 𝜖-capacity Θ 𝜖 (K) for each gain matrix 𝐾 will presented in Section 4, and the simulation results and the concluding remarks are given in Section 5.

II. Problem Statement

The linear, controlled, discrete-time systems considered in this paper have the following form

𝑥 𝑖+1 = 𝐴𝑥 𝑖 + 𝐵𝑢 𝑖 𝑥 0
where

𝑥 0 = 𝛼𝜏 0 + 𝛽 ∈ ℝ 𝑛 , 𝛽 = 𝛽 𝑗 𝑒 𝑗 ∈ ℝ 𝑛 𝑛 𝑗 =1
and 𝛼 ∈ ℝ are unknown and unavoidable disturbances that infect the initial state 𝜏 0 ∈ ℝ 𝑛 , (𝑒 𝑖 ) i , i = 1, … , n, is the canonical basis of ℝ 𝑛 . The associated output function is:

𝑦 𝑖 = 𝐶𝑥 𝑖 ∈ ℝ 𝑝
where A, B and C are real matrices of appropriate dimension. Which means that the output function is insensitive to the effects of the 𝜖-tolerable disturbances, but in the case of 𝜖-intolerable disturbances, the desired performance can be lost. The control law 𝑢 𝑖 = 𝐾𝑥 𝑖 ∈ ℝ 𝑚 is introduced in order to reduce the effects of these 𝜖-intolerable disturbances and/or makes the system insensitive to the effects of all disturbances that infect the initial state. To characterize the utility of each control law, we have the following definition Definition 2. For a given 𝜖 > 0, and a gain matrix K ∈ ℝ m×n , the set

Θ 𝜖 K = x ∈ ℝ n / C(A + BK) i x ≤ 𝜖 ∀𝑖 ≥ 0
will be called the 𝜖-capacity of the gain matrix 𝐾.

One can note that Θ 𝜖 K , if we consider 𝛼, 𝛽 = (1,0) ∈ ℝ × ℝ 𝑛 and the set of the constraints 𝑌 𝜖 = 𝑦 𝑖 ∈ ℝ 𝑝 / 𝑦 𝑖 ≤ 𝜖 ∀𝑖 ≥ 0 , is the set of all initial state such that the corresponding output function satisfies the constraints 𝑦 𝑖 ∈ 𝑌 𝜖 , in the literature, Θ 𝜖 K is called the maximal output admissible set [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF]. The concept of admissible set is very important in the field of analysis and control of linear and non linear systems.

Many papers worked on the construction of admissible sets [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF][START_REF] Rachik | Discrete systems with delays in state, control and observation: The maximal output sets with state and control constraints[END_REF][START_REF] Hirata | Exact determinations of the maximal output admissible set for a class of nonlinear systems[END_REF][START_REF] Dórea | Computation of maximal admissible sets of constrained linear systems[END_REF]. We can cite the work of Kenji Hirata and Yoshito Ohta in [START_REF] Hirata | Exact determinations of the maximal output admissible set for a class of nonlinear systems[END_REF], where the authors present necessary and sufficient conditions for fulfilling the specified state and control constraints for a class of nonlinear systems, and they propose an explicit procedure to determine the maximal output admissible set. In [START_REF] Dórea | Computation of maximal admissible sets of constrained linear systems[END_REF], an efficient algorithm is provided to compute the maximal set of admissible initial states in the case of autonomous linear discrete-time systems subject to linear constraints. In this paper, we are interested in studying some applications of this concept to linear discrete-time systems with perturbed initial state. And we will show that if this set have some properties, then the system will be insensitive to the effects of all disturbances under the corresponding control law. One can note that the 𝜖-capacity of some gain matrix is described by an infinite number of inequalities, as in [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF], the maximal output admissible set is finitely determined, i.e. is described by a finite number of inequalities under some assumptions, in this paper, we show that the controllability of (A,B) and the observability of (A+BK,C) are sufficient for reducing this writing to a finite number of inequalities, which will allow us to use an algorithmic procedure for computing Θ 𝜖 K . Definition 3. If a control law 𝑢 𝑖 = 𝐾𝑥 𝑖 makes the system insensitive to the effects of all disturbances, we say that K have an admissible 𝜖-capacity Θ 𝜖 (K).

Our goal is to characterize the set Φ of all gain matrices 𝐾 those have an admissible 𝜖-capacity Θ 𝜖 (K), to be explicit:

Φ = K ∈ ℝ m×n / y i ∈ Y α, β, ϵ , y i is the corresponding output Where Y α, β, ϵ = y i ∈ ℝ p / ∂y i ∂α ≤ ϵ and ∂y i ∂β j ≤ ϵ ∀j = 1, … , n; ∀i ≥ 0
is the set of all output functions which are relatively insensitive to the effects of all disturbances.

In the next section, we will develop the main results that will allow us to assimilate the connection between the set Θ 𝜖 (K) and the set Φ, and some necessary mathematical preliminaries will be given.

III. Preliminary Results

First, we will describe the system considered, a linear, controlled, discrete-time system

𝑥 𝑖+1 = 𝐴𝑥 𝑖 + 𝐵𝑢 𝑖 x 0 = ατ 0 + β ∈ ℝ n (4)
and the associated output function is :

𝑦 𝑖 = 𝐶𝑥 𝑖 ∈ ℝ 𝑝 (5) the control law is : 𝑢 𝑖 = 𝐾𝑥 𝑖 ∈ ℝ 𝑚
where the state variable 𝑥 i ∈ ℝ 𝑛 and A, B, C and K are respectively n, n , n, m , p, n and (m, n) matrices, and

𝜏 0 ∈ ℝ 𝑛 is given, , 𝛽 = 𝛽 𝑗 𝑒 𝑗 ∈ ℝ 𝑛 𝑛 𝑗 =1
and 𝛼 ∈ ℝ are unknown disturbances that infect the initial state, knowing that they are supposed inevitable, (𝑒 𝑖 ) i , i = 1, … , n, is the canonical basis of ℝ 𝑛 . We will say that a disturbance (𝛼, 𝛽) is tolerable if, for every i ≥ 0 the corresponding output function y i is not influenced by the disturbance's effects. That could be mathematically translated by the variation of y i is independent of (𝛼, 𝛽) i.e. the partial derivate of y i with respect to 𝛼 and with respect to 𝛽 almost vanishes. Given a positive parameter ϵ, a disturbance (𝛼, 𝛽) is said to be ϵ-tolerable if:

∂y i ∂α ≤ ϵ and ∂y i ∂β j ≤ ϵ ∀j = 1, … , n; ∀i ≥ 0 (6)
In the case of the ϵ-intolerable disturbances, it is desired to find a control law such that the corresponding output function satisfies [START_REF] Wang | Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances[END_REF]. Then the principal goal is to determine the set Φ of all gain matrices that makes the system insensitive to the effects of all disturbances (𝛼, 𝛽), to be explicit:

The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems However, the set Φ is defined by an infinite number of inequalities, which may be impossible to obtain. We proof that under hypothesis, this difficulty can be overcome and the set Φ will be described by a finite number of inequalities which allows us to use an algorithmic procedure to characterize it. Equation ( 5) can be written as follows:

𝑦 𝑖 = 𝐶 𝐴 + 𝐵𝐾 𝑖 (ατ 0 + β)
In fact

𝑥 𝑖+1 = 𝐴𝑥 𝑖 + 𝐵𝑢 𝑖 𝑥 𝑖+1 = (𝐴 + 𝐵𝐾)𝑥 𝑖 x 𝑖 = 𝐶 𝐴 + 𝐵𝐾 𝑖 x 0 𝑥 𝑖 = 𝐴 + 𝐵𝐾 𝑖 (ατ 0 + β) then 𝑦 𝑖 = 𝐶 𝐴 + 𝐵𝐾 𝑖 (ατ 0 + β)
And

∂y i ∂α = 𝐶 𝐴 + 𝐵𝐾 𝑖 τ 0 ∂y i ∂β j = 𝐶 𝐴 + 𝐵𝐾 𝑖 𝑒 𝑗 (8)
Using ( 7) and ( 8), we can write Φ as follows:

Φ = K ∈ ℝ m ×n / 𝐶 𝐴 + 𝐵𝐾 𝑖 τ 0 ≤ ϵ and 𝐶 𝐴 + 𝐵𝐾 𝑖 𝑒 𝑗 ≤ ϵ ∀j = 1, … , n; ∀i ≥ 0 (9)
In the following remark, we will show the interest of introducing the set Θ 𝜖 (K) in the characterization of Φ. For that, consider S = 1, … , n a set of indices.

Remark 4. It is clear that

Φ = K ∈ ℝ m×n / τ 0 ∈ Θ 𝜖 (K) and 𝑒 𝑗 ∈ Θ 𝜖 (K) ∀j ∈ S (10)
In fact, let 𝑦 𝑖 be the output function corresponding to an initial state x 0 = ατ 0 + β, the gain matrix K have an admissible ϵ-capacity that makes the disturbance (𝛼, 𝛽) ϵ-tolerable if:

y i ∈ Y α, β, ϵ = y i ∈ ℝ p / ∂y i ∂α ≤ ϵ and ∂y i ∂β j ≤ ϵ ∀j = 1, … , n; ∀i ≥ 0
then it is clear from ( 9) and ( 10) that:

∂y i ∂α ≤ ϵ ; ∀i ≥ 0 ⟺ 𝐶 𝐴 + 𝐵𝐾 𝑖 τ 0 ≤ ϵ; ∀i ≥ 0 ⟺ τ 0 ∈ Θ 𝜖 (K)
and

∂y i ∂β j ≤ ϵ ∀j = 1, … , n; ∀i ≥ 0 ⟺ 𝐶 𝐴 + 𝐵𝐾 𝑖 𝑒 𝑗 ≤ ϵ, ∀j ∈ S; ∀i ≥ 0 ⟺ 𝑒 𝑗 ∈ Θ 𝜖 K , ∀j ∈ S
Therefore the system (4) is insensitive to the disturbances α (respectively β) if and only if τ 0 ∈ Θ 𝜖 (K) (respectively 𝑒 𝑗 ∈ Θ 𝜖 (K) ∀j ∈ S).

We note that the ϵ-capacity of K is also defined by an infinite number of inequalities. We will establish sufficient conditions which allow us to describe it by a finite number of inequalities, so Φ. In order to characterize the set Θ 𝜖 (K) , we introduce, without loss of generality, the following notations:

Θ = Θ 𝜖 K = x ∈ ℝ n / CA i x ≤ 𝜖 ∀𝑖 ≥ 0
The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems The stop condition of the proposed algorithm in the next section is based on the following proposition.

Proposition 8. The set 𝛩 is finitely determined if and only if 𝛩 𝑘 +1 = 𝛩 𝑘 for some 𝑘 ≥ 0.

Proof: Θ is finitely determined, so there exist an integer k ≥ 0 such that Θ = Θ k the remark 5 above implies that Θ = Θ k+1 = Θ k Conversely, Θ k+1 = Θ k for some k ≥ 0, let x ∈ Θ k then CA k+1 x ≤ 𝜖 we deduce that A x ∈ Θ k , ∀x ∈ Θ k and by iteration we have

A i x ∈ Θ k , ∀i ≥ 0 then x ∈ Θ that implies
Θ k ⊂ Θ and we know that Θ ⊂ Θ k for every k ≥ 0, which complete the proof. And by remark 5 and proposition 8 we have that Θ = Θ k 0 +1 = Θ k 0 and Θ is finitely determined. Remark 10. One can note that the asymptotic stability of A is a special case of the Proposition 9.

Theorem 11. If 𝐴 is asymptotically stable ( 𝜆 < 1 for every 𝜆 eigenvalue of 𝐴 ), then there exists an integer 𝑘 0 such that the output function 𝑦 𝑖 is not sensitive to disturbance 𝛽 for every 𝑖 > 𝑘 0 .

Proof: The asymptotic stability of A implies that there exists an integer k 0 such that CA k 0 ≤ 𝜖 then 𝐶A 𝑖 𝑒 𝑗 ≤ ϵ, ∀j ∈ S; ∀i ≥ k 0 Using (9), we have ∂y i ∂β j ≤ ϵ, ∀j ∈ S; ∀i ≥ k 0 Remark 12. If the pair (A, B) is controllable, then we can choose K such that the matrix A is asymptotically stable. So the controllability is a sufficient condition for the theorem 11.

Theorem 13. Suppose the following assumptions hold: i) the pair (𝐴 , 𝐶) is observable ii) the pair (𝐴, 𝐵) is controllable Then 𝛩 is finitely determined.

Proof: By the observability of (A , C), the rank of the matrix Λ is n, where

Λ = C CA CA 2 ⋮ CA n-1
which implies that Λ T Λ is invertible, so there exists a ρ ≥ 0 such that ρ x 2 ≤ Λ T Λx, x , ∀x ∈ ℝ n (11) and we have

Λx ∈ B 0, ϵ × … × B(0, ϵ) n times , ∀x ∈ Θ n-1 where B 0, ϵ = x ∈ ℝ n / x ≤ 𝜖 . Since B 0, ϵ × … × B(0, ϵ) n times is bounded, then there exists a 𝜆 ≥ 0 such that Λ T Λx, x ≤ 𝜆 x , ∀x ∈ Θ n-1 (12)
from ( 11) and ( 12), there exists 𝛾 ≥ 0 such that x ≤ 𝛾, ∀x ∈ Θ n-1 (13) the pair (A, B) is controllable, so there exists K such that the matrix A = A + BK is asymptotically stable, then there exists an integer k 0 ≥ n -1 such that

CA k 0 ≤ 𝜖 𝛾 from (13) we have CA k 0 x ≤ 𝜖, ∀x ∈ Θ n-1 which implies that Θ n-1 ⊂ Θ k 0 and we know that Θ k 0 ⊂ Θ n-1 because k 0 ≥ n -1 then Θ k 0 = Θ n-1
and Θ is finitely determined.

The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems DOI: 10.9790/5728-113XXXXX www.iosrjournals.org Remark 14. If the conditions of theorem 13 are verified, Θ is finitely determined, so Θ is described by a finite number of inequalities, and consequently Φ is also described by a finite number of inequalities.

In the next section, we will propose an algorithmic procedure to determine such integer k 0 described above.

IV. Algorithmic Determination

Using the previous results, we can deduce an algorithm for computing the ϵ-capacity for a given gain matrix. This algorithm has almost the same theoretical convergence properties as the one proposed by Gilbert and Tan [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF].

Definition 15. Let k 0 the smallest integer which satisfies Θ k 0 = Θ, k 0 is called the index of determination.

The last theorem and proposition 8 in the previous section, suggests the following algorithm for determining the index of determination k 0 .

Algorithm 1: ____________________

Step 1: Set 𝑘 = 0

Step 2: if 𝛩 𝑘 = 𝛩 𝑘 +1 𝑘 0 = 𝑘 and stop else continue

Step 3: 𝑘 = 𝑘 + 1 and return to Step2 ____________________ It is clear that the algorithm 1 will produce k 0 if and only if the 𝜖-capacity Θ, of the chosen gain matrix K, is finitely determined. But, in the case where Θ is not finitely determined, there is no finite algorithmic procedure. Fortunately, as subsequent developments show, it is often possible to resolve the issue of finite determination by other means.

This algorithm can be considered as an initial version, because it does not describe how the test Θ k = Θ k+1 is implemented in Step 2. The main problem here is to achieve Θ k = Θ k+1 . The difficulty can be overcome if we consider the following approach: Let ℝ n be normed by

x = max x i 1 ≤ i ≤ n , ∀x = (x 1 , … , x n ) ∈ ℝ n
Then, for every k ≥ 0, the set Θ k can be written down as follows:

Θ k = x ∈ ℝ n /h s CA i x ≤ 𝜖 , ∀𝑖 = 0, … , k, ∀𝑠 = 0, … ,2n
where h s : ℝ n → ℝ are such that for every x ∈ ℝ n and for every 𝑗 = 1, … , n we have

h 2j-1 x = x j -𝜖 h 2j x = -x j -𝜖
Remark 16. We see that if Θ is finitely determined, then there exists the index of determination k 0 and Θ can be written as follows:

Θ = x ∈ ℝ n /h s CA i x ≤ 𝜖 , ∀𝑖 = 0, … , k 0 , ∀𝑠 = 0, … ,2n
From the remark 5, the test

Θ k = Θ k+1 is equivalent to Θ k ⊂ Θ k+1 It follows that Θ k = Θ k+1 ⟺ ∀x ∈ Θ k , h s CA k+1 x ≤ 𝜖, ∀𝑠 = 0, … ,2n or equivalently, sup h s CA k+1 x ≤ 𝜖, x ∈ Θ k for 𝑠 = 1, … ,2n
Consequently the test Θ k = Θ k+1 leads to a set of mathematical programming problems. We will propose below another version of the algorithm 1, the modified algorithm is given as follows.

The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems DOI: 10.9790/5728-113XXXXX www.iosrjournals.org

Algorithm 2: ________________________

Step 1: Set 𝑘 = 0

Step 2: Solve the following optimization problems for 𝑠 = 1, … ,2𝑛 Maximize 𝐽 𝑠 𝑥 = ℎ 𝑠 𝐶𝐴 𝑘+1 𝑥 Subject to the constraints ℎ 𝑗 𝐶𝐴 𝑙 𝑥 ≤ 𝜖, ∀𝑗 = 1, … ,2𝑛, ∀𝑙 = 0, … , 𝑘 Let 𝐽 * 𝑠 be the maximum value of 𝐽 𝑠 𝑥 If 𝐽 * 𝑠 ≤ 0 for 𝑠 = 1, … ,2𝑛 Set 𝑘 0 = 𝑘 and define 𝛩 by [START_REF] Dórea | Computation of maximal admissible sets of constrained linear systems[END_REF], and Stop else continue

Step 3: Set 𝑘 = 𝑘 + 1 and return to Step2 ________________________ Remark 17. i) This algorithm can never be useful, if there are no methods to solve rather large mathematical programming problems which arise in Step 2. This presents some difficulty because global optima are needed. Even when the functions h j , ∀𝑗 = 1, … ,2n are convex, the difficulty remains because the programming problems require the maximum of a convex function subject to convex constraints. ii) Assumptions of theorems 13 are sufficient but not necessary. If these conditions are not verified, there is no guarantee that Algorithm 2 will stop. If the Algorithm 2 converge then the 𝜖-capacity Θ is finitely determined, else it is not. In the case where Θ is not finitely determined, we can choose a different gain matrix K which ensures convergence of the algorithm and the assumptions of theorem 13. To illustrate this work we will give in the next section some numerical examples.

V. Examples

Using the above ideas it is possible to characterize the 𝜖-capacity Θ of every given gain matrix K that verifies the necessary conditions for the convergence of the algorithm 2. In this section, we are content to apply them to some simple examples.

Example1 : Consider the following system 𝑥 𝑖+1 = 𝐴𝑥 𝑖 + 𝐵𝑢 𝑖 x 0 = α 0,3 0,5

+ β ∈ ℝ 2
where 𝛼, 𝛽 ∈ ℝ × ℝ 2 is unknown and unavoidable disturbance which infect the initial state, τ 0 = 0,3 0,5 ∈ ℝ 2 and 𝐴, 𝐵 are described as follows : A = 0,9 0 0,6 0,3 and B = -1,5 2 1 -3

The associated output function is 𝑦 𝑖 = 𝐶𝑥 𝑖 where C = (1 1) and 𝜖 = 2. We can easily verify that the pair (𝐴, 𝐵) is controllable, so we can place the eigenvalues of A = A + BK arbitrarily such that A is asymptotically stable. For that, we choose K as follows K = 0,32 0,16 0,24 0,12 which implies that A = 0,9 0 0,2 0,1

Now, we are sure after a simple verification that the pair (A , C) is observable. The theorem 13 ensure the convergence of the algorithm 2, which gives the index of determination k 0 = 0, then the 𝜖-capacity of the chosen matrix 𝐾 can given by Θ = x y ∈ ℝ 2 / x + y ≤ 2 and 1,1x + 0,1y ≤ 2 and 1,01x + 0,01y ≤ 2 We can by a simple hand calculation, see that e 1 ∈ Θ and e 2 ∈ Θ where (𝑒 𝑖 ) i , i = 1,2 is the canonical basis of ℝ 2 , which means that the disturbance β does not influence the associated output function. We can see also that 0,3 0,5 ∈ Θ which means that the corresponding output function is not sensitive to the disturbance α. We can conclude that the selected gain matrix K have an admissible 𝜖-capacity, which means that 𝐾 belongs to Φ, which implies that the system is insensitive to all disturbances 𝛼, 𝛽 under the associated control law 𝑢 𝑖 = 𝐾𝑥 𝑖 . Using the algorithm developed in the previous section, we give in Table 1 and in Table 2 the values of k 0 corresponding to different choices of the matrix defining the system. The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems Comment 1: In example 1, the selected gain matrix K have an admissible 𝜖-capacity, the corresponding control law was able to reduce the effects of all disturbances 𝛼, 𝛽 ∈ ℝ × ℝ 2 , but for the same system, with another choice of K, example 6 shows that e 2 ∉ Θ, which means that the system is influenced by disturbances 𝛽, but not by 𝛼 because 0,3 0,5 ∈ Θ (see Figure 2). In this case, the chosen K does not belong to Φ, so it is not useful. In example 7, we can see also that the chosen control is not useful, because neither 0,3 0,5 ∈ Θ nor e 2 ∉ Θ (see Figure 3). The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems DOI: 10.9790/5728-113XXXXX www.iosrjournals.org

VI. Conclusion

In this paper, we have studied the problem of discrete-time linear systems with perturbed initial state. The 𝜖-capacity of a gain matrix 𝐾 is introduced as a new approach to attenuate the effects of disturbances that infect the initial state, by the corresponding feedback control 𝑢 𝑖 = 𝐾𝑥 𝑖 . Necessary conditions have been obtained. Numerical examples were provided to show the effectiveness.

Proposition 9 .

 9 If 𝐴 ≤ 𝛼 for some 𝛼 ∈]0,1[ then 𝛩 is finitely determined.Proof: Let 𝜖 > 0 A ≤ 𝛼 ⇒ CA i x ≤ γ i x , ∀i ≥ 0 where γ i = 𝛼 i C . Since γ i i→∞ 0then there exists an integer k 0 such thatCA i x ≤ 𝜖, ∀i ≥ k 0 ; ∀x ∈ ℝ n thenThe ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems DOI: 10.9790/5728-113XXXXX www.iosrjournals.org CA k 0 +1 x ≤ 𝜖; ∀x ∈ Θ k 0

  The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems DOI: 10.9790/5728-113XXXXX www.a representation of the set Θ corresponding to this example.

Figure 1 :

 1 Figure 1: The colored region represents the 𝝐-capacity 𝚯 of the matrix 𝐊 = 𝟎, 𝟑𝟐 𝟎, 𝟏𝟔 𝟎, 𝟐𝟒 𝟎, 𝟏𝟐 corresponding to example 1.

Table 1 : 5 Figure 2 :

 152 Figure 2: The colored region represents the 𝝐-capacity 𝚯 of the matrix 𝐊 chosen in example 6.

Figure 3 :

 3 Figure 3: The colored region represents the 𝝐-capacity 𝚯 of the matrix 𝐊 chosen in example 7.

Table 2 : 10 Comment 2 :

 2102 Data Of Examples 6-While the conditions in the theorem 13 are sufficient for the convergence of the algorithm 2, examples 3-5-7 show that they are not necessary. We can see that the matrix A is not asymptotically stable but only Lyapunov stable. Moreover, A is not stable in examples 4-8-9-10.

Figure 4 :

 4 Figure 4: The colored region represents the 𝝐-capacity 𝚯 of the matrix 𝐊 chosen in example 3.


Gianluca Perniciano

Modified Alcubierre Warp Drive II

.Here we investigate the negative energy in the Pfenning zone or warped region (appendix 3).

Introduction:

Alcubierre [1] in 1994 proposed a solution of the equations of general relativity which provides the only viable means to accelerate a spaceship up to superluminal velocities without using wormholes.

A problem was soon identified: Pfenning [4] showed that the required energy is comparable to the total energy of the universe and that it is negative.We presented a way to reduce the amount of energy involved and its spacial distribution within the warp bubble.

Note: In the following we adopt the notation used by Landau and Lifshitz in the second volume ("The Classical Theory of Fields") of their well known Course of Theoretical Physics [START_REF] Landau | Theory of Fields[END_REF]. *email:g.perniciano@gmail.com

Summary:

We start with the metric

ds 2 = ( 1-v 2 f ( x , y , z-k (t)) 2 a (x , y , z -k (t )) 2 ) dt 2 +2 v f (x , y , z-k ( t)) a( x , y , z -k (t)) dt dz-dx 2 -dy 2 -dz 2 (1)
From the components of the Einstein tensor in contravariant form for

• 1)-The Pfenning zone is the zone within the interval: R-Δ 2 <r <R+ Δ 2 where Δ≪1 R is the radius of the Warp bubble and Δ is the wall thickness of the Warp bubble R≫Δ .

• 2)-r =( x 2 + y 2 +( z -k (t)) 2 )
1 2 and dk (t) dt =v=const

• 3)-In the Pfenning zone we let a (r )=a (x , y , z -k (t ))≫1 and da (r )/ dr≤a(r

)
(there is the source of esotic matter) can be reduced by an arbitrary value [11].

Einstein Equations:

G ik = 8 π G c 4 T ik [15] T ik (energy-impulse tensor)
The functions f (r )= f ( x , y , z -k (t)) and a (r )=a (x , y , z -k (t )) can assume the following values:

• 1)-inside the warp bubble (0<r <R-Δ 2 ) f (r )=1 and a (r )=1

• 2)-outside the warp bubble (r >R+ Δ 2 ) f (r )=0 and a (r )=1

• 3)-in the Alcubierre warped region (R-Δ

2 <r <R+ Δ 2 ) 0< f (r )<1 , f (r ) is f (r )=- ( r-R-Δ 2 )
Δ (Pfenning zone [4]) and a (r )=a ( x , y , z -k (t ))≫1 and da (r )/ dr≤a( r )

• 1)-Internal metric of the Warp bubble (0<r <R-Δ 2 ) is:

ds 2 =dt 2 -( dz-v dt) 2 -dx 2 -dy 2 (2)
moving with velocity v (multiple of the speed of light c) along the z-axis.

• 2)-Metric outside of the bubble beyond the Pfenning zone (r >R+ Δ 2

) is:

ds 2 =dt 2 -dx 2 -dy 2 -dz 2 (3)
2 Computation of the negative energy in the Pfenning zone, its comparison with the Casimir effect (plane-parallel condenser) and quantum inequalities.

The energy is:

E= ∫∫∫ (-g ) 1/ 2 T tt dx 3 (4)
where the triple integral extends over all the volume and the energy density ρ is:

ρ=T tt =k G tt ( 5
)
(g determinant of the spacial metric) and k =c 4 /8π G .

In our case we get [11],in Alcubierre form:

G tt =- 1 4 v 2 x 2 + y 2 r 2 g (r ) (6)
where g ( r ) is given by:

g (r )= [ 1 a (r ) 2 ( df (r ) dr ) 2 + ( f (r ) 2 a (r ) 4 ) ( da (r ) dr ) 2 -2 df (r ) dr f (r ) a (r ) 3 da ( r) dr ] (7)
• 1)-inside the warp bubble (0<r <R-Δ 2 ) f (r )=1 and a (r )=1

• 2)-outside the warp bubble (r >R+ Δ 2 ) f (r )=0 and a (r )=1

• 3)-in the Alcubierre warped region (R-Δ

2 <r <R+ Δ 2 ) 0< f (r )<1 , f (r ) is f (r )=- ( r-R-Δ 2 )
Δ and a (r )=a ( x , y , z -k (t ))= A=constant ≫1 possessing extremely large values

2.1

The energy E in the Pfenning zone in our case (see [4] for an example):

The energy given by ( 4), taking into account the conditions set therein, becomes:

E=- 8 π k v 2 32 π ∫ x 2 + y 2 r 2 g (r) dx 3 (8) E=- 8 π k v 2 12 ∫ r 2 g ( r) dr (9) g ( r )( -1 a ( r) Δ ) 2 (10)
(( 9) is dominant term of (7) for Δ ≪1 , a (r )≫1 and da (r) /dr≤a(r )

, a (r )> 1 Δ E=- 8 π k v 2 12 ∫ R-Δ 2 R+ Δ 2 r 2 g (r ) dr=- 8 π k v 2 12 ∫ R -Δ 2 R + Δ 2 r 2 ( -1 a( r ) Δ ) 2 dr (10) E=- 8 π k v 2 12 ∫ R-Δ 2 R+ Δ 2 r 2 ( -1 A Δ ) 2 dr (11) E=- 8 π k 12 v 2 ( R 2 Δ + Δ 12 ) A 2 (12)
where R≫Δ (very likely), R being the radius of the warp bubble. If Δ is about 1.6 •10 -35 m (Planck lenght) as an absurd, setting A=10 50 and R = 100 m, the energy E gets reduced significantly. Using the chosen data the energy is E=-3 •10 -17 v 2 joule quite smaller than the U(d) due the the Casimir effect.

The energy U(d) between the two plates in a plane-parallel condenser in empty space, due

to the Casimir effect, is:

U (d )=-π 2 [ ( h 2 π ) c 720d 3 ] L 2 (13)
where d is the distance between the plates and L is the side of the square conducting plate. As can be seen, the energy is negative and this implies that the force (equal to the opposite of the derivative with respect to d) is attractive, as has been experimentally found ("Lamoreaux" [START_REF] Lamoreaux | Demonstration of the Casimir Force in the 0.6 to 6 µm Range[END_REF]). In the case d =1μ m and L = 1 m (L is chosen to be quite large, but not as large as the Pfenning zone) it is found that U (d )=-4 10 • 1)-inside the warp bubble f (r )=1 and a (r )=1

• 2)-outside the warp bubble f (r )=0 and a (r )=1

• 3)-within the Alcubierre warped region 0< f (r )<1 and a (r )≫1 , da (r ) dr ≤a (r ) f (r ) is for all r (example):

f (r )=(1/2)(1-tanh @( r-R)) , (df /dr ) 2 =(1/4)(@ 2 /[(cosh @( r-R)) 4 ]
) [17] .( 14) a (r ) is for all r (example):

a (r )=(2 P )/[1+(tanh[@(r-R)]
) 2 ] P P ≫1 [17] (15)

da (r)/dr=(-P)/((1/2)[1+(tanh [@(r -R)]) 2 ]) -( P+1) (@ tanh[@(r -R)]/(cosh [@(r -R)]) 2 ) (da (r )/ dr) 2 =(-P ) 2 /((1/ 2)[1+(tanh [ @(r-R)]) 2 ]) (-2(P +1)) (@ tanh [@ (r-R)]/(cosh [@ (r-R )]) 2 )
2 then da ( r)/dr≤a(r ) , (da ( r )/dr ) 2 ≤a (r ) 2 , R radius of the warp bubble the dominant term of ( 7) is (new case):

g ( r )( ( df ( r )/dr ) 2 a (r ) 2 )=( 1 4 ) @ 2 a (r ) 2 (cosh [@ (r -R)]) 4 ( 16
)
a (r )≫1 and da (r) /dr≤a(r ) , (da (r )/dr ) 2 ≤a (r ) 2 in Alcubierre warped region then |energy density|≤|-

1 16 k v 2 @ 2 a (r ) 2 ( cosh [@(r -R)]) 4 | . (17)
And if R=100 m P=140 , @=5000 [17],in Alcubierre warped region (there is the source of esotic matter),the energy is:

E-8.7 v 2 10 -35 joule (18)
use equations ( 9), [START_REF] Landau | Theory of Fields[END_REF],for the same values of the parameters shown above (energy is very small compared with Casimir effect).A modification of parameters P, @, in a(r) and for f (r), between those chosen, determines a greater or lesser reduction of the energy density and the energy,with the same radius value R of the warp bubble that contains the spaceship,and also changes the thickness of the warped region.

Conclusion:

The calculations seem to suggest that the modified Alcubierre propulsion system allows to reach superluminal speeds without problems in the energy density and components of the energy-impulse tensor, and that the negative energy can be arbitrarily reduced. This energy is compared with that of the Casimir effect in a parallel plane condenser, to investigate the experimental feasibility.In the Alcubierre solution a event horizon is present at v>1 while in our solution it is present at v≥1 .

-10 joule 3

 3 Appendix:The energy density,and energy E in the Alcubierre warped region in new case (no trivial):