stabilization by means of state space depending switching rules abstract in this paper we consider switched systems defined by a pair of linear systems x a 1 x x a 2 x such that there exists a neutrally stable linear combination of the matrices a 1 a 2 under an additional assumption we prove that there exists a state space depending switching rule which stabilizes the system in a very general sense the result is illustrated by some simulations and examples

closed loop stabilization of planar bilinear switched systems in this paper we address the closed loop switched stabilization problem for planar bilinear systems under the assumption that the control is one dimensional and takes only the values 0 and 1 we construct a class of state static memoryless stabilizing feedback laws which preserve the properties of open loop switching signals in order to prove the stability of the implemented system we use lyapunov techniques for differential equations with discontinuous righthand side finally we point out some possible extensions of our result and compare it with related results previously proven by other authors

entropy formulation for parabolic degenerate equations with general dirichlet boundary conditions and application to the convergence of fv methods this paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions it consists of two parts the first part is devoted to the definition of an original notion of entropy solutions to the continuous problem which can be adapted to define a notion of measure valued solutions or entropy process solutions the uniqueness of such solutions is established in the second part the convergence of the finite volume method is proved this result relies on weak estimates and on the theorem of uniqueness of the first part it also entails the existence of a solution to the continuous problem

uniform stability of switched linear systems extensions of lasalle s invariance principle this paper addresses the uniform stability of switched linear systems where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set we provide a collection of results that can be viewed as extensions of lasalle s invariance principle to certain classes of switched linear systems using these results one can deduce asymptotic stability using multiple lyapunov functions whose lie derivatives are only negative semidefinite depending on the regularity assumptions placed on the switching signals one may be able to conclude just asymptotic stability or uniform exponential stability we show by counter example that the results obtained are tight

stability results for switched controller systems there are many practical control problems where the control action is determined by switching among a given set of control laws this paper presents necessary and sufficient conditions to test for quadratic stabilizability and for robust stabilizability with a quadratic storage function for switched controller systems algorithms which can be used to construct appropriately stabilizing control laws are also presented

on the regularization of zeno hybrid automata fundamental properties of hybrid automata such as existence and uniqueness of executions are studied particular attention is devoted to zeno hybrid automata which are hybrid automata that take infinitely many discrete transitions in finite time it is shown that regularization techniques can be used to extend the zeno executions of these automata to times beyond the zeno time different types of regularization may however lead to different extensions a water tank control problem and a bouncing ball system are used to illustrate the results

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

renormalized solutions to the vlasov equation with coefficients of bounded variation we prove that weak bounded solutions to the vlasov equation with bv coefficients have the renormalization property and we show that when the renormalization property holds for a general transport equation it also holds for only lipschitz nonlinearities

brief on solution concepts and well posedness of linear relay systems in this paper we study the well posedness existence and uniqueness of solutions of linear relay systems with respect to two different solution concepts filippov solutions and forward solutions we derive necessary and sufficient conditions for well posedness in the sense of filippov of linear systems of relative degree one and two in closed loop with relay feedback to be precise uniqueness of filippov and also forward solutions follows in this case if the first non zero markov parameter is positive by means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two the influence of the zeno phenomenon an infinite number of relay switching times in a finite length time interval on well posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems the consequences of the presence of the zeno behaviour is typical for many other classes of non smooth and hybrid systems

common quadratic lyapunov like functions with associated switching regions for two unstable second order lti systems in this paper we establish quadratic lyapunov like functions for qualitative analysis of switched systems specifically for a class of switched systems consisting of two unstable second order lti linear time invariant subsystems we explore in detail some necessary and sufficient conditions for the existence of common weakly quadratic lyapunov like functions with associated switching regions in r 2 plane the existence conditions and the construction of such quadratic lyapunov like functions are established by using the conic switching laws

differential automata and their discrete simulators on presente la notion d automate differentiel pour la formulation d une classe de problemes aux valeurs initiales sur des champs vectoriels hysteretiques

existence of fixed points for some convex operators and applications to multi point boundary value problems existence results of fixed points for some convex operators are given by means of fixed point theorem of cone expansion and compression then they are applied to nonlinear multi point boundary value problems

existence of carathe spl acute odory solutions in nonlinear systems with discontinuous switching feedback controllers in this note we consider the existence of a carathe spl acute odory solution in a single input single output nonlinear system with a discontinuous switching feedback controller the main contribution is to show that if the nonlinear system can be transformed into a global normal form then we can specify the value of the discontinuous switching feedback controller on the switching hypersurface so that the closed loop system has a carathe spl acute odory solution

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

nonhomogeneous dirichlet problems for degenerate parabolic hyperbolic equations the aim of the paper is to give a formulation for the initial boundary value problem of parabolic hyperbolic type in the case of nonhomogeneous boundary data a 0 here u u x t with x t q 0 t where is a bounded domain in n with smooth boundary and t 0 the function b is assumed to be nondecreasing allowing degeneration zones where b is constant is locally lipschitz continuous and g l 0 b after introducing the definition of an entropy solution to the above problem in the spirit of otto 14 we prove uniqueness of the solution in the proposed setting moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type specifically the diffusion function b is approximated by functions b that are strictly increasing

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

stability of planar switched systems the linear single input case we study the stability of the origin for the dynamical system dot x t u t ax t 1 u t bx t where a and b are two 2 2 real matrices with eigenvalues having strictly negative real part x in mbox bf r 2 and u 0 infty to 0 1 is a completely random measurable function more precisely we find a coordinates invariant necessary and sufficient condition on a and b for the origin to be asymptotically stable for each function u the result is obtained without looking for a common lyapunov function but studying the locus in which the two vector fields ax and bx are collinear there are only three relevant parameters the first depends only on the eigenvalues of a the second depends only on the eigenvalues of b and the third contains the interrelation among the two systems and it is the cross ratio of the four eigenvectors of a and b in the projective line cp1 in the space of these parameters the shape and the convexity of the region in which there is stability are studied this bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation

the initial boundary value problems for hyperbolic conservation laws with relaxation abstract the hyperbolic conservation laws with relaxation appear in many physical models such as those for gas dynamics with thermo non equilibrium elasticity with memory flood flow with friction traffic flow etc the main concern of this article is the long time effect of the relaxations on the boundary layer behaviors in this article we investigate this problem for a simple model of 2 2 systems conditions relating the boundary data and far field states are found for the existence of the boundary layers also it is proven that the boundary layers thus obtained are nonlinearly stable

on strongly degenerate convection diffusion problems modeling sedimentation consolidation processes abstract we investigate initial boundary value problems for a quasilinear strongly degenerate convection diffusion equation with a discontinuous diffusion coefficient these problems come from the mathematical modeling of certain sedimentation consolidation processes the existence of entropy solutions belonging to bv is shown by the vanishing viscosity method the existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient new uniqueness proofs for entropy solutions are also presented these proofs rely on a recent extension to second order equations of kru kov s method of doubling the variables the application to a sedimentation consolidation model is illustrated by two numerical examples

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

stabilization of second order lti switched systems this paper studies and solves the problem of asymptotic stabilization of switched systems consisting of unstable secondorder linear time invariant lti subsystems necessary and sufficient conditi

a discrete bgk approximation for strongly degenerate parabolic problems with boundary conditions abstract we consider a class of bgk systems with a finite number of velocities depending on a positive relaxation parameter that approximate strongly degenerate hyperbolic parabolic equations with initial boundary conditions we prove a priori estimates for the solutions of the systems showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero

finite valued feedback laws and piecewise classical solutions abstract often in engineering literature we find control systems in which the open loop inputs are piecewise constant and take values in a finite set such open loop inputs cause the system to have fairly regular solutions on the other hand when acting in closed loop feedback laws taking values in a finite set may not be reinterpreted as open loop inputs of the considered type in fact pathological behaviours such as the accumulation of discontinuities may appear zeno phenomenon we give some conditions which can be used as tools for building finite valued feedback laws not causing such pathological behaviours

boundary conditions for scalar conservation laws from a kinetic point of view boundary conditions for multidimensional scalar conservation laws are obtained in the context of hydrodynamic limits from a kinetic point of view the initial boundary value kinetic problem is well posed since inward and outward characteristics of the domain can be distinguished the convergence of the first momentum of the distribution function to an entropy solution of the conservation law is established boundary conditions are obtained the equivalence with the bardos leroux and nedelec conditions is studied

convergence of diffusive bgk approximations for nonlinear strongly parabolic systems we study a class of bgk approximations of parabolic systems in one space dimension we prove stability and existence of global solutions for this model moreover under certain conditions we prove a rigorous result of convergence toward the formal limit by using compensated compactness techniques

asymptotic controllability and robust asymptotic stabilizability this paper deals with asymptotically controllable systems for which there exists no smooth stabilizing state feedback to investigate the robustness asymptotic stabilization property a new class of hybrid feedbacks with a continuous component and a discrete one is introduced the hybrid patchy feedbacks the notion of solutions is a generalization of pi solutions and euler solutions it is proved that the origin of all globally asymptotically controllable systems can be globally asymptotically stabilized via a hybrid feedback with robustness with respect to measurement noise actuator errors and external disturbances

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

analysis of zeno behaviors in a class of hybrid systems this note investigates conditions for existence of zeno behaviors where a system undergoes an unbounded number of discrete transitions in a finite length of time in a class of hybrid systems zeno behavior occurs for example when a controller unsuccessfully attempts to satisfy an invariance specification by switching the system among different configurations faster and faster two types of zeno systems are investigated 1 strongly zeno systems where all runs of the system are zeno and 2 weakly zeno systems where only some runs of the system are zeno for constant rate and bounded rate hybrid systems and some nonlinear generalizations necessary and sufficient conditions for both zenoness and strong zenoness are derived the analysis is based on studying the trajectory set of a certain equivalent continuous time system that is associated with the dynamic equations of the hybrid system the relation between the possibility of existence of zeno behaviors in a system and the problem of existence of non zeno safety controllers that keep the system in a specified region of its operating space is also examined it is shown that in certain zeno systems a minimally interventive safety controller may not exist even if a safety controller exists disproving a conjecture made earlier in the literature

kinetic approximation of a boundary value problem for conservation laws we design numerical schemes for systems of conservation laws with boundary conditions these schemes are based on relaxation approximations taking the form of discrete bgk models with kinetic boundary conditions the resulting schemes are riemann solver free and easily extendable to higher order in time or in space for scalar equations convergence is proved we show numerical examples including solutions of euler equations

characterization of well posedness of piecewise linear systems one of the basic issues in the study of hybrid systems is the well posedness existence and uniqueness of solutions problem of discontinuous dynamical systems the paper addresses this problem for a class of piecewise linear discontinuous systems under the definition of solutions of caratheodory the concepts of jump solutions or of sliding modes are not considered here in this sense the problem to be discussed is one of the most basic problems in the study of well posedness for discontinuous dynamical systems first we derive necessary and sufficient conditions for bimodal systems to be well posed in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions next its extensions to the multimodal case are discussed as an application to switching control in the case that two state feedback gains are switched according to a criterion depending on the state we give a characterization of all admissible state feedback gains for which the closed loop system remains well posed

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented