continuous selections of trajectories of hybrid systems we present results on existence of continuous selections of trajectories of hybrid systems evolving according to lipschitz nonlinear inclusions first we utilize the skorohod metric to define both a family of pseudo metrics and a metric on the set of trajectories accepted by the hybrid automaton we show that under a non zeno condition the latter metric is complete second the existence of continuous selections with respect to the family of pseudo metrics is proved under a relatively mild assumption of transversality

multiple solutions for nonlinear systems with gyroscopic terms abstract in this paper we investigate the existence of multiple solutions to some nonlinear systems with gyroscopic terms via variational methods some new results are obtained and some results from the literature are improved

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

systems of controlled functional differential equations and adaptive tracking an adaptive servomechanism is developed in the context of the problem of approximate or practical tracking with prescribed asymptotic accuracy by the system output of any admissible reference signal absolutely continuous and bounded with essentially bounded derivative for every member of a class of controlled dynamical systems modelled by functional differential equations

infinitely many nontrivial periodic solutions for damped vibration problems with asymptotically linear terms in this paper we study a class of damped vibration problem with asymptotically quadratic terms at infinity we obtain infinitely many nontrivial periodic solutions by variational method to the best of our knowledge there is no published result focusing on this class of damped vibration problem with asymptotically quadratic terms at infinity

brief paper lyapunov function design for finite time convergence analysis twisting controller for second order sliding mode realization a generalization of the zubov method of a lyapunov function design is presented it is based on the characteristic method application and is related to resolving the first order partial differential equation of a special type a successful resolution of this equation guaranties a finite time convergence for the corresponding dynamics given by an ordinary differential equation with a discontinuous right hand side the suggested method is illustrated by its application to the so called twisting controller stability analysis the constructed lyapunov function as well as its level line sections is graphically illustrated

on a class of damped vibration problems with super quadratic potentials abstract the main purpose of this paper is to study the following damped vibration problems 1 u t q t i n n b u t f t u t a t u t 1 2 q t b u t a e t 0 t u 0 u t u t u t 0 where t 0 q l 1 0 t r with 0 t q t d t 0 a t a i j t is a symmetric n n matrix valued function defined in 0 t with a i j l 0 t for all i j 1 2 n b b i j is an antisymmetry n n constant matrix by establishing a proper variational set one existence result and two multiplicity results of non trivial periodic solutions of 1 are obtained

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

periodic solutions of superquadratic damped vibration problems in this paper we study a class of damped vibration problems with superquadratic terms at infinity by using variational methods we obtain infinitely many nontrivial periodic solutions under conditions weaker than those previously assumed to the best of our knowledge there is no published result focusing on this case by the method used in this paper

semistability finite time stability differential inclusions and discontinuous dynamical systems having a continuum of equilibria this paper focuses on semistability and finite time semistability for discontinuous dynamical systems having a continuum of equilibria semistability is the property whereby the solutions of a dynamical system converge to lyapunov stable equilibrium points determined by the system initial conditions in this paper we extend the theory of semistability to discontinuous autonomous dynamical systems in particular lyapunov based tests for semistability and finite time semistability for autonomous differential inclusions are established

stability and stabilization of switched linear systems with mode dependent average dwell time in this paper the stability and stabilization problems for a class of switched linear systems with mode dependent average dwell time mdadt are investigated in both continuous time and discrete time contexts the proposed switching law is more applicable in practice than the average dwell time adt switching in which each mode in the underlying system has its own adt the stability criteria for switched systems with mdadt in nonlinear setting are firstly derived by which the conditions for stability and stabilization for linear systems are also presented a numerical example is given to show the validity and potential of the developed techniques

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

an extension of lasalle s invariance principle and its application to multi agent consensus in the paper an extension of lasalle s invariance principle to a class of switched linear systems is studied one of the motivations is the consensus problem in multi agent systems unlike most existing results in which each switching mode in the system needs to be asymptotically stable this paper allows that the switching modes are only lyapunov stable under certain ergodicity assumptions an extension of lasalle s invariance principle for global asymptotic stability is obtained then it is used to solve the consensus reaching problem of certain multi agent systems in which each agent is modeled by a double integrator and the associated interaction graph is switching and is assumed to be only jointly connected

necessary optimality conditions for infinite dimensional state constrained control problems abstract this paper is concerned with first order necessary optimality conditions for state constrained control problems in separable banach spaces assuming inward pointing conditions on the constraint we give a simple proof of pontryagin maximum principle relying on infinite dimensional neighboring feasible trajectories theorems proved in 20 further we provide sufficient conditions guaranteeing normality of the maximum principle we work in the abstract semigroup setting but nevertheless we apply our results to several concrete models involving controlled pdes pointwise state constraints as positivity of the solutions are allowed

brief paper on the algebraic characterization of invariant sets of switched linear systems in this paper a suitable lasalle principle for continuous time linear switched systems is used to characterize invariant sets and their associated switching laws an algorithm to determine algebraically these invariants is proposed the main novelty of our approach is that we require no dwell time conditions on the switching laws by not focusing on restricted control classes we are able to describe the asymptotic properties of the considered switched systems observability analysis of a flying capacitor converter is proposed as an illustration

observer design for switched recurrent neural networks an average dwell time approach this paper is concerned with the problem of observer design for switched recurrent neural networks with time varying delay the attention is focused on designing the full order observers that guarantee the global exponential stability of the error dynamic system based on the average dwell time approach and the free weighting matrix technique delay dependent sufficient conditions are developed for the solvability of such problem and formulated as linear matrix inequalities the error state decay estimate is also given then the stability analysis problem for the switched recurrent neural networks can be covered as a special case of our results finally four illustrative examples are provided to demonstrate the effectiveness and the superiority of the proposed methods

stability analysis of discontinuous quantum control systems with dipole and polarizability coupling closed quantum systems under the influence of a laser field whose interaction is modeled by a schrodinger equation with a coupling control operator containing both a linear dipole and a quadratic polarizability term are analyzed discontinuous feedbacks obtained by a lyapunov trajectory tracking procedure have recently been proposed to control these types of system the purpose of this paper is to study the asymptotic stability by considering the solutions in the filippov sense the analysis is developed by applying a variant of the lasalle invariance principle for differential inclusions numerical simulations are included to illustrate the efficiency of the discontinuous control

global stability results for switched systems based on weak lyapunov functions in this paper we study the stability of nonlinear and time varying switched systems under restricted switching we approach the problem by decomposing the system dynamics into a nominal like part and a perturbation like one most stability results for perturbed systems are based on the use of strong lyapunov functions i e functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state however switched systems under restricted switching may not admit strong lyapunov functions even when asymptotic stability is uniform over the set of switching signals considered the main contribution of the current paper consists in providing stability results that are based on the stability of the nominal like part of the system and require only a weak lyapunov function these results may have wider applicability than results based on strong lyapunov functions the results provided follow two lines first we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal like and the perturbation like parts of the system second we provide input to state stability iss results for the case when the nominal like part is switched linear time varying we provide two types of iss results standard iss that involves the essential supremum norm of the input and a modified iss that involves a power type norm

a viability theorem for set valued states in a hilbert space abstract applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact possibly non convex sets rather than vectors this leads to evolutions in a basic set which can be supplied with a metric like the well established pompeiu hausdorff distance but it does not have an obvious linear structure this article extends differential inclusions with state constraints to compact valued states in a separable hilbert space h the focus is on sufficient conditions such that a given constraint set of compact subsets is viable a k a weakly invariant our main result extends the tangential criterion in the well known viability theorem usually for differential inclusions in a vector space to the metric space of non empty compact subsets of h

analyzing the stability of switched systems using common zeroing output systems this paper introduces the notion of common zeroing output systems czos to analyze the stability of switched systems the concept of czos allows one to verify weak zero state detectability it characterizes a common behavior of any individual subsystem when the output signal for each subsystem is approaching zero heuristically speaking it removes the effect of switching behavior and thus enables one to analyze stability properties in systems with complex switching signals with the help of czos the krasovskii lasalle theorem can be extended to switched nonlinear time varying systems with both arbitrary switching and more general restricted switching cases for switched nonlinear time invariant systems the needed detectability condition is further simplified leading to several new stability results particularly when a switched linear time invariant system is considered it is possible to generate a recursive method which combines a krasovskii lasalle result and a nested matrosov result to find a czos if it exists the power of the proposed czos is demonstrated by consensus problems in literature to obtain a stronger convergence result with weaker conditions

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

non autonomous systems asymptotic behaviour and weak invariance principles abstract results pertaining to asymptotic behaviour of solutions of non autonomous ordinary differential equations with locally integrably bounded right hand sides are presented ramifications for weakly asymptotically autonomous systems and adaptively controlled systems are highlighted

lower semi continuity of the solution set for semilinear differential inclusions abstract in this paper we provide sufficient conditions for the solution set of a semilinear differential inclusion to be lower semi continuous we give an application to the propagation of continuity of the minimum time function associated with the given differential inclusion and the target zero

brief paper stability of switched positive linear systems with average dwell time switching in this paper the stability analysis problem for a class of switched positive linear systems splss with average dwell time switching is investigated a multiple linear copositive lyapunov function mlclf is first introduced by which the sufficient stability criteria in terms of a set of linear matrix inequalities are given for the underlying systems in both continuous time and discrete time contexts the stability results for the splss under arbitrary switching which have been previously studied in the literature can be easily obtained by reducing mlclf to the common linear copositive lyapunov function used for the system under arbitrary switching those systems finally a numerical example is given to show the effectiveness and advantages of the proposed techniques

periodic solutions for some second order systems abstract in this paper we use the minimax theorem to investigate periodic solutions of some second order systems and give some sufficient conditions for the existence for periodic solutions the results generalize and improve some known results

a general stability criterion for time varying systems using a modified detectability condition uniform asymptotic stability for general nonlinear time varying systems is investigated from the output to state viewpoint a unified criterion is proposed using a modified detectability condition and certain integral inequalities the criterion extends the krasovskii lasalle theorem to nonperiodic time varying systems an example taken from the tracking control of mobile robots is given to illustrate that much as the invariance principle of lasalle is used in studying the stability of time invariant systems our results can be used to study the stability of nonperiodic time varying systems by checking the modified detectability condition

nonpathological lyapunov functions and discontinuous caratheodory systems differential equations with discontinuous right hand side and solutions intended in caratheodory sense are considered for these equations sufficient conditions which guarantee both lyapunov stability and asymptotic stability in terms of nonsmooth lyapunov functions are given an invariance principle is also proven

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

relaxation theorem for the evolution differential inclusions abstract in this paper we use an improved version of baire category method to prove the existence of solutions to the evolution differential inclusion ax ext f t x under the hausdorff noncompactness measure condition we also obtain a density result for the evolution differential inclusion

a recurrence principle for stochastic difference inclusions the invariance principle is extended to a recurrence principle and is developed for stochastic difference inclusions for these systems random solutions are not unique under appropriate lyapunov like conditions it is established for every random solution that almost every complete sample path converges to the largest weakly totally recurrent set contained in a level set of the lyapunov like function such a set is not larger and is sometimes smaller than the largest weakly invariant set contained in the level set the principle is useful for establishing robust uniform asymptotic stability in probability or robust uniform strong recurrence under weak lyapunov conditions for stochastic discrete time control systems that employ discontinuous feedback laws examples demonstrate the achieved results

global asymptotic stability of a pid control system with coulomb friction for a point mass subject to coulomb friction in feedback with a pid controller we consider a model based on a differential inclusion comprising all the possible magnitudes of static friction during the stick phase and having unique solutions we study the set of all equilibria and we establish its global asymptotic stability using a discontinuous lyapunov like function and a suitable lasalle s invariance principle the well posedness of the proposed model allows to establish useful robustness results including an iss property from a suitable input in a perturbed context simulation results are also given to illustrate our statements

a razumikhin type lemma for functional differential equations with application to adaptive control a lemma of razumikhin type pertaining to boundedness of solutions of functional differential equations is presented this lemma is subsequently invoked in the analysis of adaptive strategies that guarantee approximate tracking with prescribed asymptotic error bound of reference signals of sobolev class w 1 for two classes of uncertain systems modelled by controlled nonlinear functional differential equations

an integral invariance principle for differential inclusions with applications in adaptive control the byrnes martin integral invariance principle for ordinary differential equations is extended to differential inclusions on bbb r n the extended result is applied in demonstrating the existence of adaptive stabilizers and servomechanisms for a variety of nonlinear system classes

contingent cones to reachable sets of control systems high order necessary conditions for optimality for an optimal control problem are studied via properties of contingent cones to reachable sets along the optimal trajectory it is shown that the adjoint vector of pontryagin s maximum principle is normal to the set of variations of reachable sets results are applied to study optimal control problems for dynamical systems described by 1 closed loop control systems 2 nonlinear implicit systems 3 differential inclusions 4 control systems with jumps

baire s category and the bang bang property for evolution differential inclusions of contractive type abstract the baire category method is employed in order to establish that the bang bang property holds for a class of evolution differential inclusions of contractive type in reflexive and separable real banach spaces

uniform asymptotic stability of switched nonlinear time varying systems and detectability of reduced limiting control systems this paper is concerned with the study of both local and global uniform asymptotic stability for switched nonlinear time varying nltv systems through the detectability of output maps with this aim the notion of reduced limiting control systems for switched nltv systems whose switchings verify time state dependent constraints and the concept of weak zero state detectability for those reduced limiting systems are introduced necessary and sufficient conditions for the global uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this detectability property these sufficient conditions in conjunction with the existence of multiple weak lyapunov functions yield a criterion for the global uniform asymptotic stability of families of trajectories of the switched system this criterion can be seen as an extension of the classical krasovskii lasalle theorem an interesting feature of the results is that no dwell time assumptions are made moreover they can be used for establishing the global uniform asymptotic stability of the switched nltv system under arbitrary switchings the effectiveness of the proposed results is illustrated by means of various interesting examples including the stability analysis of a semiquasi z source inverter

the minimum time function for semilinear evolutions we study the bellman equation for the minimum time problem associated with a semilinear evolution system from the point of view of contingent solutions for this purpose new contingent derivatives are defined which involve functions instead of vectors as directions our analysis is based on flow invariance results for appropriate differential inclusions

using fuzzy logic to design separation function in flocking algorithms flocking algorithms essentially consist of three components alignment cohesion and separation to track a desired trajectory the flock center should move along the desired trajectory and thus the fourth component navigation is necessary the alignment cohesion and navigation components are well implemented through consensus protocols and tracking controls while the separation component is designed through heuristic based approaches this paper proposes a fuzzy logic solution to the separation component the ts rules and gaussian membership functions are used in fuzzy logic for fixed network flocking a standard stability proof by using lasalle s invariance principle is provided for dynamic network flocking a filipov solution definition is given for nonsmooth dynamics then a lasalle s invariance principle for nonsmooth dynamics is used to prove the stability a group of mobile robots with double integrator dynamics is simulated for the flocking algorithms in a 2 d environment

lasalle yoshizawa corollaries for nonsmooth systems in this technical note two generalized corollaries to the lasalle yoshizawa theorem are presented for nonautonomous systems described by nonlinear differential equations with discontinuous right hand sides lyapunov based analysis methods that achieve asymptotic convergence when the candidate lyapunov derivative is upper bounded by a negative semi definite function in the presence of differential inclusions are presented a design example illustrates the utility of the corollaries

fast homoclinic solutions for a class of damped vibration problems in this paper we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problemsu t q t u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t our approach is variational and it is based on the critical point theory we prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function moreover some open problems proposed by zhang and ruan are resolved our theorems appear to be the first such result

a new product integral representation for differential equations in separable banach spaces abstract letxbe a separable banach space consider the problem formula where the continuous functionf 0 x xis locally lipschitz continuous inx uniformly inton bounded intervals and continuous intuniformly w r t x the product integral formula x t lim n i 0 n i t n f i t n x 0 0 t t max for the solutionx t of d has been shown to converge we also show that iff t is lipschitz continuous onxwith constantl then the mappingx0 x t is lipschitz continuous with constanteltfor anyt 0 this formula has been recently developed for differential inclusions inrnby wolenski but the infinite dimensional case is considerably more involved

existence and multiplicity of solutions for a class of elliptic boundary value problems abstract in this paper we investigate the existence and multiplicity of solutions for the following elliptic boundary value problems u a x u g x u in u 0 on where g x u k u x u w u x u by using the symmetric mountain pass theorem we obtain two results about infinitely many solutions when g x u is odd in u k satisfies the pinching condition and w has a super quadratic growth moreover when the condition g x u is odd is not assumed by using the mountain pass theorem we also obtain two existence results of one nontrivial weak solution one of these results generalizes a recent result in mao zhu and luan 2012 10

brief paper a unified dissipativity approach for stability analysis of piecewise smooth systems the main objective of this paper is to present a unified dissipativity approach for stability analysis of piecewise smooth pws systems with continuous and discontinuous vector fields the filippov definition is considered for the solution of these systems using the concept of generalized gradients for nonsmooth functions sufficient conditions for the stability of a pws system are formulated based on lyapunov theory the importance of the proposed approach is that it does not need any a priori information about attractive sliding modes on switching surfaces which is in general difficult to obtain a section on application of the main results to piecewise affine pwa systems followed by a section with extensive examples clearly show the usefulness of the proposed unified methodology in particular we present an example with a stable sliding mode where the proposed method works and previously suggested methods fail

an invariance principle for nonlinear hybrid and impulsive dynamical systems in this paper we develop an invariance principle for dynamical systems possessing left continuous flows specifically we show that left continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems as a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems namely state dependent impulsive dynamical systems these results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems

a priori estimates for operational differential inclusions we prove a set valued gronwall lemma and a relaxation theorem for the semilinear differential inclusion x ax f t x x 0 x0 where a is the infinitesimal generator of a c0 semigroup on a separable banach space x and f 0 t x x is a set valued map this allows us to investigate infinitesimal generators of reachable sets and variational inclusions the results are applied to a semilinear optimal control problem with end point constraints

towards a stability theory of general hybrid dynamical systems in recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time for such systems we introduced the usual concepts of lyapunov and lagrange stability we showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties but defined on real time r 0 the motions of this class of systems are in general discontinuous this class of systems may be finite or infinite dimensional for the above discontinuous dynamical systems and hence for the above hybrid dynamical systems we established the principal lyapunov stability theorems as well as lagrange stability theorems for some of these we also established converse theorems we demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems in the present paper we continue the work described above in doing so we first develop a general comparison theory for the class of hybrid dynamical systems resp discontinuous dynamical systems considered herein making use of stability preserving mappings we then show how these results can be applied to establish some of the principal lyaponov stability theorems for the latter we also state and prove a converse theorem not considered previously finally to demonstrate the applicability of our results we consider specific examples throughout the paper

smooth lyapunov functions for hybrid systems part i existence is equivalent to robustness hybrid systems are dynamical systems where the state is allowed to either evolve continuously flow on certain subsets of the state space or evolve discontinuously jump from other subsets of the state space for a broad class of such systems characterized by mild regularity conditions on the data we establish the equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth lyapunov function this result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right hand sides furthermore we give a description of forward completeness of a hybrid system in terms of a smooth lyapunov like function

meagre functions and asymptotic behaviour of dynamical systems a measurable function x j r x x a metric space is said to be c meagre if c x is non empty and for every closed set k x with k c x 1 k has 5nite lebesgue measure this concept of meagreness is shown to provide a unifying framework which facilitates a variety of characterizations extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems by way of motivation consider the initial value problem f x x 0 rn with f rn rn locally lipschitz and denote its unique solution by x t t de5ned on its maximal interval of existence i 0 let g rn r be continuous assume that x is a bounded solution of the initial value problem in which case i 0 then g x is uniformly continuous if in addition g x l1 then we may conclude by an elementary observation frequently referred to as barb alat s lemma 5 see also 18 section 21 lemma 1 that g x t 0 as t 0 since by boundedness x t approaches as t its non empty compact invariant limit set b we also conclude that b g 1 0 it follows that x t tends as t to the largest invariant subset of g 1 0 this is a short proof of theorem 1 2 of 6 the proof in 6 is based on arguments involving properties of the dow generated by the dieerential

on variational methods for a class of damped vibration problems abstract in the present paper the following damped vibration problems 1 1 u t q t u t a t u t f t u t a e t 0 t u 0 u t u 0 e q t u t 0 and 1 1 u t q t u t a t u t f t u t a e t 0 t u 0 u t u 0 e q t u t 0 are studied where t 0 0 q l 1 0 t r q t 0 t q s d s a t a i j t is a symmetric n n matrix valued function defined in 0 t with a i j l 0 t for all i j 1 2 n and there exists a positive constant such that a t 2 for all r n and a e t 0 t the variational principles are given and an existence theorem and three multiplicity theorems of periodic solutions are obtained

on weak invariance principles for nonlinear switched systems in this technical note we develop two weak invariance principles for nonlinear switched systems we first present a union weak invariance principle for switched systems which includes as a special case the integral invariance principle it is shown that the switched solution approaches the largest weakly invariant set of the combined zero loci of the output functions then we extend the union weak invariance principle to an intersection weak invariance principle which greatly reduces the convergence region unlike the existing results in which the constructions of lyapunov functions are inevitable our principles do not require the existence of lyapunov functions numerical examples are presented to demonstrate the feasibility of our principles as well as applications to multi agent consensus problems

generalized lyapunov and invariant set theorems for nonlinear dynamical systems in this paper we develop generalized lyapunov and invariant set theorems for nonlinear dynamical systems wherein all regularity assumptions on the lyapunov function and the system dynamics are removed in particular local and global stability theorems are given using lower semicontinuous lyapunov functions furthermore generalized invariant set theorems are derived wherein system trajectories converge to a union of largest invariant sets contained in intersections over finite intervals of the closure of generalized lyapunov level surfaces the proposed results provide transparent generalizations to standard lyapunov and invariant set theorems

the local linking theorem with an application to a class of second order systems in the present paper some existence theorems are obtained concerning periodic and homoclinic solutions for a class of second order systems by means of a local linking theorem

synchronization of coupled harmonic oscillators with local interaction this paper studies synchronization of coupled second order linear harmonic oscillators with local interaction we analyze convergence conditions over respectively directed fixed and switching network topologies by using tools from algebraic graph theory matrix theory and nonsmooth analysis it is shown that the coupled harmonic oscillators can be synchronized under mild network connectivity conditions examples are given to validate the convergence conditions the theoretical result is also applied to synchronized motion coordination of multi agent systems as a proof of concept

an extension of the invariance principle for dwell time switched nonlinear systems abstract in this paper we propose an extension of the invariance principle for nonlinear switched systems under dwell time switched solutions this extension allows the derivative of an auxiliary function v also called a lyapunov like function along the solutions of the switched system to be positive on some sets the results of this paper are useful to estimate attractors of nonlinear switched systems and corresponding basins of attraction uniform estimates of attractors and basin of attractions with respect to time invariant uncertain parameters are also obtained results for a common lyapunov like function and multiple lyapunov like functions are given illustrative examples show the potential of the theoretical results in providing information on the asymptotic behavior of nonlinear dynamical switched systems

stability theorems for delay differential inclusions this technical note addresses stability problems of delay differential inclusions a chain rule is proposed for a wide class of lyapunov krasovskii functionals which are not necessarily invariantly differentiable we also obtain an invariance like theorem where the derivative of the candidate functional is bounded above by a continuous negative semidefinite function several examples are given to show the effectiveness of the results

a generalization of krasovskii lasalle theorem for nonlinear time varying systems converse results and applications this paper presents a practically applicable characterization of uniform global asymptotic stability uas and ugas for general nonlinear time varying systems under certain output dependent conditions in the spirit of the krasovskii lasalle theorem the celebrated krasovskii lasalle theorem is extended from two directions one is using the weak zero state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time derivative of the lyapunov function is the zero set another one is using an almost bounded output energy condition to relax the assumption that the time derivative of the lyapunov function is negative semi definite then the uas and ugas properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly lyapunov stable systems the proposed conditions turn out to be also necessary under some mild assumptions and thus give a new characterization of ugas and uas through an equivalence relation the proposed detectability condition can also be verified in terms of usual pe condition to validate the proposed results the obtained stability criteria are applied to a class of time varying passive systems and to revisit a tracking control problem of nonholonomic chained systems for the latter under certain persistency of excitation conditions the k exponential stability is achieved based on our approach

relaxation of control systems under state constraints in this paper we provide a relaxation result for control systems under both equality and inequality constraints involving the state and the control in particular we show that the mangasarian fromowitz constraint qualification allows us to rewrite constrained systems as differential inclusions with locally lipschitz right hand side then the filippov wazewski relaxation theorem may be applied to show that ordinary solutions are dense in the set of relaxed solutions if besides agreeing with the above constraints the state has to remain in a control independent set k then the mangasarian fromowitz condition cannot hold this case is investigated as well by means of a condition on the feasible velocities on the boundary of k

recurrence and lasalle invariance principle abstract in this paper we consider three different notions of recurrence chain recurrence strong chain recurrence and generalized recurrence where the generalized recurrence is introduced by auslander 1964 and the strong chain recurrence is derived from the idea of a strong chain defined by easton 1978 we prove that for a quasi gradient flow in a compact metric space these three recurrences are equivalent finally we establish an invariance principle i e the limit set of a precompact orbit is contained in the generalized recurrent component of zero derivative set of a lyapunov function

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

stabilization of nonholonomic integrators via logic based switching this paper explains how to stabilize a nonholonomic integrator using a hybrid control law employing switching and logic results concerning asymptotic stability and exponential convergence to the origin are derived the notion of a positively invariant set is extended to hybrid systems and sufficient conditions for invariance are presented the verification of these conditions does not require the computation of the state trajectories and their use goes beyond the analysis of the system presented in this paper

hybrid feedback control and robust stabilization of nonlinear systems in this paper we show for any nonlinear system that is asymptotically controllable to a compact set that a logic based hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise actuator error and external disturbance the construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed loop hybrid system the robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties auxiliary results give uniformity of convergence and of overshoots for hybrid systems and give a characterization of asymptotic stability of compact sets

an invariance principle for nonlinear switched systems abstract in this paper we address the problem of extending lasalle invariance principle to switched system we prove an extension of the invariance principle relative to dwell time switched solutions and a second one relative to constrained switched systems

invariance like results for nonautonomous switched systems this paper generalizes the lasalle yoshizawa theorem to switched nonsmooth systems the filippov and krasovskii regularizations of a switched system are shown to be contained within the convex hull of the filippov and krasovskii regularizations of the subsystems respectively a common candidate lyapunov function that has a negative semidefinite generalized time derivative along the trajectories of the subsystems is shown to be sufficient to establish lasalle yoshizawa like results for the switched system of independent interest are the results on approximate continuity and filippov regularization of set valued maps reduction of differential inclusions using lipschitz continuous regular functions and comparative remarks on different generalizations of the time derivative along the trajectories of a nonsmooth system

extensions of weak invariance principle for nonlinear switched systems with time invariant and time varying subsystems abstract this paper addresses the stability problem for nonlinear switched systems with both time invariant and time varying subsystems given that lyapunov like functions are difficult to construct for nonlinear switched systems generalized invariance principles are established based on observer functions for nonlinear switched systems where the subsystems share lyapunov like functions the generalized invariance principles can be specialized to lyapunov based invariance principles where accurate convergent region can be obtained in addition to the above efforts the definitions of p limit system and limit system set are presented under which the proposed invariance principles are extended to nonlinear switched systems with time varying subsystems illustrative examples show the effectiveness of the theoretical results

the maximum principle for an optimal solution to a differential inclusion with end points constraints we derive pontryagin s maximum principle for a general optimal control problem using the set valued version of variational equation we achieve this aim by exploiting an adequate differential calculus of set valued maps furthermore the calmness condition is replaced by a surjectivity condition involving reachable sets of the set valued linearization of the initial control problem duality then provides both the adjoint differential inclusion and the maximum principle

local controllability and infinitesimal generators of semigroups of set valued maps we study the semigroup properties of reachable sets of a differential inclusion x in f x using the derivative of the set valued map which associates to each initial state the set of solutions the results are applied to the local controllability problem

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control

existence and multiplicity of homoclinic solutions for a class of damped vibration problems abstract the main purpose of this paper is to study the following damped vibration problems 1 1 u t b u t a t u t f t u t a e t r u t 0 u t 0 as t where a a i j t c r r n 2 is an n n symmetric matrix valued function b b i j is an antisymmetry n n constant matrix f c 1 r r n r and f t u u f t u by a symmetric mountain pass theorem and a generalized mountain pass theorem an existence result and a multiplicity result of homoclinic solutions of 1 1 are obtained