convergence of time stepping schemes for passive and extended linear complementarity systems generalizing recent results in m k camlibel complementarity methods in the analysis of piecewise linear dynamical systems ph d thesis center for economic research tilburg university tilburg the netherlands 2001 m k camlibel w p m h heemels and j m schumacher ieee trans circuits systems i fund theory appl 49 2002 pp 349 357 and j s pang and d stewart math program ser a 113 2008 pp 345 424 this paper provides an in depth analysis of time stepping methods for solving initial value and boundary value non lipschitz linear complementarity systems lcss under passivity and broader assumptions the novelty of the methods and their analysis lies in the use of least norm solutions in the discrete time linear complementarity subproblems arising from the numerical scheme these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets among the principal results it is shown that using such least norm solutions of the discrete time subproblems an implicit euler scheme is convergent for passive initial value lcss generalizations under a strict copositivity assumption and for boundary value lcss are also established

conewise linear systems non zenoness and observability conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation this class of dynamical systems represents a large number of piecewise linear systems most notably linear complementarity systems with the p property and their generalizations to affine variational systems which have many applications in engineering systems and dynamic optimization the challenges of dealing with this type of hybrid system are due to two major characteristics mode switchings are triggered by state evolution and states are constrained in each mode in this paper we first establish the absence of zeno states in such a system based on this fundamental result we then investigate and relate several state observability notions short time and t time or finite time local global observability for the short time observability notions constructive finitely verifiable algebraic both sufficient and necessary conditions are derived due to their long time mode transitional behavior which is very difficult to predict only partial results are obtained for the t time observable states nevertheless we completely resolve the t time local observability for the bimodal conewise linear system for finite t and provide numerical examples to illustrate the difficulty associated with the long time observability

necessary conditions of asymptotic stability for unilateral dynamical systems in this paper we develop a mathematical tool that can be used to state necessary conditions of asymptotic stability of isolated stationary solutions of a class of unilateral dynamical systems more precisely nonlinear evolution variational inequalities are considered instability criteria are also given applications can be found in mechanics or electrical circuits

quasi polynomials linear diophantine equations and semi linear sets we investigate the family of semi linear sets of n t and z t we study the growth function of semi linear sets and we prove that such a function is a piecewise quasi polynomial on a polyhedral partition of n t moreover we give a new proof of combinatorial character of a famous theorem by dahmen and micchelli on the partition function of a system of diophantine linear equations

theoretical approach to biological aging abstract we present a model for biological aging that considers the number of individuals whose inherited genotype determines the maximum age for death each individual may die before that age due to some external factor but never after that limit the genotype of the offspring is inherited from the parent with some mutations described by a transition matrix the model can describe different strategies of reproduction and it is exactly soluble we applied our method to the bit string model for aging and the results are in perfect agreement with numerical simulations

analysis on the smoothing method for the p linear complementarity systems abstract a smoothing method for solving the linear complementarity systems lcs has been proposed by zheng in his thesis in zheng s algorithm the lcs is reformulated as a parameterized differential algebraic equation dae however the existence of the solution of the parameterized dae is not proved the aim of this paper is to establish the existence of the solution of the parameterized dae and to demonstrate the continuous dependence of the solution on the parameter and the initial value of the parameterized dae

strongly regular differential variational systems a differential variational system is defined by an ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution of a finite dimensional variational inequality containing the state variable of the system this paper addresses two system theoretic topics for such a nontraditional nonsmooth dynamical system namely non zenoness and local observability of a given state satisfying a blanket strong regularity condition for the former topic which is of contemporary interest in the study of hybrid systems we extend the results in our previous paper where we have studied zeno states and switching times in a linear complementarity system lcs as a special case of the differential variational inequality dvi the lcs consists of a linear time invariant ode and a linear complementarity problem the extension to a nonlinear complementarity system ncs with analytic inputs turns out to be non trivial as we need to use the lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state further extension to a differential variational inequality is obtained via its equivalent karush kuhn tucker formulation for the second topic which is classical in system theory we use the non zenoness result and the recent results in a previous paper pertaining to the b differentiability of the solution operator of a nonsmooth ode to obtain a sufficient condition for the short time local observability of a given strongly regular state of an ncs refined sufficient conditions and necessary conditions for local observability of the lcs satisfying the p property are obtained

numerical aspects of the sweeping process the sweeping process introduced some time ago by the author with motivation in plasticity theory today remains an object of mathematical research it is considered in this paper as the prototype of an evolution conditioned by inequality constraints since the governing differential requirements are only of order one with respect to time this provides a simplified setting for analysing some numerical and theoretical features also present in unilateral dynamics the latter is governed by differential inclusions of order two for the numerical handling of which the existing literature proposes diverse strategies briefly discussed the paper is especially intended to offer an introduction to the numerical approach called contact dynamics

nonsmooth lyapunov pairs for differential inclusions governed by operators with nonempty interior domain the general theory of lyapunov stability of first order differential inclusions in hilbert spaces has been studied by the authors in the previous paper adly et al in nonlinear anal 75 3 985 1008 2012 this new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty this includes in a natural way the finite dimensional case the current setting leads to simplified more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials some consequences of the viability of closed sets are given our analysis makes use of standard tools from convex and variational analysis

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

higher order moreau s sweeping process mathematical formulation and numerical simulation in this paper we present an extension of moreau s sweeping process for higher order systems the dynamical framework is carefully introduced qualitative dissipativity stability existence regularity and uniqueness results are given the time discretization of these nonsmooth systems with a time stepping algorithm is also presented this differential inclusion can be seen as a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary slackness variables applications of such high order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes

an euler newton continuation method for tracking solution trajectories of parametric variational inequalities a finite dimensional variational inequality parameterized by t in 0 1 is studied under the assumption that each point of the graph of its generally set valued solution mapping is a point of strongly regularity it is shown that there are finitely many lipschitz continuous functions on 0 1 whose graphs do not intersect each other such that for each value of the parameter the set of values of the solution mapping is the union of the values of these functions moreover the property of strong regularity is uniform with respect to the parameter along any such function graph an euler newton continuation method for tracking a solution trajectory is introduced and demonstrated to have l infty accuracy of order o h 4 thus generalizing a known error estimate for equations two examples of tracking economic equilibrium parametrically illustrate the theoretical results a correction is attached

on a new class of differential variational inequalities and a stability result this paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering operations research and physical sciences this new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right hand sides differential algebraic systems with constraints dynamic complementarity systems and evolutionary variational systems the purpose of this paper is two fold firstly we show that these differential variational inequalities when considering slow solutions and the more general level of a hilbert space contain projected dynamical systems another recent subclass of general differential inclusions this relation follows from a precise geometric description of the directional derivative of the metric projection in hilbert space which is based on the notion of the quasi relative interior secondly we are concerned with stability of the solution set to this class of differential variational inequalities here we present a novel upper set convergence result with respect to perturbations in the data including perturbations of the associated set valued maps and the constraint set here we impose weak convergence assumptions on the perturbed set valued maps use the monotonicity method of browder and minty and employ mosco convergence as set convergence also as a consequence we obtain a stability result for linear complementarity systems

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

passivity and complementarity this paper studies the interaction between the notions of passivity of systems theory and complementarity of mathematical programming in the context of complementarity systems these systems consist of a dynamical system given in the form of state space representation and complementarity relations we study existence uniqueness and nature of solutions for this system class under a passivity assumption on the dynamical part a complete characterization of the initial states and the inputs for which a solution exists is given these initial states are called consistent states for the inconsistent states we introduce a solution concept in the framework of distributions

necessary optimality conditions for optimal control problems with equilibrium constraints this paper introduces and studies the optimal control problem with equilibrium constraints ocpec the ocpec is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint and it can be seen as a dynamic mathematical program with equilibrium constraints it provides a powerful modeling paradigm for many practical problems such as bilevel optimal control problems and dynamic principal agent problems in this paper we propose weak clarke mordukhovich and strong stationarities for the ocpec moreover we give some sufficient conditions to ensure that the local minimizers of the ocpec are fritz john type weakly stationary mordukhovich stationary and strongly stationary unlike pontryagain s maximum principle for the classical optimal control problem with equality and inequality constraints a counterexample shows that for general ocpecs there may exist two sets of multipliers for complementarity constraints a condition under which these

convergence of regularized time stepping methods for differential variational inequalities this paper provides convergence analysis of regularized time stepping methods for the differential variational inequality dvi which consists of a system of ordinary differential equations and a parametric variational inequality pvi as the constraint the pvi often has multiple solutions at each step of a time stepping method and it is hard to choose an appropriate solution for guaranteeing the convergence in l han a tiwari m k camlibel and j s pang siam j numer anal 47 2009 pp 3768 3796 the authors proposed to use least norm solutions of parametric linear complementarity problems at each step of the time stepping method for the monotone linear complementarity system and showed the novelty and advantages of the use of the least norm solutions however in numerical implementation when the pvi is not monotone and its solution set is not convex finding a least norm solution is difficult this paper extends the tikhonov regularization approximation to the p 0 function dvi whi

a high accuracy method for solving odes with discontinuous right hand side ordinary differential equations with discontinuities in the state variables require a differential inclusion formulation to guarantee existence 8 from this formulation a high accuracy method for solving such initial value problems is developed which can give any order of accuracy and tracks the discontinuities the method uses an active set approach and determines appropriate active sets from solutions to linear complementarity problems convergence results are established under some non degeneracy assumptions the method has been implemented and results compare favourably with previously published methods 7 21

well posedness and output regulation for implicit time varying evolution variational inequalities a class of evolution variational inequalities evis which comprises ordinary differential equations odes coupled with variational inequalities vis associated with time varying set valued mappings is proposed in this paper we first study the conditions for existence and uniqueness of solutions the central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single valued lipschitz map and a time dependent maximal monotone operator regularity assumptions on the set valued mapping determine the regularity of the resulting solutions complementarity systems with time dependence are studied as a particular case we then use this result to study the problem of designing state feedback control laws for output regulation in systems described by evis the derivation of control laws for output regulation is based on the use of internal model principle and two cases are treated first a static feedback control law is derived when full state feedback is available in the second case only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed as applications we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints another application is seen in designing control inputs for regulation in power converters

stability and instability matrices for linear evolution variational inequalities this paper deals with the characterization of the stability and instability matrices for a class of unilaterally constrained dynamical systems represented as linear evolution variational inequalities levi such systems can also be seen as a sort of differential inclusion or in special cases as linear complementarity systems which in turn are a class of hybrid dynamical systems examples show that the stability of the unconstrained system and that of the constrained system may drastically differ various criteria are proposed to characterize the stability or the instability of levi

existence results of partial differential mixed variational inequalities without lipschitz continuity abstract the aim of this paper is to use order theoretic approaches to investigate a class of partial differential mixed variational inequalities pdmvis for short which are receiving much attention recently to this end the order preservation properties of proximity mapping are explored applying these order preservation properties and combining with the knaster tarski fixed point theorem we prove the existence of solutions and the order preservation properties of solution mapping for mixed variational inequalities on hilbert lattices relying on the above obtained results as well as some order theoretic techniques the existence of mild solutions to pdmvi is established in contrast to many existing results on pdmvis the order theoretic approaches are new and the results obtained in this paper do not require the lipschitz continuity of the mapping associated to nonlinear evolution equation

nonsmooth contact dynamics for the large scale simulation of granular material for the large scale simulation of granular material the nonsmooth contact dynamics method nscd is examined first the equations of motion of nonsmooth mechanical systems are introduced and classified as a differential variational inequality dvi that has a structure similar to differential algebraic equations daes using a galerkin projection in time we derive nonsmooth extensions of the shake and rattle schemes a matrix free interior point method ipm is used for the complementarity problems that need to be solved in each time step we demonstrate that in this way the nscd approach yields highly accurate results and is competitive compared to the discrete element method dem

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

linear passive systems and maximal monotone mappings this paper deals with a class of dynamical systems obtained from interconnecting linear systems with static set valued relations we first show that such an interconnection can be described by a differential inclusions with a maximal monotone set valued mappings when the underlying linear system is passive and the static relation is maximal monotone based on the classical results on such differential inclusions we conclude that such interconnections are well posed in the sense of existence and uniqueness of solutions finally we investigate conditions which guarantee well posedness but are weaker than passivity

uniqueness for solutions of differential complementarity problems in this paper we consider the question of which matrices m give unique solutions for differential complementarity problems mandelbaum 1989 unpublished manuscript of the form begin array ll frac dw dt m z q t quad w 0 w 0 k ni z t perp w t in k quad rm for all t end array with applications to complementarity and discretized l1 regularization problems assuming semi smoothness it is shown that super linearly convergent newton methods can be globalized if appropriate descent directions are used for the merit function f x 2 special attention is paid to directions obtained from the primal dual active set strategy

nonsmooth neural network for convex time dependent constraint satisfaction problems this paper introduces a nonsmooth ns neural network that is able to operate in a time dependent td context and is potentially useful for solving some classes of ns td problems the proposed network is named nonsmooth time dependent network ntn and is an extension to a td setting of a previous ns neural network for programming problems suppose c t t ge 0 is a nonempty td convex feasibility set defined by td inequality constraints the constraints are in general ns nondifferentiable functions of the state variables and time ntn is described by the subdifferential with respect to the state variables of an ns td barrier function and a vector field corresponding to the unconstrained dynamics this paper shows that for suitable values of the penalty parameter the ntn dynamics displays two main phases in the first phase any solution of ntn not starting in c 0 at t 0 is able to reach the moving set c cdot in finite time t h whereas in the second phase the solution tracks the moving set i e it stays within c t for all subsequent times t ge t h ntn is thus able to find an exact feasible solution in finite time and also to provide an exact feasible solution for subsequent times this new and peculiar dynamics displayed by ntn is potentially useful for addressing some significant td signal processing tasks as an illustration this paper discusses a number of examples where ntn is applied to the solution of ns td convex feasibility problems

solutions to affine generalized equations using proximal mappings the normal map has proven to be a powerful tool for solving generalized equations of the form find z e c with 0 e f z nc z where c is a convex set and nc z is the normal cone to c at z in this paper we use the t map a generalization of the normal map to solve equations of the more general form find z e dom t with 0 e f z t z where t is a maximal monotone multifunction we present a path following algorithm that determines zeros of coherently oriented piecewise affine functions and we use this algorithm together with the t map to solve the generalized equation for affine coherently oriented functions f and polyhedral multifunctions t the path following algorithm we develop here extends the piecewise linear homotopy framework of eaves to the case where a representation of a subdivided manifold is unknown

examining the smooth and nonsmooth discrete element approaches to granular matter the smooth and nonsmooth approaches to the discrete element method dem are examined from a computational perspective the main difference can be understood as using explicit versus implicit time

on the commutative equivalence of semi linear sets of n k given two subsets s 1 s 2 of n k we say that s 1 is commutatively equivalent to s 2 if there exists a bijection f s 1 s s 2 from s 1 onto s 2 such that for every v s 1 v f v where v denotes the sum of the components of v we prove that every semi linear set of n k is commutatively equivalent to a recognizable subset of n k

a numerical method based on the complementarity and optimal control formulations for solving a family of zero sum pursuit evasion differential games abstract this paper presents an efficient numerical method for a class of zero sum pursuit evasion differential games zspedgs the aim of the presented method is to resolve the drawbacks of the indirect methods in solving zspedgs in the indirect methods the solution of zspedg is found by solving a two point boundary value problem tpbvp derived from the necessary conditions the indirect methods are accurate and fast however they are very sensitive to initial guess and when the control is bounded some non smooth equations appear in the resulting tpbvp these drawbacks restrict the use of indirect methods for solving zspedgs to overcome these drawbacks at first we reformulate the discontinuous equations of tpbvp by complementarity conditions and as a result a differential complementarity system dcs is obtained then the resulting dcs is reformulated to an optimal control problem which can be solved by a well developed direct or indirect method the efficiency and robustness of the method are reported by means of two benchmarks and two real life zspedg problems

efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded this work analyzes the initial value problem in ordinary differential equations with a parametric lexicographic linear program lp embedded the lp is said to be embedded since the dynamics depend on the solution of the lp which is in turn parameterized by the dynamic states this problem formulation finds application in dynamic flux balance analysis which serves as a modeling framework for industrial fermentation reactions it is shown that the problem formulation can be intractable numerically which arises from the fact that the lp induces an effective domain that may not be open a numerical method is developed which reformulates the system so that it is defined on an open set the result is a system of semi explicit index one differential algebraic equations which can be solved with efficient and accurate methods it is shown that this method addresses many of the issues stemming from the original problem s intractability the application of the method to examples of industrial fermentation processes demonstrates its effectiveness and efficiency

a continuous time linear complementarity system for dynamic user equilibria in single bottleneck traffic flows this paper formally introduces a linear complementarity system lcs formulation for a continuous time multi user class dynamic user equilibrium due model for the determination of trip timing decisions in a simplified single bottleneck model existence of a lipschitz solution trajectory to the model is established by a constructive time stepping method whose convergence is rigorously analyzed the solvability of the time discretized subproblems by lemke s algorithm is also proved combining linear complementarity with ordinary differential equations and being a new entry to the mathematical programming field the lcs provides a computational tractable framework for the rigorous treatment of the due problem in continuous time this paper makes a positive contribution in this promising research venue pertaining to the application of differential variational theory to dynamic traffic problems

reformulations of measure differential inclusions and their closed graph property abstract new formulations are given for measure differential inclusions dx dt k t r n where x is a function of bounded variation and k is a set valued map with closed convex values and has closed graph measure differential inclusions were first named by j j moreau for studying rigid body with impacts shocks and coulomb friction and assumed that k t is always a cone the new formulations are used to show that the graph of the solution operator is closed under point wise convergence of the solutions x and weak convergence of the differential measures dx provided that the minimum norm points of k t are bounded and the asymptotic cones k t are always pointed

on the commutative equivalence of bounded context free and regular languages the semi linear case this is the third paper of a group of three where we prove the following result let a be an alphabet of t letters and let a nt be the corresponding parikh morphism given two languages l1 l2 a we say that l1 is commutatively equivalent to l2 if there exists a bijection f l1 l2 from l1 onto l2 such that for every u l1 u f u then every bounded context free language is commutatively equivalent to a regular language

evolutionary problems driven by variational inequalities abstract in this paper we introduce the differential system obtained by mixing an evolution equation and a variational inequality eevi for short first by using kkm theorem and monotonicity arguments we prove the superpositional measurability and upper semicontinuity for the solution set of a general variational inequality then we establish that the solution set of eevi is nonempty and compact our approach is based on the theory of semigroups filippov implicit function lemma and fixed point theory for set valued mappings

on the structure of the counting function of sparse context free languages we give an exact description of the counting function of a sparse context free language let l be a sparse context free language and let fl be its counting function then there exist polynomials p0 p1 pk 1 with rational coefficients and an integer constant k0 such that for any n k0 one has fl n pj n where j is such that j n mod k as a consequence one can easily show the decidability of some questions concerning sparse context free languages finally we show that for any sparse context free language l there exists a regular language l such that for any n 0 one has fl n fl n and therefore fl is rational

convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities in this paper we analyze and discuss the well posedness of two new variants of the so called sweeping process introduced by moreau in the early 70s moreau in sem anal convexe montpellier 1971 with motivation in plasticity theory the first new variant is concerned with the perturbation of the normal cone to the moving convex subset c t c t supposed to have a bounded variation by a lipschitz mapping under some assumptions on the data we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation the second variant for which a large analysis is made concerns a first order sweeping process with velocity in the moving set c t c t this class of problems subsumes as a particular case the evolution variational inequalities widely used in applied mathematics and unilateral mechanics duvaut and lions in inequalities in mechanics and physics springer berlin 1976 assuming that the moving subset c t c t has a continuous variation for every t in 0 t t 0 t with c 0 c 0 bounded we show that the problem has at least a lipschitz continuous solution the well posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator we also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model krej i in eur j appl math 2 281 292 1991 to the planning procedure in mathematical economy henry in j math anal appl 41 179 186 1973 and cornet in j math anal appl 96 130 147 1983 and to nonregular electrical circuits containing nonsmooth electronic devices like diodes acary et al nonsmooth modeling and simulation for switched circuits lecture notes in electrical engineering springer new york 2011 the theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof our methodology is based only on tools from convex analysis like other papers in this collection we show in this presentation how elegant modern convex analysis was influenced by moreau s seminal work

non zenoness of a class of differential quasi variational inequalities the zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time the absence of this phenomenon is crucial to the numerical simulation of such a system by time stepping methods and to the understanding of the behavior of the system trajectory extending a previous result for a strongly regular differential variational inequality this paper establishes that a certain class of non strongly regular differential variational inequalities is devoid of the zeno phenomenon the proof involves many supplemental results that are of independent interest specialized to a frictional contact problem with local compliance and polygonal friction laws this non zenoness result is of fundamental significance and the first of its kind

optimization based simulation of nonsmooth rigid multibody dynamics we present a time stepping method to simulate rigid multibody dynamics with inelastic collision contact and friction the method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program we prove that a solution sequence of the method converges to the solution of a measure differential inclusion we present numerical results for a few examples and we illustrate the difference between the results from our scheme and previous linear complementarity based time stepping schemes

the parikh counting functions of sparse context free languages are quasi polynomials let l be a sparse context free language over an alphabet of t letters and let f l n t n be its parikh counting function we prove the following two results 1 there exists a partition of n t into a finite family of polyhedra such that the function f l is a quasi polynomial on each polyhedron of the partition 2 there exists a partition of n t into a finite family of rational subsets such that the function f l is a polynomial on each set of the partition

shape restricted smoothing splines via constrained optimal control and nonsmooth newton s methods shape restricted smoothing splines receive considerable attention motivated by many important applications in science and engineering in this paper we consider smoothing splines subject to general linear dynamics and control constraints and formulate them as finite horizon constrained linear optimal control problems with unknown initial state and control by exploring techniques from functional and variational analyses optimality conditions are developed in terms of variational inequalities due to the control constraints the optimality conditions give rise to a nonsmooth b differentiable equation of an optimal initial condition whose unique solution completely determines the shape restricted smoothing spline a modified nonsmooth newton s algorithm with line search is used to solve this equation detailed convergence analysis of the proposed algorithm is presented using techniques from nonsmooth analysis and polyhedral theory we show the global convergence of the algorithm for shape restricted smoothing splines subject to general polyhedral control constraints

convergence of a class of semi implicit time stepping schemes for nonsmooth rigid multibody dynamics in this work we present a framework for the convergence analysis in a measure differential inclusion sense of a class of time stepping schemes for multibody dynamics with contacts joints and friction this class of methods solves one linear complementarity problem per step and contains the semi implicit euler method as well as trapezoidal like methods for which second order convergence was recently proved under certain conditions by using the concept of a reduced friction cone the analysis includes for the first time a convergence result for the case that includes joints an unexpected intermediary result is that we are able to define a discrete velocity function of bounded variation although the natural discrete velocity function produced by our algorithm may have unbounded variation

robust non zenoness of piecewise affine systems with applications to linear complementarity systems piecewise affine systems pass constitute an important class of nonsmooth switching dynamical systems subject to state dependent mode transitions arising from control and dynamic optimization a fundamental issue in dynamics analysis of switching systems pertains to the possible occurrence of infinitely many switchings in finite time referred to as the zeno behavior there has been a growing interest in characterization of zeno free switching systems different from the recent non zeno analysis of switching systems the present paper studies non zeno properties of pass subject to system parameter and or initial state perturbations inspired by sensitivity and uncertainty analysis of pass specifically by exploiting the geometry of polyhedral subdivisions and dynamical system techniques this paper establishes a uniform bound on the number of mode switchings for a family of lipschitz pass under mild uniform conditions on system parameters and associated polyhedral subdivisions this result is employed t

switching relay and complementarity systems a tutorial on their well posedness and relationships this paper reviews the relationships between switching systems defined from a partition of the state space into convex cells and relay or complementarity dynamical systems which are other classes of discontinuous systems well posedness results i e results on the existence and the uniqueness of solutions for different classes of relay and complementarity systems which are also switching systems are presented the reverse issue when can a switching system be rewritten equivalently as a relay or a complementarity system is also tackled many examples from mechanics circuits and biology illustrate the developments throughout the paper the paper focuses on systems with continuous solutions i e with no state jumps convexity is the central property

differential mixed variational inequalities in finite dimensional spaces in this paper a differential vector variational inequality is introduced and studied in finite dimensional euclidean spaces the existence of a caratheodory weak solution for the differential vector variational inequality is presented under some suitable conditions furthermore the upper semicontinuity and the lower semicontinuity of the solution sets for the differential variational inequality are established when both the mapping and the constraint set are perturbed by two different parameters

on the commutative equivalence of bounded context free and regular languages this is the first paper of a group of three where we prove the following result let a be an alphabet of t letters and let a s n t be the corresponding parikh morphism given two languages l 1 l 2 a we say that l 1 is commutatively equivalent to l 2 if there exists a bijection f l 1 s l 2 from l 1 onto l 2 such that for every u l 1 u f u then every bounded context free language is commutatively equivalent to a regular language

a matrix free cone complementarity approach for solving large scale nonsmooth rigid body dynamics this paper proposes an iterative method that can simulate mechanical systems featuring a large number of contacts and joints between rigid bodies the numerical method behaves as a contractive mapping that converges to the solution of a cone complementarity problem by means of iterated fixed point steps with separable projections onto convex manifolds since computational speed and robustness are important issues when dealing with a large number of frictional contacts we have performed special algorithmic optimizations in order to translate the numerical scheme into a matrix free algorithm with o n space complexity and easy implementation a modified version that can run on parallel computers is discussed a multithreaded version of the method has been used to simulate systems with more than a million contacts with friction

differential variational inequality approach to dynamic games with shared constraints the dynamic nash equilibrium problem with shared constraints nepsc involves a dynamic decision process with multiple players where not only the players cost functionals but also their admissible control sets depend on the rivals decision variables through shared constraints for a class of the dynamic nepsc we propose a differential variational inequality formulation using this formulation we show the existence of solutions of the dynamic nepsc and develop a regularized smoothing method to find a solution of it we prove that the regularized smoothing method converges to the least norm solution of the differential variational inequality which is a solution of the dynamic nepsc as the regularization parameter lambda and smoothing parameter mu go to zero with the order mu o lambda o numerical examples are given to illustrate the existence and convergence results

normal maps inducted by linear transformations we study a certain piecewise linear manifold which we call the normal manifold associated with a polyhedral convex set and a family of continuous functions called normal maps that are induced on this manifold by continuous functions from rn to rn these normal maps occur frequently in optimization and equilibrium problems and the subclass of normal maps induced by linear transformations plays a key role our main result is that the normal map induced by a linear transformation is a lipschitzian homeomorphism if and only if the determinant of the map in each n cell of the normal manifold has the same nonzero sign

optimal control of the sweeping process over polyhedral controlled sets abstract the paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given bolza type functional which depends on control and state variables as well as their velocities besides the highly non lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types all of this creates serious challenges for deriving necessary optimality conditions we develop here the method of discrete approximations and combine it with advanced tools of first order and second order variational analysis and generalized differentiation this approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions as a by product of the developed approach we prove the strong w 1 2 convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous time system and derive necessary optimality conditions for the discrete counterparts the established necessary optimality conditions for the sweeping process are illustrated by several examples

dynamics of the bit string model of age structured populations abstract the dynamics of age structured populations in the framework of the bit string model developed by t j p penna and collaborators is analyzed in terms of the spectral properties of a time evolution generator this is a functional of inclusive properties of the current state of the population which is closely related to the well known leslie operators the analysis of the generator allows for the identification of propagation modes associated with invariant subspaces of the propagator matrix to which in general different gain factors are associated restrictions on the reproductive age span translate into the number structure and relative dominance of the propagation modes the time evolution of semelparous populations is found to be associated in general with oscillatory steady regimes through the appearance of a dominant family of propagation modes with identical gain factors in complete analogy with the semelparous regime of the leslie model explicit results are given and discussed for the case in which one deleterious trait is lethal

on two parameter global bifurcation of periodic solutions to a class of differential variational inequalities abstract in this paper by using the method of integral guiding functions a multiparameter global bifurcation theorem for differential inclusions with the periodic condition is proved it is shown how the abstract result can be applied to the study of the two parameter global bifurcation of periodic solutions for a class of differential variational inequalities

using krylov subspace and spectral methods for solving complementarity problems in many body contact dynamics simulation summary many body dynamics problems are expected to handle millions of unknowns when for instance investigating the three dimensional flow of granular material unfortunately the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds this is typically the case when the equations of motion embed a differential variational inequality problem that captures contact and possibly frictional interactions between rigid and or flexible bodies as the size of the physical system increases the speed and or the quality of the numerical solution decreases this paper describes three methods the gradient projected minimum residual method the preconditioned spectral projected gradient with fallback method and the modified proportioning with reduced gradient projection method that demonstrate better scalability than the projected jacobi and gauss seidel methods commonly used to solve contact problems that draw on a differential variational inequality based modeling approach copyright 2013 john wiley sons ltd

nonlocal problem for differential complementarity systems in this paper by using the topological degree theory for multivalued maps and the bounding function method we give sufficient conditions for the existence of solutions to a nonlocal problem of differential complementarity systems an example is given

pullback attractor for differential evolution inclusions with infinite delays we analyze the existence of pullback attractor for non autonomous differential inclusions with infinite delays in banach spaces by using measures of noncompactness the obtained results can be applied to control systems driven by semilinear partial differential equations and multivalued feedbacks

partial differential variational inequalities involving nonlocal boundary conditions in banach spaces abstract in this paper we firstly introduce a complicated system obtained by mixing a nonlinear evolutionary partial differential equation and a mixed variational inequality in infinite dimensional banach spaces in the case where the set of constraints is not necessarily bounded and the problem is driven by nonlocal boundary conditions which is called partial differential variational inequality pdvi for short then we show that the solution set of the mixed variational inequality involved in problem pdvi is nonempty bounded closed and convex moreover the upper semicontinuity and measurability properties for set valued mapping u 0 t e 2 c b v e 1 see 3 7 below are also established finally several existence results for pdvi are obtained by using a fixed point theorem for condensing set valued operators and theory of measure of noncompactness

a sequential homotopy method for mathematical programming problems we propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract hilbert spaces under the guignard constraint qualification the method is equivalent to performing projected backward euler timestepping on a projected gradient antigradient flow of the augmented lagrangian the projected backward euler equations can be interpreted as the necessary optimality conditions of a primal dual proximal regularization of the original problem the regularized problems are always feasible satisfy a strong constraint qualification guaranteeing uniqueness of lagrange multipliers yield unique primal solutions provided that the stepsize is sufficiently small and can be solved by a continuation in the stepsize we show that equilibria of the projected gradient antigradient flow and critical points of the optimization problem are identical provide sufficient conditions for the existence of global flow solutions and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient antigradient flow practically eradicating convergence to saddle points and maxima the sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework we demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth newton approach for the regularized subproblems

existence result for differential variational inequality with relaxing the convexity condition abstract in this paper a class of differential variational inequalities are studied and a new approach is introduced to relax the convexity condition firstly an existence theorem of caratheodory weak solution for the differential variational inequalities is established secondly an algorithm for solving the problem is developed and the convergence analysis for the algorithm is given finally a numerical example is reported to illustrate the proposed algorithm

lyapunov stability of complementarity and extended systems a linear complementarity system lcs is a piecewise linear dynamical system consisting of a linear time invariant ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution to a finite dimensional linear complementarity problem lcp whose constant vector is a linear function of the differential variable continuing the authors recent investigation of the lcs from the combined point of view of system theory and mathematical programming this paper addresses the important system theoretic properties of exponential and asymptotic stability for an lcs with a c 1 state trajectory the novelty of our approach lies in our employment of a quadratic lyapunov function that involves the auxiliary algebraic variable of the lcs when expressed in the state variable alone the lyapunov function is piecewise quadratic and thus nonsmooth the nonsmoothness feature invalidates standard stability analysis that is based on smooth lyapunov functions in addition to providing sufficient conditions for exponential stability we establish a generalization of the well known lasalle invariance theorem for the asymptotic stability of a smooth dynamical system to the lcs which is intrinsically a nonsmooth system sufficient matrix theoretic copositivity conditions are introduced to facilitate the verification of the stability properties properly specialized the latter conditions are satisfied by a passive like lcs and certain hybrid linear systems having common quadratic lyapunov functions we provide numerical examples to illustrate the stability results we also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities based on a new converse theorem for odes with b differentiable right hand sides the latter theorem asserts that the existence of a b differentiable lyapunov function is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system

bifurcations from nondegenerate families of periodic solutions in lipschitz systems abstract the paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations the approach to solve this problem can be described as transforming by a lyapunov schmidt reduction the initial system into one which is in the standard form of averaging and subsequently applying the averaging principle this approach encounters a fundamental problem when the perturbation is only lipschitz nonsmooth as we do not longer have smooth lyapunov schmidt projectors the situation of lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions motivated by recent challenges in control theory we are interested in the uniqueness problem we achieve this in the case when the lipschitz constant of the perturbation obeys a suitable estimate

linear complementarity systems zeno states a linear complementarity system lcs is a hybrid dynamical system defined by a linear time invariant ordinary differential equation coupled with a finite dimensional linear complementarity problem lcp the present paper is the first of several papers whose goal is to study some fundamental issues associated with an lcs specifically this paper addresses the issue of zeno states and the related issue of finite number of mode switches in such a system the cornerstone of our study is an expansion of a solution trajectory to the lcs near a given state in terms of an observability degree of the state on the basis of this expansion and an inductive argument we establish that an lcs satisfying the p property has no strongly zeno states we next extend the analysis for such an lcs to a broader class of problems and provide sufficient conditions for a given state to be weakly non zeno while related mode switch results have been proved by brunovsky and sussmann for more general hybrid systems our analysis exploits the special structure of the lcs and yields new results for the latter that are of independent interest and complement those by these two and other authors

convergence of a time stepping scheme for rigid body dynamics and resolution of painleve s problem this paper gives convergence theory for a new implicit time stepping scheme for general rigid body dynamics with coulomb friction and purely inelastic collisions and shocks an important consequence of this work is the proof of existence of solutions of rigid body problems which include the famous counterexamples of painleve the mathematical basis for this work is the formulation of the rigid body problem in terms of measure differential inclusions of moreau and monteiro marques the implicit time stepping method is based on complementarity problems and is essentially a particular case of the algorithm described in anitescu potra 2 which in turn is based on the formulation in stewart trinkle 47

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

a time stepping complementarity approach for frictionless systems of rigid bodies this paper presents a time stepping complementarity approach for computing a numerical trajectory of a system of rigid bodies under frictionless contact such a system is formulated as a time dependent ordinary differential complementarity problem dcp which consists of an ordinary differential equation ode coupled with a complementarity condition the phenomenon of impact plays an essential role when modeling a mechanical system of this type within the time stepping scheme which uses a high order ode discretization we include an impact detection routine and an impact law governing the behavior of the system over the duration of an impact we report computational results for the application of the overall numerical scheme to computing the solution trajectories of several realistic mechanical systems

gompertz mortality law and scaling behavior of the penna model the penna model is a model of evolutionary ageing through mutation accumulation where traditionally time and the age of an organism are treated as discrete variables and an organism s genome is represented by a binary bit string we reformulate the asexual penna model and show that a universal scale invariance emerges as we increase the number of discrete genome bits to the limit of a continuum the continuum model introduced by almeida and thomas int j mod phys c 11 1209 2000 can be recovered from the discrete model in the limit of infinite bits coupled with a vanishing mutation rate per bit finally we show that scale invariant properties may lead to the ubiquitous gompertz law for mortality rates for early ages which is generally regarded as being empirical

second order differential variational inequalities involving anti periodic boundary value conditions abstract this paper deals with a new class of second order nonlinear differential variational inequalities firstly we study the properties of a solution set for a variational inequality by using the kkm technique and a monotonicity property then we prove the existence of solutions to a second order nonlinear differential variational inequality with anti periodic boundary value conditions not assuming the lipschitz continuity of the nonlinear term our approaches which are essentially different from previous research methods in differential variational inequalities are based on the topological degree theory and the filippov implicit function lemma

convergence of time stepping method for initial and boundary value frictional compliant contact problems beginning with a proof of the existence of a discrete time trajectory this paper establishes the convergence of a time stepping method for solving continuous time boundary value problems for dynamic systems with frictional contacts characterized by local compliance in the normal and tangential directions our investigation complements the analysis of the initial value rigid body model with one frictional contact encountering inelastic impacts by stewart arch ration mech anal 145 1998 pp 215 260 and the recent analysis by anitescu optimization based simulation for nonsmooth rigid multibody dynamics argonne national laboratory argonne il 2004 using the framework of measure differential inclusions in contrast to the measure theoretic approach of these authors we follow a differential variational approach and address a broader class of problems with multiple elastic or inelastic impacts applicable to both initial and affine boundary value problems our main convergence result pertains to the case where the compliance in the normal direction is decoupled from the compliance in the tangential directions and where the friction coefficients are sufficiently small

difference methods for differential inclusions a survey the main objective of this survey is to study convergence properties of difference methods applied to differential inclusions it presents in a unified way a number of results scattered in the li

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field

time stepping for three dimensional rigid body dynamics traditional methods for simulating rigid body dynamics involves determining the current contact arrangement e g each contact is either a rolling or sliding contact the development of this approach is most clearly seen in the work of haug et al mech machine theory 21 1986 401 425 and pfeiffer and glocker multibody dynamics with unilateral contacts wiley 1996 however there has been a controversy about the status of rigid body dynamics as a theory due to simple problems in the area which do not appear to have solutions the most famous if not the earliest is due to paul painleve c r acad sci paris 121 1895 112 115 recently a number of time stepping methods have been developed to overcome these difficulties these time stepping methods use integrals of the forces over time steps rather than the actual forces this allows impulsive forces without the need for a separate formulation or special procedures to cover this case the newest of these methods are developed in terms of complementarity problems the complementarity problems that define the time stepping procedure are solvable unlike previous methods for simulating rigid body dynamics with friction proof of the existence of solutions to the continuous problem can be shown in the sense of measure differential inclusions in terms of these methods in this paper a number of these variants will be discussed and their essential properties proven

dependence of solutions of nonsmooth differential algebraic equations on parameters abstract the well posedness of nonsmooth differential algebraic equations daes is investigated more specifically semi explicit daes with caratheodory style assumptions on the differential right hand side functions and local lipschitz continuity assumptions on the algebraic equations the daes are classified as having differential index one in a generalized sense solution regularity is formulated in terms of projections of generalized clarke jacobians existence of solutions is derived under consistency and regularity of the initial data uniqueness of a solution is guaranteed under analogous caratheodory ordinary differential equation uniqueness assumptions the continuation of solutions is established and sufficient conditions for continuous and lipschitzian parametric dependence of solutions are also provided to accomplish these results a theoretical tool for analyzing nonsmooth daes is provided in the form of an extended nonsmooth implicit function theorem the findings here are a natural extension of classical results and lay the foundation for further theoretical and computational analyses of nonsmooth daes

mortality curves using a bit string aging model abstract in this work we studied the shape of mortality curves for a population under different conditions using a bit string model for biological aging more specifically we studied how the model parameters affect the shape of the mortality curves although these curves are in reasonable agreement with the gompertz law for the increase of mortality for a extended range of ages other parameters play an important role to determine the overall behavior of the mortality s curves

sweeping process with regular and nonregular sets abstract differential inclusions involving the normal cone to a moving set are investigated a special attention is paid to the sweeping process associated with sets for which no regularity assumption is required