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

on the covering number of a matroid element abstract let s be a nonempty finite set with cardinality m let m s i m be a matroid on s with no loops we present results connecting the structure of the bases of m k the k th power of m and the covering number of elements of s in m

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

on initial instantaneous jumps of singular systems this paper is concerned with instantaneous jumps in the state trajectory of a singular system it is shown that such jumps generically exist at the initial time due to inconsistent initial states the formula for the jumps is derived which indicates that the jumps linearly depend on the initial state moreover the set of all initial states which do not cause any state jumps for any output feedback is characterized and is revealed to be a linear subspace in the meantime the necessary and sufficient conditions are given for the existence of an output feedback to suppress completely the jumps for a given initial state finally we formulate the problem of designing an output feedback not only to stabilize the plant but also to achieve minimum initial jumps as a minimization problem

when is a linear multi modal system disturbance decoupled in this paper we study the question under which conditions a linear multi modal system is disturbance decoupled we establish necessary and sufficient geometric conditions from which the existing results on switched linear systems and conewise linear systems can be recovered as special cases also we apply these conditions to a class of linear complementarity systems in order to obtain a more crisp characterization

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

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

a gauss seidel type method for dynamic nonlinear complementarity problems the dynamic nonlinear complementarity problem dncp consisting of a nonlinear differential system and a complementarity system has been used to formulate and study many dynamic problems in a gaus

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

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

an lpcc approach to nonconvex quadratic programs filling a gap in nonconvex quadratic programming this paper shows that the global resolution of a feasible quadratic program qp which is not known a priori to be bounded or unbounded below can be accomplished in finite time by solving two linear programs with linear complementarity constraints i e lpccs specifically this task can be divided into two lpccs the first confirms whether the qp is bounded below on the feasible set and if not computes a feasible ray on which the qp is unbounded the second lpcc computes a globally optimal solution if it exists by identifying a stationary point that yields the best quadratic objective value in turn the global resolution of these lpccs can be accomplished by a parameter free mixed integer programming based finitely terminating algorithm developed recently by the authors which can be enhanced in this context by a new kind of valid cut derived from the second order conditions of the qp and by exploiting the special structure of the lpccs throughout our treatment makes no boundedness assumption of the qp this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption the general theory is illustrated by 3 classes of indefinite problems qps with simple upper and lower bounds existence of optimal solutions is guaranteed same qps with an additional inequality constraint extending the case of simple bound constraints and nonnegatively constrained copositive qps no guarantee of the existence of an optimal solution we also present numerical results to support the special cuts obtained due to the qp connection

quantum field theory and phylogenetic branching a calculational framework is proposed for phylogenetics using nonlocal quantum field theories in hypercubic geometry quadratic terms in the hamiltonian give the underlying markov dynamics while higher degree terms represent branching events the spatial dimension l is the number of leaves of the evolutionary tree under consideration momentum conservation modulo 2 l in l 1 scattering corresponds to tree edge labelling using binary l vectors the bilocal quadratic term allows for momentum dependent rate constants only the tree or trees compatible with selected nonzero edge rates contribute to the branching probability distribution applications to models of evolutionary branching processes are discussed

brief paper a full characterization of stabilizability of bimodal piecewise linear systems with scalar inputs this paper studies open loop stabilization problem for bimodal systems with continuous vector field it is based on the earlier work of the authors on the controllability problem for the same class of systems a full characterization of stabilizability is established by presenting algebraic necessary and sufficient conditions it is also shown that controllability implies stabilizability for these systems in a very similar fashion to the linear case

the dynamics of discrete mechanical systems with perfect unilateral constraints the dynamics of discrete mechanical systems with perfect unilateral constraints is formulated in a very general setting the well posedness of the resulting evolution problem is studied it is proved that existence and uniqueness of a maximal solution is ensured provided strong assumptions are made on the regularity of the data they are supposed to be analytic simple examples show that this regularity assumption may not be relaxed sufficient conditions to ensure that the maximal solution is defined for all time are supplied the continuous dependence of the solution on initial conditions is also studied and the numerical computation of the solution is discussed

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

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

state constraints in optimal control the degeneracy phenomenon in this paper we study the degeneracy phenomenon arising in optimal control problems with state constraints it is shown that this phenomenon occurs because of the incompleteness of the standard variants of pontryagin s maximum principle for problems with state constraints the new maximum principle containing some additional information about the behavior of the hamiltonian at the endtimes is developed as application we obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle the results obtained envelope the optimal control problems with systems described by differential inclusions and ordinary differential equations

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

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

some results on the controllability of planar variational inequalities abstract this note deals with the controllability of a class of planar complementarity dynamical systems which can also be viewed as planar evolution variational inequalities it is shown that the complementarity conditions influence the controllability of the system a lot

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

algebraic necessary and sufficient conditions for the controllability of conewise linear systems the problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable this paper focuses on conewise linear systems i e systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions for this class of systems we present algebraic necessary and sufficient conditions for controllability we also show that the classical results of controllability of linear systems and input constrained linear systems can be recovered from our main result our treatment employs tools both from geometric control theory and mathematical programming

multiple time scale decomposition in cheap control problems singular control a linear quadratic regulator problem where a small parameter 2multiplies the control cost is considered as mu rightarrow 0 it exhibits a dynamic behavior with multiple time scales whereas for mu 0 it is a singular control problem at first a multiparameter singularly perturbed model of the given problem is constructed by transferring the singularity that exists in the performance index to the dynamic equations then the problem is decomposed into several subproblems of minimal order each pertaining to only one time scale the method developed permits under a single framework the characterization of asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as mu rightarrow 0 moreover the asymptotic solution brings into focus 1 the nature of singular control and its connection to transmission zeros and 2 the presence of various types of impulses and other higher order distributions in both state and control the approach unifies and extends significantly several ideas previously somewhat disjoint

on the powers of a vectorial matroid an equivalence relation abstract let v be a nonzero vector space over the field f of characteristic zero let x be a finite family of nonzero vectors of v for each a v we denote by ma the vectorial matroid of x a for each k n we denote by ma k the kth power of ma we define an equivalence relation k on v such that a kb if and only if the matroids ma k and mb k are on the same set and have the same set of independent sets we present results on this equivalence relation

an extension and analysis of the shu osher representation of runge kutta methods in the context of solving nonlinear partial differential equations shu and osher introduced representations of explicit runge kutta methods which lead to stepsize conditions under which the numerical process is total variation diminishing tvd much attention has been paid to these representations in the literature in general a shu osher representation of a given runge kutta method is not unique therefore of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them several basic questions are still open notably regarding the following issues 1 the formulation of a simple and general strategy for finding a best possible shu osher representation for any given runge kutta method 2 the question of whether the tvd property of a given runge kutta method can still be guaranteed when the stepsize condition corresponding to a best possible shu osher representation of the method is violated 3 the generalization of the shu osher approach to general possibly implicit runge kutta methods in this paper we give an extension and analysis of the original shu osher representation by means of which the above questions can be settled moreover we clarify analogous questions regarding properties which are referred to in the literature by the terms monotonicity and strong stability preserving ssp

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 survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

efficiently implementable algebraically stable runge kutta methods recent work by butcher bit 16 1976 pp 237 240 on the implementation of implicit runge kutta methods has shown that if his technique is used the most efficiently implementable methods are those whose runge kutta matrix has a single real eigenvalue in this paper therefore we analyze the algebraic stability of two families of methods with this property namely singly implicit and diagonally implicit methods

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

direct singular perturbation analysis of high gain and cheap control problems a new and direct approach to the analysis and design of high gain feedback and cheap control problems is developed transformation of such singular problems to singular perturbation models of conventional type early at the onset of analysis or design allows a direct application of the readily available singular perturbation literature the approach permits a simple characterization of a multivariable root loci under a high gain feedback and b asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as the control cost coefficient in the performance index goes to zero

implicit runge kutta methods for lipschitz continuous ordinary differential equations implicit runge kutta irk methods for solving the nonsmooth ordinary differential equation ode involve a system of nonsmooth equations we show superlinear convergence of the slanting newton method for solving the system of nonsmooth equations we prove the slanting differentiability and give a slanting function for the involved function we develop a new code based on the slanting newton method and the irk method for nonsmooth odes arising from structural oscillation and pounding we show that the new code is efficient for solving a nonsmooth ode model for the collapse of the tacoma narrows suspension bridge and simulating 13 different earthquakes

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

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

on the solutions of linear differential equations with singular coefficients abstract equations of the form e ax u with e and a square matrices and e singular are considered the controversy that exists in the literature concerning the solutions of such equations is investigated solutions are arrived at through an application of singular perturbation theory

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

path integral formulation and feynman rules for phylogenetic branching models a dynamical picture of phylogenetic evolution is given in terms of markov models on a state space comprising joint probability distributions for character types of taxonomic classes phylogenetic branching is a process which augments the number of taxa under consideration and hence the rank of the underlying joint probability state tensor we point out the combinatorial necessity for a second quantized or fock space setting incorporating discrete counting labels for taxa and character types to allow for a description in the number basis rate operators describing both time evolution without branching and also phylogenetic branching events are identified a detailed development of these ideas is given using standard transcriptions from the microscopic formulation of non equilibrium reaction diffusion or birth death processes these give the relations between stochastic rate matrices the matrix elements of the corresponding evolution operators representing them and the integral kernels needed to implement these as path integrals the free theory without branching is solved and the correct trilinear interaction terms representing branching events are presented the full model is developed in perturbation theory via the derivation of explicit feynman rules which establish that the probabilities pattern frequencies of leaf colourations arising as matrix elements of the time evolution operator are identical with those computed via the standard analysis simple examples phylogenetic trees with two or three leaves are discussed in detail further implications for the work are briefly considered including the role of time reparametrization covariance

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

popov belevitch hautus type tests for the controllability of linear complementarity systems it is well known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems this paper deals with the controllability problem of a class of piecewise linear systems known as linear complementarity systems by exploiting the underlying structure and employing the results on the controllability of the so called conewise linear systems we present a set of inequality type conditions as necessary and sufficient conditions for controllability of linear complementarity systems the presented conditions are of popov belevitch hautus type in nature

on preserving dissipativity properties of linear complementarity dynamical systems with the θ method in this work we study the following problem given a numerical method an extended theta method named the theta gamma method find the class of dissipative linear complementarity systems such that their discrete time counterpart is still dissipative with the same storage energy function supply rate reciprocal variables and dissipation function systems with continuous solutions and with state jumps are studied the notion of numerical dissipation is given a rigorous meaning

yosida moreau regularization of sweeping processes with unbounded variation lett c t be a hausdorff continuous multifunction with closed convex values in a hilbert spacehsuch thatc t has nonempty interior for allt we show that the yosida moreau regularizations of the sweeping process with moving setc t i e the solutions of formula are strongly pointwisely convergent as 0 to the solution of the corresponding sweeping process formally written as formula

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

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

the rational complementarity problem abstract an extension of the linear complementarity problem lcp of mathematical programming is the so called rational complementarity problem rcp this problem occurs if complementarity conditions are imposed on input and output variables of linear dynamical input state output systems the resulting dynamical systems are called linear complementarity systems since the rcp is crucial both in issues concerning existence and uniqueness of solutions to complementarity systems and in time simulation of complementarity systems it is worthwhile to consider existence and uniqueness questions of solutions to the rcp in this paper necessary and sufficient conditions are presented guaranteeing existence and uniqueness of solutions to the rcp in terms of corresponding lcps using these results and proving that the corresponding lcps have certain properties we can show uniqueness and existence of solutions to linear mechanical systems with unilateral constraints electrical networks with diodes and linear dynamical systems subject to relays and or coulomb friction

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

complementarity modeling of hybrid systems a complementarity framework is described for the modeling of certain classes of mixed continuous discrete dynamical systems the use of such a framework is well known for mechanical systems with inequality constraints but we give a more general formulation which also applies for instance to switching control systems the main theoretical results in the paper are concerned with uniqueness of smooth continuations the solution of this problem requires the construction of a map from the continuous state to the discrete state a crucial technical tool is the so called linear complementarity problem from mathematical programming and we introduce various generalizations of this problem

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

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

the covering number of the elements of a matroid and generalized matrix functions abstract let s be a nonempty finite set with cardinality m let m be a matroid on s with no loops the covering number of an element x in s is the smallest positive integer k such that x is a coloop of the union of k copies of m we investigate connections between the structure of m and the values of the covering numbers of elements of s applications to the study of the rank partition and generalized matrix functions are presented

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

u 1 u 1 u 1 symmetry of the kimura 3st model and phylogenetic branching processes an analysis of the kimura 3st model of dna sequence evolution is given on the basis of its continuous lie symmetries the rate matrix commutes with a u 1 u 1 u 1 phase subgroup of the group gl 4 of 4 4 invertible complex matrices acting on a linear space spanned by the four nucleic acid base letters the diagonal branching operator representing speciation is defined and shown to intertwine the u 1 u 1 u 1 action using the intertwining property a general formula for the probability density on the leaves of a binary tree under the kimura model is derived which is shown to be equivalent to established phylogenetic spectral transform methods

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

a high order path following method for locating the least 2 norm solution of monotone lcps a high order path following method is proposed for finding the least 2 norm solution of a monotone lcp this method follows the regularized central paths introduced in y b zhao and d li siam j control optim 40 2001 pp 898 924 by using the analyticity of these paths we showed the global convergence under the assumption that the lcp has at least one solution and the superlinear rate of local convergence under the assumption that the least 2 norm solution is maximally complementary