Robust Non-Zenoness of Piecewise Affine Systems with Applications to Linear Complementarity Systems

Abstract 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 to show robust non-Zenoness of several classes of Lipschitz linear complementarity systems in different switching notions. The paper also develops partial results for robust non-Zenoness of non-Lipschitz PASs, particularly well-posed bimodal non-Lipschitz PASs

Stability analysis of partial differential variational inequalities in Banach spaces

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques

A class of differential hemivariational inequalities in Banach spaces

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function u↦f(t,x,u) and compactness of C0-semigroup eA(t)

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.</p

Differential variational inequalities

International audienceThis 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-dimensiona

Design of observers and certainty equivalence controllers for differential inclusions: a passivity approach

This paper deals with the design of state observers for a class of systems with set-valued right-hand-sides. Two main classes of set-valued mappings are considered: maximal monotone and time-invariant; or normal cones to possibly time-varying sets (not necessarily convex). Such controlled systems can be embedded in various kinds of differential inclusions with absolutely continuous and bounded variation solutions (inclusions with maximal monotone multivalued mappings, and the perturbed Moreau's sweeping process), using a suitable state space transformation. The well-posedness properties of the observers dynamics are carefully analyzed. The stability of the observation error is proved. Then the problem of stabilization with a feedback control using the estimated state is studied in a particular case. Most of the results rely on rendering some suitable operator dissipative (or SPR in the linear case)

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers