Linear passive systems and maximal monotone mappings

Abstract

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. Mathematics Subject Classification

Differential-algebraic inclusions with maximal monotone operators

“…In such cases, it often happens that the set-valued mappings that describe the maximal monotone relation are multliplied by non-square matrices which do not render the resulting map maximal monotone. This problem is solved in the literature by imposing a passivity relation between the variables constrained by the maximal monotone relation [2], [5]. This allows for rewriting the entire system dynamics as a DI with maximal monotone relation.…”

“…When E = I, it was shown in [5] that under passivity assumption on the quadruple (A, B, C, D), the system (1) can be rewritten as a differential inclusion of the formẋ ∈ −M(x) for some maximal monotone M. One can then invoke the classical results from [3] to obtain existence and uniqueness of the solutions. Trying to generalize this idea for a singular E matrix does not always lead to a differential inclusion with maximal monotone operators.…”

“…It is shown in this paper that under appropriate passivity assumptions on the matrix quintuple (E, A, B, C, D), system (2) can be rewritten in the form of inclusion (1), hence the solution theory for system (1) is also very relevant for studying the dynamical system (2). While system (2) already presents a generalization from the system class studied in [5], the utility of system class (2) can also be found in the modeling of electrical circuits with nonsmooth devices such as diodes and switches. An example is included in Section V for this purpose.…”

Set-Valued Sliding-Mode Control of Uncertain Linear Systems: Continuous and Discrete-Time Analysis

“…In the case when the function γ is constant, the differential inclusion (23) belongs to the class of differential inclusions with maximal monotone right-hand side for which numerous results have been proposed, see e.g., [4,6,9,10,12,33,35] and it embraces several mathematical formulations [11]. The existence and uniqueness of solutions of (23) for the case where γ is constant has been studied for several conditions imposed on the termŵ +φ, see e.g., [9,12,15].…”

“…The existence and uniqueness of solutions of (23) for the case where γ is constant has been studied for several conditions imposed on the termŵ +φ, see e.g., [9,12,15]. For a solution of (23) we mean an absolutely continuous function σ : R + → R m that satisfies σ(0) = σ 0 ∈ Dom M together with (23) almost everywhere on [0, +∞), that is, we consider solutions of differential inclusion (23) in the sense of Caratheodory [18].…”