Abstract
This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in optimization algorithms and the modeling of physical systems. The differential inclusion is described by a time-dependent set-valued mapping having the property that, for a given time instant, the set-valued mapping describes a maximal monotone operator. By successive application of a proximal operator, we construct a sequence of functions parameterized by the sampling time that corresponds to the discretization of the continuous-time system. Under certain mild assumptions on the regularity with respect to the time argument, and using appropriate tools from functional and variational analysis, this sequence is then shown to converge to the unique solution of the original differential inclusion. The result is applied to develop conditions for well-posedness of differential equations interconnected with nonsmooth time-dependent complementarity relations, using passivity of underlying dynamics (equivalently expressed in terms of linear matrix inequalities).
Highlights
The theory of monotone operators emerged as an important area of research within the field of nonlinear analysis in early 1960’s [32,40,67]
Applications of dynamical systems with maximal monotone operators range from modeling traffic equilibrium [44] to electronics [1]
This article is focused on studying maximal monotone operators in the context of mathematical models for dynamical systems, and the central object of our study is to investigate conditions for existence of solutions to the differential inclusion x ∈ −F(t, x), x(0) ∈ dom F(0, ·), (1)
Summary
The theory of monotone operators emerged as an important area of research within the field of nonlinear analysis in early 1960’s [32,40,67]. Because of the relaxed nature of assumptions, our results provide a constructive framework for studying differential equations with complementarity relations Such nonsmooth relations form a particular subclass of maximal monotone operators, and have been useful in modeling systems with piecewise affine characteristics [11,13,18]. Inspired by the result in [19], we provide conditions under which it is possible to recast the interconnection of an ordinary differential equation with time-dependent complementarity relation in the form of a differential inclusion with time-dependent maximal monotone operator, for which the existence of solutions is being studied in this article.
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