A Proximal Point Based Approach to Optimal Control of Affine Switched Systems

Abstract

This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider switched control systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the dynamical models under consideration makes it possible to establish some continuity/approximability properties and to specify these models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problem (OCP) corresponds to a conventional convex optimization problem in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on the gradient-type method and classic proximal point techniques.

Optimal fixed-levels control for nonlinear systems with quadratic cost-functionals

“…Note that OCP constitutes a convex minimization problem in a real Hilbert space. We refer to for the general theory of convex OCPs with ordinary differential equations. The approximability property of the fully relaxed OCP can be expressed as follows : This is simply a consequence of the following simple facts: where J ∗ ( u (·)) is a conjugate of J ( u (·)).…”

“…The nonlinear systems we study can be interpreted as a particular family of the general switched systems with the time‐driven location transitions. We refer to for the basic concepts and some technical details of the generic switched systems theory.…”

Optimal LQ-Type Switched Control Design for a Class of Linear Systems with Piecewise Constant Inputs

“…As we can see (8) is a convex relaxation of the initial OCP (6). The proved convexity of OCP (8) makes it possible to apply the powerful numerical convex programming approaches to this auxiliary optimization problem.…”

“…The non-stationary linear systems we study in this paper include a particular family of switched systems with the time-driven location transitions. We refer to [6,11,27,35,39] for the basic concepts and some technical details.…”

Some results of studies on the stability and controlability properties of the switched systems of differential equations

“…However, there have also been dedicated research efforts. In this sense we can highlight the paper [1], which focuses in the regularization technique of the proximal point for a class of optimal control processes, governed by affine switched systems. Here switching of the control systems described by nonlinear ordinary differential equations, which are affine respect to the input is considered.…”