Approximately Reachable Directions for Piecewise Linear Switched Systems

TL;DR

This paper investigates the approximate reachability of states in piecewise linear switched systems with stochastic elements, introducing novel backward stochastic Riccati equations for control synthesis.

Contribution

It introduces a new approach using backward stochastic Riccati equations for controllability analysis in stochastic switched systems with non-deterministic coefficients.

Findings

Characterization of almost reachable states at fixed time T.

Existence and uniqueness of solutions to backward stochastic Riccati equations.

Development of a control synthesis method for stochastic systems.

Abstract

This paper deals with some reachability issues for piecewise linear switched systems with time-dependent coefficients and multiplicative noise. Namely, it aims at characterizing data that are almost reachable at some fixed time T > 0 (belong to the closure of the reachable set in a suitable L 2-sense). From a mathematical point of view, this provides the missing link between approximate controllability towards 0 and approximate controllability towards given targets. The methods rely on linear-quadratic control and Riccati equations. The main novelty is that we consider an LQ problem with controlled backward stochastic dynamics and, since the coefficients are not deterministic (unlike some of the cited references), neither is the backward stochastic Riccati equation. Existence and uniqueness of the solution of such equations rely on structure arguments (inspired by [7]). Besides solvability, Riccati representation of the resulting control problem is provided as is the synthesis of optimal (non-Markovian) control. Several examples are discussed.