Nonequivalence of Controllability Properties for Piecewise Linear Markov Switch Processes

TL;DR

This paper investigates controllability properties of switch-dependent piecewise linear Markov processes, revealing that exact null-controllability is stronger than approximate null-controllability and that no hierarchy exists between approximate and exact controllability.

Contribution

It establishes a controllability metric linked to backward stochastic Riccati equations and demonstrates the non-equivalence of various controllability notions in switch processes.

Findings

Exact null-controllability induces a controllability metric.

Exact null-controllability is strictly stronger than approximate null-controllability.

No hierarchy exists between approximate and exact null-controllability.

Abstract

In this paper we study the exact null-controllability property for a class of controlled PDMP of switch type with switch-dependent, piecewise linear dynamics and multiplicative jumps. First, we show that exact null-controllability induces a con-trollability metric. This metric is linked to a class of backward stochastic Riccati equations. Using arguments similar to the euclidian-valued BSDE in [4], the equation is shown to be equivalent to an iterative family of deterministic Riccati equations that are solvable. Second, we give an example showing that, for switch-dependent coefficients, exact null-controllability is strictly stronger than approximate null-controllability. Finally, we show by convenient examples that no hierarchy holds between approximate (full) controllability and exact null-controllability. The paper is intended as a complement to [15] and [14].