Nonequivalence of Controllability Properties for Piecewise Linear Markov Switch Processes *

E-ISSN： 2267-3059|57|issue|37-47

ISSN： 2267-3059

Source： ESAIM: Proceedings and surveys, Vol.57, Iss.issue, 2017-10, pp. : 37-47

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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 controllability 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 [11] and [10].