Approximate and approximate null-controllability of a class of piecewise linear Markov switch systems

Abstract

We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in [13]. The proof relies on recent results in [4] allowing to reduce the dual stochastic backward system to a family of ordinary differential equations. Second, we prove by examples that the notion of approximate controllability is strictly stronger than approximate null-controllability. A sufficient criterion for this stronger notion is also provided. The results are illustrated on a model derived from repressed bacterium operon (given in [19] and reduced in [5]).

Nonequivalence of Controllability Properties for Piecewise Linear Markov Switch Processes

“…The proof is quasi-identical to the duality arguments in [ In the remaining of the section, unless stated otherwise, we assume the control matrix B to be modeindependent (constant). Using the explicit construction of BSDE with respect to marked-point processes, an invariance (algebraic) necessary and sufficient criterion for approximate null-controllability has been given in [15,Theorem 6]. We recall the following invariance concepts (cf.…”

“…We construct a mode-indexed family of linear subspaces of R N denoted by V M,n Here, Π V denotes the orthogonal projection operator onto the linear space V ⊂ R N . The explicit criterion is the following In the same paper [15], the property of approximate null-controllability for general systems is shown (using convenient examples) to be strictly weaker than approximate controllability. The following sufficient criterion is proven to guarantee the approximate controllability.…”

“…The controllability notions (exact null, approximate null, approximate) are given in Section 2.3. We gather in the same section some useful results on approximate and approximate null-controllability given in [14] and [15]. The main results on Riccati BSDEs are given in Section 3.1.…”

“…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]. Résumé Nousétudions la propriété de zéro-contrôlabilité exacte pour une classe de processus de type Markovienà switch ayant des dynamiques linéaires par morceauxà coefficients switchés et bruit multiplicatif.…”

Approximately reachable directions for piecewise linear switched systems

“…[12], [28], [1], [22], etc.). In the recent papers [15] and [14] we have addressed the approximate controllability problem with switched dynamics (much like those presented at the very beginning). In particular, [14] gives a complete criterion allowing to control the system around 0 (approximate null-controllability).…”