Algebraic invariance conditions in the study of approximate (null-)controllability of Markov switch processes

Abstract

We aim at studying approximate null-controllability properties of a particular class of piecewise linear Markov processes (Markovian switch systems). The criteria are given in terms of algebraic invariance and are easily computable. We propose several necessary conditions and a sufficient one. The hierarchy between these conditions is studied via suitable counterexamples. Equivalence criteria are given in abstract form for general dynamics and algebraic form for systems with constant coefficients or continuous switching. The problem is motivated by the study of lysis phenomena in biological organisms and price prediction on spikedriven commodities.

Approximately reachable directions for piecewise linear switched systems

“…However, approximate reachability (or approximate terminal controllability) intrinsically requires the initial data to vary. For the frameworks studied in [15], we show, by means of examples, that approximate terminal controllability (to every target) can hold without (global) approximate null-controllability. This seems to suggest that while approximate controllability (at least for the given frameworks) is captured by deterministic-related concepts (Kalman criterion, invariance, etc.…”

“…Second, for continuous switching systems (cf. [15,Section 4.2]), approximate null-controllability (starting from every x ∈ R n ) and approximate controllability are equivalent. The criterion is a Kalman-type condition holding true for every deterministic component of the dynamics i.e.…”

“…For constant coefficients A, B, C, approximate and approximate null-controllability (for all initial data) are also equivalent (cf. [15,Section 4.2]). The criterion is again deterministic (reducing to invariant subspaces of the kernel ker (B * )).…”

“…[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).…”

“…Links with the LQ problem and synthesis of the optimal control are given in the first main result Theorem 4.3. The proofs rely on the structural representations (via a system of iterated deterministic Riccati equations (15)) in Section 5 (see Proposition 5.2). The second main result (Theorem 5.3) provides qualitative properties (existence, uniqueness, positiveness, lower and upper-bounds, monotony) of the equations intervening in (15).…”