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Approximately reachable directions for piecewise linear switched systems

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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 \({\mathbb {L}}^2\)-sense). From a mathematical point of view, this provides the missing link between approximate controllability toward 0 and approximate controllability toward 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 Confortola (Ann Appl Probab 26(3):1743–1773, 2016)]. Besides solvability, Riccati representation of the resulting control problem is provided as is the synthesis of optimal (non-Markovian) control. Several examples are discussed.

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Notes

  1. The current literature on BSDE is highly rich but, for our linear context, it seems superfluous to mention recent advances on discontinuity, terminal irregularity, super-linear coefficient growth, etc.

  2. The reader is invited to note that we make a slight abuse of notation by considering a measurable function \(A: E \longrightarrow {\mathbb {R}}\) and then setting \(A(\omega ,t):=A\left( \varGamma _t(\omega )\right) \). This kind of abuse of notation will be employed several times in the sequel (particularly for examples).

  3. Or, equivalently, the system (6) is approximately terminal controllable in time t to \(\xi \).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University at Weihai, Weihai, 264209, P.R. China

    Dan Goreac

  2. LAMA (UMR 8050), UPEMLV, UPEC, CNRS, Université Paris-Est, 77454, Marne-la-Vallée, France

    Dan Goreac

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Correspondence to Dan Goreac.

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Goreac, D. Approximately reachable directions for piecewise linear switched systems. Math. Control Signals Syst. 31, 333–362 (2019). https://doi.org/10.1007/s00498-019-0240-x

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