Averaged controllability in a long time horizon

Martin Lazar (1) , Jérôme Lohéac (2)

1 University of Dubrovnik
2 IMT Atlantique - DAPI - Département Automatique, Productique et Informatique

In this article we study a linear control system with unknown parameter. However, we assume that the possible realisations of this parameter are finite and each realisation can appear with a known probability. We aim to design a control independent of the parameter such that the expectation of the system's realisation reach a given target at a final time T > 0. This problem is now well-known as averaged controllability. We extend this notion to the one of long time averaged controllability. More precisely, we consider the question: is it possible to steer the averaged of the system to some prescribed value in time T > 0 and then keep the averaged of the system at this prescribed value for all times t > T ? We provide a necessary and sufficient condition for this property to hold. Once the condition is satisfied, one can apply a feedback control that keeps the average fixed during a given time period. We also address the L 2-norm optimality of such controls. Relations between the introduced and previously existing different control notions of parameter dependent systems are discussed, accompanied by numerical examples.

Mots clés: Parameter dependent ODEs; Averaged control; Controlled invariant subspace; Linear optimal control; Open-loop control systems

Domaines: Systèmes dynamiques [math.DS]; Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Martin Lazar, Jérôme Lohéac. Averaged controllability in a long time horizon. 2018. ⟨hal-01888043v1⟩