Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels

Abstract

Abstract.We consider a distributed system in which the state q is governed by a parabolic equation and a pair of controls v = (h, k) where h and k play two different roles: the control k is of controllability type while h expresses that the state q does not move too far from a given state.Therefore, it is natural to introduce the control point of view. In fact, there are several ways to state and solve optimal control problems with a pair of controls h and k, in particular the Least Squares method with only one criteria for the pair (h, k) or the Pareto Optimal Control for multicriteria problems.We propose here to use the notion of Hierarchic Control. This notion assumes that we have two controls h, k where h will be the leader while k will be the follower. The main tool used to solve the null-controllability problem with constraints on the follower is an observability inequality of Carleman type which is "adapted" to the constraints.The obtained results are applied to the sentinels theory of Lions [Masson (1992)].Mathematics Subject Classification. 35K05, 35K15, 35K20, 49J20, 93B05.

Hierarchic control of a linear heat equation with missing data

“…The results were achieved by means of a Carleman inequality adapted to the constraint. Mercan revisited the notion of controllability in the sense of Stackelberg given by Nakoulima, choosing the follower of the minimal norm. This new notion is then applied by Mercan and Nakoulima for the controllability of a two‐stroke problem with the constraint on the state.…”

“…The follower acts on the system in order to bring the state not far from the desired state while the leader has to steer the state at the final time to a small neighborhood of a given state. Nakoulima used this concept control for a backward heat equation involving two controls to be determined: one of the null controllability type with a constraint on the control, called follower, and the other of optimal control type, called leader. The results were achieved by means of a Carleman inequality adapted to the constraint.…”

Control of Stackelberg for Coupled Parabolic Equations

“…The notion of controllability in the sense of Stackelberg was initiated by O. Nakoulima [8]. Actually, the author was interested in the concept of hierarchical control for a backward heat equation.…”

“…The results were achieved by means of a Carleman inequality adapted to the constraint and were applied to a problem of discriminating sentinels (see [4] for the notion of sentinel). In [9,10], M. Mercan revisited the notion of controllability in the sense of Stackelberg given by O. Nakoulima [8] by choosing the Follower of minimal norm. This new notion is then applied by M. Mercan and O. Nakoulima in [13] on the controllability of a two-stroke problem with constraint on the states.…”

“…is the solution of the null controllability problem with constraints on the state (1), (7), (8). Moreover,…”