Optimal control problem for exact synchronization of parabolic system

Abstract

This paper studies a kind of minimal time control problems related to the exact synchronization for a controlled linear system of parabolic equations. Each problem depends on two parameters: the bound of controls and the initial state. The purpose of such a problem is to find a control (from a constraint set) synchronizing components of the corresponding solution vector for the controlled system in the shortest time. In this paper, we build up a necessary and sufficient condition for the optimal time and the optimal control; we also obtain how the existence of optimal controls depends on the above mentioned two parameters.2010 Mathematics Subject Classifications. 49K20, 93B05, 93B07, 93C20 y(t; y 0 , u) = (y 1 (t; y 0 , u), y 2 (t; y 0 , u), . . . , y n (t; y 0 , u)) ⊤ for the solution of the system (1.1). (Here and throughout this paper, we denote the transposition of a matrix J by J ⊤ .) It is well known that for each T > 0, y(·; y 0 , u) ∈ W 1,2 (0, T ; H −1 (Ω) n ) ∩ L 2 (0, T ; H 1 0 (Ω) n ) ⊆ C([0, T ]; L 2 (Ω) n ). We will treat this solution as a function from [0, +∞) to L 2 (Ω) n .We next define control constraint set U M (with M > 0) and the target set S as follows:S {(y 1 , y 2 , . . . , y n ) ⊤ ∈ L 2 (Ω) n : y 1 = y 2 = · · · = y n }.Given M > 0, y 0 ∈ L 2 (Ω) n , we define the minimal time control problem (T P ) y 0 M :T (M, y 0 ) inf u∈U M {T ≥ 0 : u(·) = 0 and y(·; y 0 , u) ∈ S over [T, +∞)}.About Problem (T P ) y 0 M , several notes are given in order: (a 1 ) We call T (M, y 0 ) the optimal time; we call u ∈ U M an admissible control if there is T ≥ 0 so that u(·) = 0 and y(·; y 0 , u) ∈ S over [T, +∞); we call u * ∈ U M an optimal control if T (M, y 0 ) < +∞, u * (·) = 0 and y(·; y 0 , u * ) ∈ S over [T (M, y 0 ), +∞); we agree that T (M, y 0 ) = +∞ if the problem (T P ) y 0 M has no admissible control.(a 2 ) One can easily check that if y = (y 1 , y 2 , . . . , y n ) ⊤ ∈ L 2 (Ω) n , then y ∈ S if and only if Dy = 0, where and throughout this paper,

Minimal time control of exact synchronization for parabolic systems

“…The exact synchronization in the PDEs case was first studied for a coupled system of wave equations both for the higher-dimensional case in the framework of weak solutions in [15,16,19], and for the one-dimensional case in the framework of classical solutions in [11,17]. Recently, Pontryagin's maximum principle of optimal control problems for the exact synchronization of parabolic systems was studied in [34].…”

“…There is a pair (A, B) satisfying (H 2 ). For example, In [34], we showed that the system (1.1) is exactly synchronizable at each time T if and only if either the following ( H 1 ) or ( H 2 ) is true: ( H 1 ) The pair (A, B) satisfies (1.6).…”

“…( H 2 ) The pair (A, B) does not satisfy (1.6), but satisfies (1.4) and (1.3). The difference between [34] and this paper is as follows: In [34], the research is carried out from the perspective of rank condition: rank(B, AB, . .…”

Convergence of simultaneous distributed-boundary parabolic optimal control problems

“…In [7] and [8], were studied control problems on the source g and the flux q respectively, for parabolic variational inequalities of second kind. Other papers on the subject are [12,13,14,18,19,20,21,25,26,27]. Our interest is the convergence when α → ∞, which is related to [4,22,23].…”

Minimal Time Impulse Control of an Evolution Equation

“…In most papers concerning minimal time control problems, people can provide necessary conditions for optimal controls, i.e., Pontryagin's maximum principle (see, for instance, [2], [11] and [14]). In some specific situations, people can also give characteristics for the optimal time, as well as the optimal control for a minimal time control problem (see, for instance, [12], [23] and [25]). We refer the reader to Remark 2.2 for a characteristic for the optimal control of the problem (T P ) τ M .…”