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,
In this paper, we derive a bang-bang property of a kind of time optimal control problem for some semilinear heat equation on bounded C 2 domains (of the Euclidean space), with homogeneous Dirichlet boundary condition and controls distributed on an open and nonempty subset of the domain where the equation evolves.2010 Mathematics Subject Classifications. 35K58, 49J20, 49J30, 93B07Keywords. time optimal controls, bang-bang property, semilinear heat equation, observability estimate from measurable sets 1 IntroductionIn this paper, we assume that Ω is a bounded domain in R d , d ≥ 1, with a C 2 boundary ∂Ω; ω is an open and non-empty subset of Ω; χ ω is the characteristic function of the set ω. Define, for an arbitrarily fixed M > 0, q ∈ (d, ∞) if d ≥ 2 and q ∈ [2, ∞) if d = 1, a control constraint set:Let f : R → R be a function holding the following properties:We arbitrarily fix a y 0 ∈ C 0 (Ω) \ {0}. Here C 0 (Ω) {y ∈ C(Ω) : y = 0 on ∂Ω}.
This paper is concerned with the bang-bang property of time optimal controls, governed by a semilinear heat equation in a bounded domain with switching controls acting locally into two open subsets. The proofs rely on an observability estimate from a positive measurable set in time for the linear heat equation, and a Kakutani fixed point argument. Keywords Time optimal control • Bang-bang property • Semilinear heat equation • Observability estimate from measurable sets • Switching control Mathematics Subject Classifications 35K58 • 49J20 • 49J30 • 93B07 Communicated by Viorel Barbu.
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.
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