On the Equivalence of Minimal Time and Minimal Norm Controls for Internally Controlled Heat Equations

Abstract

We prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the euclidean space with homogeneous Dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves. We consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time. As a consequence of this equivalence, using the well-known variational characterization of minimal norm controls, we establish necessary and sufficient conditions for the minimal time and the corresponding control.

Time optimal control of the heat equation with pointwise control constraints

“…In these works, necessary conditions for time optimal control were given. To the best of our knowledge, for time optimal control problems governed by parabolic equations, there are very few results on sufficient conditions for optimal time and optimal controls, see however [8,9,22]. In [8,9] the control acted globally onto the equation and the target set was a point.…”

“…Moreover, the initial value of the state equation or the target point satisfied some special properties. In [22] the authors obtained necessary and sufficient conditions for the optimal time associated controls for the heat equation, by establishing connections between time and norm optimal control problems. The above-mentioned contributions are concerned with the second version of the time optimal control problem.…”

“…The above-mentioned contributions are concerned with the second version of the time optimal control problem. The idea of our paper utilizes the approach from [22]. More precisely, for (P 1 ) and (P 2 ), we introduce the norm optimal control problems…”

“…By establishing the connections between (P 1 ) and (P τ * nm ), (P 2 ) and (P nmT * ), respectively, as well as strict monotonicity of N * ∞ (τ ) andN * ∞ (T ), necessary and sufficient conditions for optimal time and optimal control of (P i ) are obtained. However, there are some main differences between [22] and our paper: (i) the time optimal control problem in [22] is of the second version, while we consider two versions of time optimal control problems. (ii) The methods for the study of the connections between time and norm optimal control problems are different.…”

“…(ii) The methods for the study of the connections between time and norm optimal control problems are different. In [22], the analysis builds on the study of the optimal time T * as a function of control bound M , while we start by studying the relation of M i (i = 1, 2) and the minimum of the corresponding norm optimal control problem of (P i ). (iii) In our paper, the control constraint is in pointwise form and the target set is a closed ball in C 0 (Ω), while in [22], the control constraint is in integral form and the target set is 0.…”