Time optimal sampled-data controls for the heat equation

Abstract

In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls. Our design is reasonable from perspective of sampled-data controls. And it might provide a right way for the numerical approach of a time optimal distributed control problem, via the corresponding semi-discretized (in time variable) time optimal control problem.The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem. And obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but also prove that such estimates are optimal in some sense.

Output feedback stabilization for heat equations with sampled-data controls

“…(iii) A similar observability inequality to (2.12) was obtained in [36] (see Lemma 2.3 in [36]): There exists C = C(Ω, ω) so that for any T , S, with 0 < S < T ,…”

“…where the functional G : X M → R is defined by Proof. (i) By (2.13), we can use a standard method (see, for instance, the proof of Theorem 4.3 in [36]) to prove that G is coercive and strictly convex. This implies that G has an unique minimizer (see Theorem 2D in [37]).…”

Minimal Time Impulse Control of an Evolution Equation

“…Among the existing literature on the minimal time control problem for the evolution system, the control inputs are usually distributed in the whole time interval, i.e., they may affect the control system at each instant of time (see, for instance, [8], [16], [17], [20], [21], [24] and [28]). We also refer the readers to the minimal time sampled control problem of the heat equation (see, for instance, [1], [7], [9], [10], [13] and [22]). However, in many practical applications, it is much more convenient to use impulse controls (see, for instance, [4], [19], [26] and [27]).…”