Minimal Time Impulse Control of an Evolution Equation

Abstract

The paper is concerned with a kind of minimal time control problem for the heat equation with impulse controls. The purpose of such a problem is to find an optimal impulse control (among certain control constraint set) steering the solution of the heat equation from a given initial state to a given target set as soon as possible. We will first study the existence and uniqueness of optimal solution for this problem. In the formulation of this problem, there are two parameters: one is the upper bound of the control constraint and the other one is the moment of impulse time. Then, we will establish the continuity of the minimal time function of this problem with respect to the above mentioned two parameters. Moreover, the convergence of the optimal control is also discussed.2010 Mathematics Subject Classifications. 49K20, 49J20, 93C20

Error estimates of semidiscrete finite element approximation for minimal time impulse control problems

“…Pontryagin's Maximum Principle. Similar to Theorem 2.1 in [4], we know that there exists at least one optimal control f * = {f j } ∞ j=1 ∈ F for (T P). In the following, we will present necessary and sufficient conditions of the optimal control for (T P).…”

“…We first show the proof of ( 1). The proof of the necessity is similar to Theorem 4.1 of Chapter 7 in [13] or Theorem 2.1 in [4], and we omit its proof. For the bang-bang and unique properties, we refer the readers to Theorem 2.1 in [4].…”

“…In [10], Kunisch and Rao investigated the Hamilton-Jacobi-Bellman approach for a minimal time problem with impulse controls governed by ordinary differential equations. In [4] and [24], Duan and Wang considered optimal time impulse control problems of the linear and semilinear heat equations, and proved that the optimal controls exhibit the bang-bang property.…”

Exponential stabilization on infinite dimensional system with impulse controls

“…Recently, the impulse control system has attracted a lot of attention. For example, we can refer to [5,6,15,17,18,21]. In particular, in [18], the authors studied the exponential stabilizability of finite dimensional system with impulse controls and obtained some equivalence results.…”

Constrained approximate null controllability of coupled heat equation with periodic impulse controls

“…However, in many cases, impulse control can give an efficient way to deal with systems, which can not endure distributed control inputs, or in some applications, it is impossible to provide distributed control inputs. Impulse control systems have attracted the attention of many researchers (see, for instance, [5,6,15,16,20]). In these works, controllability, stabilizability and optimal control problems of impulse control systems were studied.…”