Equivalence between Minimal Time and Minimal Norm Control Problems for the Heat Equation

Abstract

This paper presents the equivalence between minimal time and minimal norm control problems for internally controlled heat equations. The target is an arbitrarily fixed bounded, closed and convex set with a nonempty interior in the state space. This study differs from [G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012Optim., 50 ( ), pp. 2938Optim., 50 ( -2958 where the target set is the origin in the state space. When the target set is the origin or a ball, centered at the origin, the minimal norm and the minimal time functions are continuous and strictly decreasing, and they are inverses of each other. However, when the target is located in other place of the state space, the minimal norm function may be no longer monotonous and the range of the minimal time function may not be connected. These cause the main difficulty in our study. We overcome this difficulty by borrowing some idea from the classical raising sun lemma (see, for instance, Lemma 3.5 and Figure 5 on Pages 121-122 in [E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005]). Then the following conclusions are true: (i) When (M, T ) ∈ (GT ) y0,Q \ (KN ) y0,Q , problems (T P ) M,y0 Q and (N P ) T,y0 Q are equivalent and the null controls (over R + and (0, T ), respectively) are not the minimal time control and the minimal norm control to these two problems, respectively. (ii) When (M, T ) ∈ (KN ) y0,Q , problems (T P ) M,y0 Q and (N P ) T,y0 Q are equivalent and the null controls (over R + and (0, T ), respectively) are the unique minimal time control and the unique minimal norm control to these two problems, respectively. (iii) When (M, T ) ∈ [0, +∞) × (0, +∞) \ (GT ) y0,Q ∪ (KN ) y0,Q , problems (T P ) M,y0 Q and (N P ) T,y0 Q are not equivalent.Several notes are given in order. (a) Minimal time control problems and minimal norm control problems are two kinds of important optimal control problems in control theory. The equivalence between these two kinds of problems plays an important role in the studies of these problems. To our best knowledge, in the existing literatures on such equivalence (see,

Equivalence between minimal time control problems and minimal norm control problems

“…这两个系统除了定义区 间不同外, 长得一样. 关于上述系统, 我们作如下假设: [22] 给出, 其中条件比 (H 1 )-(H 3 ) 强, 例如, 要求 A 生成一个解 析半群. 后来, 我们在撰写专著 [24] 时, 认识到 (H 1 )-(H 3 ) 可以推出定理 2.2 中的等价性结果.…”

“…第三步, 通过对这个泛函性质 的深入研究得到 (T P ) M y0 问题所需性质. 这一策略在文献 [12] 中首次被提出, 后来被逐步发展 [14][15][16][17][18][19][20][21][22][23][24] . 为说明以上策略的效益, 我们考虑具有内部控制的热传导方程.…”

“…Qin 和 Wang [22] 给出了一个新方法, 由此导出一般抽象空间中时不变发展方程在目标为一般有界、凸 闭且含内点时的最小范数控制问题与最短时间控制问题的等价性定理. 当控制系统为时变且目标集为 原点 (或不含内点的集合) 时, 问题要复杂许多.…”

Time-optimality by distance-optimality for parabolic control systems

“…To explain the approach, we consider the time optimal control problem (P ρ ) inf To follow much of the literature, we adapted the control constraints to be chosen in L ∞ ((0, T ); L 2 (ω)) rather than L ∞ ((0, T ) × ω). Under appropriate controllability assumptions these two problems have been shown to be equivalent; see [17,8,10,36,30] for parabolic equations, [38] for time-varying ordinary differential equations, [34,Chapter 5] for abstract evolution equations, and [39] for the Schrödinger equation. Note that typically these publications consider the case of exact controllability, i.e.…”

“…see [30,Section 4], compare also [9,Section 1.7]. Ifφ T is the minimizer of (1.1), then the minimum norm control is given byū…”