Departmentof Mathematics

Introduction

We have a simple example of a time-optimal control problem subject to the linear heat equation and pointwise bound constraints on the control. The goal is to steer the heat equation into an $L^2$-ball centered at some desired state in the shortest time possible by an appropriate choice of the control. The time-optimal control problem can be transformed to a fixed time interval and both versions are given below.

This particular problem utilizes a control function distributed in space and time. It features an analytically given optimal solution.

The problem has been used as a numerical test in [Bonifacius et al., 2018, Example 5.1].

Variables & Notation

Unknowns

$$\begin{array}{ccc}\hfill q\in Q=L^2\left(\left(0,T\right);L^2\left(\Omega \right)\right)& \text{control variable}\hfill & \hfill \\ \hfill u\in U=H^1\left(\left(0,T\right);L^2\left(\Omega \right)\right)\cap L^2\left(\left(0,T\right);H_0^1\left(\Omega \right)\cap H^2\left(\Omega \right)\right)& \text{state variable}\hfill \\ \hfill T& \text{terminal time}\hfill \end{array}$$

Given Data

$$\begin{array}{ccccc}\hfill \Omega & ={\left(0,1\right)}^2\hfill & \hfill & \text{spatial domain}\hfill & \hfill \\ \hfill u_d& =-2\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_2\right)\in H_0^1\left(\Omega \right)\hfill & \hfill & \text{desired state}\hfill \\ \hfill {\delta }_0& =\frac{1}{2}>0\hfill & \hfill & \text{tolerance to desired state}\hfill \\ \hfill \alpha & >0\hfill & \hfill & \text{control cost parameter (arbitrary)}\hfill \\ \hfill u_0& =\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_1\right)\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_2\right)\hfill & \hfill & \text{initial data}\hfill \\ \hfill c& =\frac{1}{2{\pi }^2}\hfill & \hfill & \text{coefficient in the PDE}\hfill \end{array}$$

Problem Description

($P$)

The state equation can be transformed to the reference time interval $\left(0,1\right)$ in order to deal with the variable time horizon; see [Bonifacius et al., 2018, Section 3.1] for details. Thus, the transformed version of ($P$) reads

($\hat{P}$)

Note that the problems ($P$) and ($\hat{P}$) are equivalent. The unknowns for the transformed problem ($\hat{P}$) are $\hat{q}\in \hat{Q}=L^2\left(\left(0,1\right);L^2\left(\Omega \right)\right)$ and $\hat{u}\in \hat{U}=H^1\left(\left(0,1\right);L^2\left(\Omega \right)\right)\cap L^2\left(\left(0,1\right);H_0^1\left(\Omega \right)\cap H^2\left(\Omega \right)\right)$.

Optimality System

The first-order necessary optimality conditions for ($\hat{P}$) are formally given as follows: for given local minimizers $\overline{q}\in \hat{Q}$, $\overline{u}\in \hat{U}$, $\overline{T}>0$ there exists Lagrange multipliers $\overline{\mu }>0$ and $\overline{z}\in W\left(0,1\right)=\left\{v\in L^2\left(0,1;H_0^1\left(\Omega \right)\right):{\partial }_{t}v\in L^2\left(0,1;H^{-1}\left(\Omega \right)\right)\right\}$ such that

$$ ∫ 011 + α 2 ∥q¯(t)∥L2(Ω)2 +(q¯(t) + c△u¯(t),z¯(t)) L2(Ω)dt = 0, αq¯(t) + z¯(t) = 0, ∥u¯(1) − ud∥L2(Ω) = δ0, $$

and the adjoint state $\overline{z}\in W\left(0,1\right)$ is determined by

$$-{\partial }_t\overline{z}\left(t\right)-\overline{T}△\overline{z}\left(t\right)=0,t\in \left(0,1\right)\overline{z}\left(1\right)=\overline{\mu }\left(\right.\overline{u}\left(1\right)-u_d\left)\right..$$

It can be shown that the above optimality conditions are satisfied in the given example, see, [Bonifacius et al., 2018, Theorem 3.10].

Supplementary Material

For this example, a local optimal solution (with verified second-order sufficient optimality conditions) is known analytically and given in [Bonifacius et al., 2018, Example 5.1]. The optimal state, adjoint state, and control for the transformed problem ($\hat{P}$) are

$$\begin{array}{ccc}\hfill \overline{u}\left(t,x\right)& =\left(c_0{e}^{-\overline{T}t}+c_1{e}^{\overline{T}\left(t-1\right)}\right)u_0\left(x\right),\hfill & \hfill \\ \hfill \overline{z}\left(t,x\right)& =\overline{\mu }{e}^{\overline{T}\left(t-1\right)}{u}_0\left(x\right),\hfill \\ \hfill \overline{q}\left(t,x\right)& =\frac{-\overline{\mu }}{\alpha }{e}^{\overline{T}\left(t-1\right)}{u}_0\left(x\right),\hfill \end{array}$$

where the parameters $c_0$ and $c_1$, the optimal time $\overline{T}$ and multiplier $\overline{\mu }$ are given by

$$\begin{array}{ccccc}\hfill c_0& =\frac{1}{2}\left(\sqrt{\frac{\alpha +8}{\alpha }}+1\right),\hfill & \hfill c_1& =\frac{1}{2}\left(-\sqrt{\frac{\alpha +8}{\alpha }}-1\right),\hfill & \hfill \\ \hfill \overline{T}& =\log <!--FUNCTION\; APPLICATION-->\left(\frac{\sqrt{\frac{\alpha +8}{\alpha }}+1}{\sqrt{\frac{\alpha +8}{\alpha }}-1}\right),\hfill & \hfill \overline{\mu }& =\left(\sqrt{\frac{8}{\alpha }+1}+1\right)\alpha .\hfill \end{array}$$

References