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 varying in time only. The exact solution is unknown, but numerical values are provided.

The problem has been used as numerical test in [Bonifacius et al., 2018a, Example 5.2].

Variables & Notation

Unknowns

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

Given Data

$$\begin{array}{ccccc}\hfill \Omega & ={\left(0,1\right)}^2\hfill & \hfill & \text{spatial domain}\hfill & \hfill \\ \hfill u_d& =0\in H_0^1\left(\Omega \right)\hfill & \hfill & \text{desired state}\hfill \\ \hfill {\delta }_0& =\frac{1}{10}>0\hfill & \hfill & \text{tolerance to desired state}\hfill \\ \hfill \alpha & \ge 0\hfill & \hfill & \text{control cost parameter (arbitrary)}\hfill \\ \hfill u_0& =4\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_1^2\right)\sin <!--FUNCTION\; APPLICATION-->\left(\pi x_2^3\right)\hfill & \hfill & \text{initial state}\hfill \\ \hfill c& =0.03\hfill & \hfill & \text{coefficient in the PDE}\hfill \\ \hfill q_a& =-1.5\hfill & \hfill & \text{lower control bound}\hfill \\ \hfill q_b& =0\hfill & \hfill & \text{upper control bound}\hfill \\ \hfill {\omega }_1& =\left(0,0.5\right)\times \left(0,1\right)\hfill & \hfill & \text{control domain 1}\hfill \\ \hfill {\omega }_2& =\left(0.5,1\right)\times \left(0,0.5\right)\hfill & \hfill & \text{control domain 2}\hfill \\ \hfill \end{array}$$

The control-action operator is defined as

$$\begin{array}{llll}\hfill B:{ℝ}^2& \to L^2\left(\Omega \right),& \hfill & \\ \hfill q=\left(q_1,q_2\right)& \mapsto Bq=q_1{\chi }_{{\omega }_1}+q_2{\chi }_{{\omega }_2}& \hfill & \end{array}$$

where ${\chi }_{{\omega }_1}$ and ${\chi }_{{\omega }_2}$ denote the characteristic functions on ${\omega }_1$ and ${\omega }_2$.

Problem Description

($P$)

The state equation is transformed to the reference time interval $\left(0,1\right)$ in order to deal with the variable time horizon; see [Bonifacius et al., 2018a, 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 $\hat{q}\in \hat{Q}=L^{\infty }\left(\left(0,1\right);{ℝ}^2\right)$ 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

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

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

(0.1)

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

Supplementary Material

For the example, no analytical solution is known. However, numerical values from [Bonifacius et al., 2018a, Example 5.2] are provided. The state and adjoint state equations are discretized by means of the discontinuous Galerkin scheme in time (corresponding to a version of the implicit Euler method) and linear finite elements in space. This scheme is guaranteed to converge with a rate $|\log <!--FUNCTION\; APPLICATION-->k|\left(k+h^2\right)$ with $k$ denoting the temporal mesh size and $h$ the spatial mesh size; cf. [Bonifacius et al., 2018a, Corollary 4.16]. For further details on the implementation we refer to [Bonifacius et al., 2018a, Section 5].

The following table provides results for [Bonifacius et al., 2018a, Example 5.2] and they were provided by the authors for different values of the control cost parameter $\alpha$, number of time steps $M$ and number of spatial nodes $N$. The analysis for the case $\alpha =0$ can be found in Bonifacius et al. [2018b].

References