## todist1 details:

Keywords: time optimal, analytic solution

Global classification: nonlinear

Functional: nonconvex nonlinear

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Heat:
• linear parabolic operator of order 2.
• Defined on a 2-dim domain in 2-dim space
• Time dependent.

Design constraints:

• box of order 0

State constraints:

• nonlinear convex, local of order 1

Mixed constraints:

• none

• Description (pdf)
• Bibliography (bib)
• No MATLAB data available.

Submitted on 2018-04-20 by Lucas Bonifacius. Published on 2019-03-11

## todist1 description:

### 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].

### 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

 ($\stackrel{̂}{P}$)

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

### Optimality System

The ﬁrst-order necessary optimality conditions for ($\stackrel{̂}{P}$) are formally given as follows: for given local minimizers $\overline{q}\in \stackrel{̂}{Q}$, $\overline{u}\in \stackrel{̂}{U}$, $\overline{T}>0$ 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

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

 $-{\partial }_{t}\overline{z}\left(t\right)-\overline{T}△\overline{z}\left(t\right)=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right)\phantom{\rule{1em}{0ex}}\overline{z}\left(1\right)=\overline{\mu }\phantom{\rule{0.3em}{0ex}}\left(\right\overline{u}\left(1\right)-{u}_{d}\left)\right.$

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 ($\stackrel{̂}{P}$) are

 $\begin{array}{ccc}\hfill \overline{u}\left(t,x\right)& =\left({c}_{0}{e}^{-\overline{T}t}+{c}_{1}{e}^{\overline{T}\left(t-1\right)}\right){u}_{0}\left(x\right),\hfill & \hfill \\ \hfill \overline{z}\left(t,x\right)& =\overline{\mu }\phantom{\rule{0.3em}{0ex}}{e}^{\overline{T}\left(t-1\right)}{u}_{0}\left(x\right),\hfill \\ \hfill \overline{q}\left(t,x\right)& =\frac{-\overline{\mu }}{\alpha }{e}^{\overline{T}\left(t-1\right)}{u}_{0}\left(x\right),\hfill \end{array}$

where the parameters ${c}_{0}$ and ${c}_{1}$, the optimal time $\overline{T}$ and multiplier $\overline{\mu }$ are given by

 $\begin{array}{ccccc}\hfill {c}_{0}& =\frac{1}{2}\left(\sqrt{\frac{\alpha +8}{\alpha }}+1\right),\hfill & \hfill {c}_{1}& =\frac{1}{2}\left(-\sqrt{\frac{\alpha +8}{\alpha }}-1\right),\hfill & \hfill \\ \hfill \overline{T}& =log\left(\frac{\sqrt{\frac{\alpha +8}{\alpha }}+1}{\sqrt{\frac{\alpha +8}{\alpha }}-1}\right),\hfill & \hfill \overline{\mu }& =\left(\sqrt{\frac{8}{\alpha }+1}+1\right)\alpha .\hfill \end{array}$

### References

L. Bonifacius, K. Pieper, and B. Vexler. A priori error estimates for space-time ﬁnite element discretization of parabolic time-optimal control problems. ArXiv e-prints, February 2018. URL https://arxiv.org/abs/1802.00611.