## todist2 details:

Keywords: time optimal

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

## todist2 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 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

#### Given Data

The control-action operator is deﬁned as

$\begin{array}{llll}\hfill B:{ℝ}^{2}& \to {L}^{2}\left(\Omega \right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill q=\left({q}_{1},{q}_{2}\right)& ↦Bq={q}_{1}{\chi }_{{\omega }_{1}}+{q}_{2}{\chi }_{{\omega }_{2}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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

 ($\stackrel{̂}{P}$)

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

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

 $-{\partial }_{t}\overline{z}\left(t\right)-\overline{T}△\overline{z}\left(t\right)=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right)\phantom{\rule{1em}{0ex}}\overline{z}\left(1\right)=\overline{\mu }\phantom{\rule{0.3em}{0ex}}\left(\right\overline{u}\left(1\right)-{u}_{d}\left)\right.$ (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 $|\mathrm{log}k|\left(k+{h}^{2}\right)$ 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].

$\overline{T}$
 $\alpha =10$ $\alpha =1$ $\alpha =0.1$ $\alpha =0.01$ $\alpha =0.001$ $\alpha =0$ $M$ $N$ $640$ $289$ $2.605661$ $2.075153$ $1.845201$ $1.808456$ $1.808257$ $1.808255$ $1280$ $1089$ $2.593450$ $2.061039$ $1.830766$ $1.794457$ $1.794261$ $1.794260$ $2560$ $4225$ $2.589968$ $2.057095$ $1.826762$ $1.790567$ $1.790372$ $1.790370$ $5120$ $16641$ $2.588884$ $2.055897$ $1.825559$ $1.789395$ $1.789200$ $1.789198$ $10240$ $16641$ $2.588670$ $2.055684$ $1.825355$ $1.789193$ $1.788998$ $1.788997$

### 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 2018a. URL https://arxiv.org/abs/1802.00611.

L. Bonifacius, K. Pieper, and B. Vexler. Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls. ArXiv e-prints, September 2018b. URL https://arxiv.org/abs/1809.04886.