todist1 details:

Keywords: time optimal, analytic solution

Global classification: nonlinear

Functional: nonconvex nonlinear

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



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 L2-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

q Q = L2((0,T);L2(Ω)) control variable u U = H1((0,T);L2(Ω)) L2((0,T);H 01(Ω) H2(Ω))state variable T terminal time

Given Data

Ω = (0,1)2 spatial domain ud = 2sin(πx1)sin(πx2) H01(Ω)desired state δ0 = 1 2 > 0 tolerance to desired state α > 0 control cost parameter (arbitrary) u0 = sin(πx1)sin(πx2) initial data c = 1 2π2 coefficient in the PDE

Problem Description

Minimize j(T,q) := T + α 2 0T q(t) L2(Ω)2dt, subject to T > 0, tu cu = q, in (0,T) × Ω, u = 0, on (0,T) × Ω, u(0) = u0,in Ω, 1 2u(T) udL2(Ω)2 δ02 2 0. (P)

The state equation can be transformed to the reference time interval (0,1) 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

Minimize ĵ(T,q̂) := T 1 + α 2 01q̂(t) L2(Ω)2dt, subject to T > 0, tû Tcû = Tq̂,in (0,1) × Ω, û = 0, on (0,1) × Ω, û(0) = u0, in Ω, 1 2û(1) udL2(Ω)2 δ02 2 0. (P̂)

Note that the problems (P) and (P̂) are equivalent. The unknowns for the transformed problem (P̂) are q̂ Q̂ = L2((0,1);L2(Ω)) and û Û = H1((0,1);L2(Ω)) L2((0,1);H01(Ω) H2(Ω)).

Optimality System

The first-order necessary optimality conditions for (P̂) are formally given as follows: for given local minimizers q¯ Q̂, u¯ Û, T¯ > 0 there exists Lagrange multipliers μ¯ > 0 and z¯ W(0,1) = {v L2(0,1;H01(Ω)) : tv L2(0,1;H1(Ω))} such that

011 + α 2 q¯(t)L2(Ω)2 +(q¯(t) + cu¯(t),z¯(t)) L2(Ω)dt = 0, αq¯(t) + z¯(t) = 0, u¯(1) udL2(Ω) = δ0,

and the adjoint state z¯ W(0,1) is determined by

tz¯(t) T¯z¯(t) = 0,t (0,1)z¯(1) = μ¯u¯(1) ud.

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 (P̂) are

u¯(t,x) = c0eT¯t + c 1eT¯(t1) u 0(x), z¯(t,x) = μ¯eT¯(t1)u 0(x), q¯(t,x) = μ¯ α eT¯(t1)u 0(x),

where the parameters c0 and c1, the optimal time T¯ and multiplier μ¯ are given by

c0 = 1 2 α + 8 α + 1,c1 = 1 2 α + 8 α 1, T¯ = log α+8 α + 1 α+8 α 1,μ¯ = 8 α + 1 + 1α.

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.