todist2 details:

Keywords: time optimal

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


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 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 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

q Q = L((0,T); 2) 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 = 0 H01(Ω) desired state δ0 = 1 10 > 0 tolerance to desired state α 0 control cost parameter (arbitrary) u0 = 4sin(πx12)sin(πx 23)initial state c = 0.03 coefficient in the PDE qa = 1.5 lower control bound qb = 0 upper control bound ω1 = (0,0.5) × (0,1) control domain 1 ω2 = (0.5,1) × (0,0.5) control domain 2

The control-action operator is defined as

B: 2 L2(Ω), q = (q1,q2) Bq = q1χω1 + q2χω2

where χω1 and χω2 denote the characteristic functions on ω1 and ω2.

Problem Description

Minimize j(T,q) := T + α 2 0T q(t) 22dt, subject to T > 0, tu cu = Bq, in (0,T) × Ω, u = 0, on (0,T) × Ω, u(0) = u0, in Ω, 1 2u(T) udL2(Ω)2 δ02 2 0, qa q(t) qb,t (0,T). (P)

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

Minimize ĵ(T,q̂) := T 1 + α 2 01q̂(t) 22dt, subject to T > 0, tû Tcû = TBq, in (0,1) × Ω, û = 0, on (0,1) × Ω, û(0) = u0, in Ω, 1 2û(1) udL2(Ω)2 δ02 2 0, qa q̂(t) qb,t (0,1). (P̂)

Note that the problems (P) and (P̂) are equivalent. The unknowns for the transformed problem (P̂) are q̂ Q̂ = L((0,1); 2) 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)22 +(Bq¯(t) + cu¯(t),z¯(t)) L2(Ω)dt = 0, 01T¯(αq¯(t) + Bz¯(t),q(t) q¯(t)) 2dt 0, qa q(t) qb, u¯(1) udL2(Ω) = δ0,

where 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. (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 |logk|(k + h2) 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 α, number of time steps M and number of spatial nodes N. The analysis for the case α = 0 can be found in Bonifacius et al. [2018b].

T¯
α = 10α = 1α = 0.1α = 0.01α = 0.001α = 0
MN
6402892.6056612.0751531.8452011.8084561.8082571.808255
128010892.5934502.0610391.8307661.7944571.7942611.794260
256042252.5899682.0570951.8267621.7905671.7903721.790370
5120166412.5888842.0558971.8255591.7893951.7892001.789198
10240166412.5886702.0556841.8253551.7891931.7889981.788997

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.