mpdist2 details:

Keywords: analytic solution

Global classification: linear-quadratic, convex

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



Submitted on 2017-06-02 by John Pearson. Published on 2017-12-11


mpdist2 description:


Introduction

Here we present a simple distributed optimal control problem of the heat equation. Problems of this type are examined in detail within [Tröltzsch2010, Chapter 3]. The problem was derived as a test for the paper Güttel and Pearson [2017], which required optimal states and controls that are not polynomial in spatial or time variables. The problem is generically usable in dimensions 1 through 3 and maintains a parameter dependence for the regularization parameter β to serve as a test case for the β dependence of solvers.

The 2d version of this problem and analytical solution appear in [Güttel and Pearson2017, Section 6.1], where computations for β = 0.05 and final times T = 1 were conducted. The implementation is provided for d = 2 as well.

Variables & Notation

Unknowns

u L2(Ω × I) control variable y W(0,T) := L2(I;H 01(Ω)) H1(I;H1(Ω))state variable

Given Data

d {1,2,3} dimension of the problem T > 0 length of time interval Ω = (1,1)d spatial domain I = (0,T) time interval Q = (1,1)d × (0,T) space-time domain Σ = Ω × (0,T) lateral boundary of Q β > 0 regularization parameter yd,T = dπ2 4 + 4 dπ2βeT + 1 dπ2 4 4 (4 + dπ2)βetaux. function for desired state yd = yd,T (t) k=1d cos πxk 2 desired state y0 = 4 dπ2βeT 4 (4 + dπ2)β k=1d cos πxk 2 initial state

Problem Description

Minimize1 2IΩ(y yd)2dxdt + β 2 IΩu2dxdt s.t. yt y = u in Q y = 0 on Σ y = y0at t = 0

Optimality System

The following optimality system for the control u L2(Ω × I), the state y W(0,T), and the adjoint state p W(0,T), given in the strong form, characterizes the unique minimizer.

yt y = u in Q y = 0 on Σ y = y0 at t = 0 pt p = y ydin Q p = 0 on Σ p = 0 at t = T u = 1 βp in Q

Supplementary Material

The optimal state, adjoint state, and control are known analytically:

y = 4 dπ2βeT 4 (4 + dπ2)βet k=1d cos πxk 2 p = (eT et) k=1d cos πxk 2 u = 1 β(eT et) k=1d cos πxk 2

Notice that the sign of p is reversed in [Güttel and Pearson2017, Section 6.1]. Consequently, the control law reads u = 1 βp in [Güttel and Pearson2017, Section 6.1].

References

   S. Güttel and J. W. Pearson. A rational deferred correction approach to parabolic optimal control problems. IMA Journal of Numerical Analysis, online-first, 2017. doi: 10.1093/imanum/drx046. URL https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drx046/4372128.

   F. Tröltzsch. Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, 2010.