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mpdist2 details:
Keywords: analytic solution
Global classification: linear-quadratic, convex
Functional: convex quadratic
Geometry: easy, fixed
Design: coupled via volume data
Differential operator:
- Heat:
- linear parabolic operator of order 2.
- Defined on a 1-3-dim domain in 1-3-dim space
- Time dependent.
Design constraints:
- none
State constraints:
- none
Mixed constraints:
- none
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 [Tröltzsch, 2010, 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 to serve as a test case for the dependence of solvers.
The 2d version of this problem and analytical solution appear in [Güttel and Pearson, 2017, Section 6.1], where computations for and final times were conducted. The implementation is provided for as well.
Variables & Notation
Unknowns
Given Data
Problem Description
Optimality System
The following optimality system for the control , the state , and the adjoint state , given in the strong form, characterizes the unique minimizer.
Supplementary Material
The optimal state, adjoint state, and control are known analytically:
Notice that the sign of is reversed in [Güttel and Pearson, 2017, Section 6.1]. Consequently, the control law reads in [Güttel and Pearson, 2017, 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.