Welcome to the OPTPDE Problem Collection
mpdist3 details:
Keywords: flow control, analytic solution
Global classification: linear-quadratic, convex
Functional: convex quadratic
Geometry: easy, fixed
Design: coupled via volume data
Differential operator:
- Stokes:
- linear parabolic operator of order 2.
- Defined on a 2-dim domain in 2-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
mpdist3 description:
Introduction
Here we present a distributed optimal control problem of the time-dependent Stokes equations. 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 maintains a parameter dependence for the regularization parameter to serve as a test case for the dependence of solvers. This problem and its analytical solution appear in [Güttel and Pearson, 2017, Section 6.2], with computations for final time and control cost parameter .
Variables & Notation
Unknowns
where
Given Data
Problem Description
Optimality System
The following optimality system for the state , the control and the adjoint state , given in the strong form, characterizes the minimizer.
Supplementary Material
The optimal state, adjoint state, and control are known analytically, noting that the pressure and the adjoint pressure are normalized by having mean-value zero:
Notice that the sign of is reversed in [Güttel and Pearson, 2017, Section 6.2]. Consequently, the control law reads in [Güttel and Pearson, 2017, Section 6.2].
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.