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:

Design constraints:

State constraints:

Mixed constraints:



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 β to serve as a test case for the β dependence of solvers. This problem and its analytical solution appear in [Güttel and Pearson2017, Section 6.2], with computations for final time T = 1 and control cost parameter β = 102.

Variables & Notation

Unknowns

u L2(Ω × I)2 control variable y = (v,p) state variable (velocity, pressure) v L2(I,V (Ω)) H1(I,V (Ω))velocity component of state p L2(I,L 02(Ω)) pressure component of state

where

V (Ω) = v H01(Ω)2 : divv = 0, L02(Ω) = p L2(Ω) :Ωpdx.

Given Data

T > 0 length of time interval Ω = (1,1)2 spatial domain I = (0,T) time interval Q = (1,1)2 × (0,T) space-time domain Σ = Ω × (0,T) lateral boundary of Q β > 0 regularization parameter ζ = eT 4π2β parameter η = 1 (1 + 4π2)β parameter z = 4π2(ζ + ηet)cos(2πx 1)sin(2πx2),0 uncontrolled source term vd = vd,1,vd,2 desired state vd,1 = 1 + et + (ζ + ηet)sin2(πx 1)sin(2πx2) + 2π2(eT et) 1 4sin2(πx 1)sin(2πx2) component of desired state vd,2 = 1 + et (ζ + ηet)sin(2πx 1)sin2(πx 2) + 2π2(eT et)sin(2πx 1) 4sin2(πx 2) 1 component of desired state v0 = v0,1,v0,2 initial state v0,1 = (ζ + η)sin2(πx 1)sin(2πx2) component of initial state v0,2 = (ζ + η)sin(2πx1)sin2(πx 2) component of initial state

Problem Description

Minimize1 2IΩ|v vd|2dxdt + β 2 IΩ|u|2dxdt s.t. vt v + p = u + zin Q divv = 0 in Q v = 0 on Σ v = v0 at t = 0

Optimality System

The following optimality system for the state y = (v,p) L2(I,V (Ω)) H1(I,V (Ω)) × L2(I,L02(Ω)), the control u L2(Ω × I)2 and the adjoint state q = (λ,μ) L2(I,V (Ω)) H1(I,V (Ω)) × L2(I,L02(Ω)), given in the strong form, characterizes the minimizer.

vt v + p = u + z in Q divv = 0 in Q v = 0 on Γ v = v0 at t = 0 λt λ + μ = v vdin Q divλ = 0 in Q λ = 0 on Γ λ = 0 at t = T u = 1 βλ in Q

Supplementary Material

The optimal state, adjoint state, and control are known analytically, noting that the pressure p and the adjoint pressure μ are normalized by having mean-value zero:

v = (ζ + ηet)sin2(πx 1)sin(2πx2), (ζ + ηet)sin(2πx 1)sin2(πx 2) p = π(ζ + ηet)sin(2πx 1)sin(2πx2) λ = (eT et)sin2(πx 1)sin(2πx2),(eT et)sin(2πx 1)sin2(πx 2) μ = (x1 + x2) u = 1 β (eT et)sin2(πx 1)sin(2πx2),(eT et)sin(2πx 1)sin2(πx 2)

Notice that the sign of (λ,μ) is reversed in [Güttel and Pearson2017, Section 6.2]. Consequently, the control law reads u = 1 βλ in [Güttel and Pearson2017, 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.