ccparfin1 details:

Keywords: analytic solution

Global classification: nonlinear-quadratic

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



Submitted on 2013-05-27 by Fredi Tröltzsch. Published on 2013-06-21


ccparfin1 description:


Introduction

This example is an optimal control problem for a semilinear heat equation with cubic nonlinearity in a two dimensional domain. There are four time-dependent control functions restricted by box constraints. The example is constructed such that a locally optimal solution is explicitly known. It was used in the context of model reduction by POD to test an a posteriori error estimator for optimality, and appears in [Kammann et al.2013, Section 4.2].

Variables & Notation

Unknowns

ui L2(0,T),i = 1,,4 control functions y L2(0,T;H1(Ω)) H1(0,T;H1(Ω)) L(Q)state variable

Given Data

Ω = (0,π)2 spatial domain Q = Ω × (0,T) computational domain T = 1 terminal time Σ = Ω × (0,T) boundary ν outward unit normal on Ω y0(x) = cos(x1)cos(x2), initial function yQ(x,t) = cos(x1)cos(x2)1 + 0.02(t t2) 0.03t2 cos2(x 1)cos2(x 2) desired state ay(x) = 0.01cos(x1)cos(x2) weight of terminal state w1(x) = max0,10 50(x1 π 4 )2 50(x2 π 4 )2) control weight1 w2(x) = max0,10 50(x1 3 4π)2 50(x2 π 4 )2 control weight1 w3(x) = max0,10 50(x1 3 4π)2 50(x2 3 4π)2 control weight1 w4(x) = max0,10 50(x1 π 4 )2 50(x2 3 4π)2 control weight1 βi(t) = proj [1,1] t2Ωwi(x)cos(x1)cos(x2)dx coefficient, i = 1,,4 d(x,t) = i=14w i(x)βi(t) 2cos(x 1)cos(x2) cos3(x 1)cos3(x 2) distributed source term

The graphs of the functions w1,,w4 are shown in Figure 0.1.


PIC

Figure 0.1: Weight functions w1,,w4 for the controls ui.

Problem Description

Minimize 1 2Q(y yQ)2dxdt +Ωay(x)y(x,T)dx + 1 200 i=140T u i(t)2dt s.t. y t (x,t) y(x,t) + y3(x,t) + d(x,t) = i=14w i(x)ui(t)in Q y ν(x,t) = 0 on Σ y(x,0) = y0(x) in Ω and |ui(t)| 1in (0,T),i = 1,,4.

Optimality System

The following optimality system for the state y, the control u, and the adjoint state p, given in the strong form, represents first-order necessary optimality conditions.

y t (x,t) y(x,t) + y3(x,t) + d(x,t) = i=14w i(x)ui(t)in Q y ν(x,t) = 0 on Σ y(x,0) = y0(x) in Ω p t (x,t) p(x,t) + 3y2(x,t)p(x,t) = y(x,t) y Q(x,t) in Q p ν(x,t) = 0 on Σ p(x,T) = ay(x) in Ω ui(t) = proj[1,1] 100Ωwi(x)p(x,t)dx,i = 1,,4 in (0,T).

Supplementary Material

A set of locally optimal controls ui with associated state y and adjoint state p are known analytically.

ȳ(x,t) = cos(x1)cos(x2) in Q p̄(x,t) = t2 100cos(x1)cos(x2) in Q ūi(t) = proj[1,1] t2Ωwi(x)cos(x1)cos(x2)dx,i = 1,,4in (0,T).

The controls are shown in Figure 0.2.

To show the local optimality of this solution, we verify that (ȳ,ū1,,ū4) obeys the standard second-order sufficient optimality conditions.1 1This analysis is not presented in Kammann et al. [2013] but it was carried out by the authors in 2014 (unpublished). To this end, we introduce the Lagrangian; writing u := (u1,,u4), we define

(y,u,p) := J(y,u) Q y t y + y3 + d i=14w iui pdxdt.

The second-order derivative of with respect to (y,u) at (ȳ,ū,p̄) is

(ȳ,ū,p̄)(y,u)2 =Q y2 6ȳp̄y2 dxdt + 1 100 i=140T u i(t)2dt Q0.94y2dxdt + 1 100 i=140T u i(t)2dt 1 100 i=140T u i(t)2dtfor all y W(0,T),u L2(0,T)4.

Notice that |ȳ(x,t)| 1 and |p̄(x,t)| 0.01 is satisfied. Invoking [Tröltzsch2010, Theorem 5.17], we obtain that ū is locally optimal with respect to the topology of L(0,T)4. (The quoted theorem is formulated for a control function u : Q , but it obviously extends to the case u : Q 4.)


PICPIC

Figure 0.2: Optimal controls ū1,ū2. By symmetry, ū1 = ū3 and ū2 = ū4 holds.

Revision History

References

   E. Kammann, F. Tröltzsch, and S. Volkwein. A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD. ESAIM. Mathematical Modelling and Numerical Analysis, 47(2):555–581, 2013. ISSN 0764-583X. doi: 10.1051/m2an/2012037.

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