ccparbnd1 details:

Keywords:

Global classification: linear-quadratic, convex

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via boundary values 1st order

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



Submitted on 2013-01-16 by Arnd Rösch. Published on 2013-02-11


ccparbnd1 description:


Introduction

This is a very classical parabolic Robin boundary control problem in one space dimension with control constraints. Originally it was posed as a time-optimal control problem, see Schittkowski [1979]. Several years later in Tröltzsch [1984] this problem was modified to an optimal control problem with fixed final time. In Eppler and Tröltzsch [1986] an additional Tikhonov regularization was introduced. We present the example in that form but change the notation to standard variables. The same example is studied in many publications sometimes with small modifications. If the regularization parameter α is zero, then the optimal control has bang-bang structure with one switching point.

Variables & Notation

Unknowns

u L2(0,T) control variable y C([0,T],L2(0,1))state variable

Given Data

No analytic solution is known for the given data. The numerical experiments in the literature show results for α = 0 (no regularization) with a bang-bang structure as well as results for α > 0. In particular one can find results for α = 10ν with different ν in Eppler and Tröltzsch [1986].

Ω = (0,1) computational domain T = 1.58 final time yd(x) = 1 2(1 x2)desired final state b = 1 heat exchange coefficient α 0 Tikhonov parameter

Problem Description

Minimize1 2y(T,) ydL2(Ω)2 + α 2 uL2(0,T)2 s.t. y t y = 0 in (0,T) × Ω y(0,x) = 0 in Ω y x(t,0) = 0 in (0,T) y x(t,1) = b(u(t) y(t,1))in (0,T) and 1 u(t) 1in (0,T).

Optimality System

The following optimality system for the state y C([0,T],L2(0,1)), the control u L2(0,T) and the adjoint state p C([0,T],L2(0,1)), given in the strong form, characterizes the unique minimizer.

y t y = 0 in (0,T) × Ω p t y = 0 in (0,T) × Ω y(0,x) = 0 in Ω p(T,x) = y(T,x) yd(x)in Ω y x(t,0) = 0 in (0,T) p x(t,0) = 0 in (0,T) y x(t,1) = b(u(t) y(t,1))in (0,T) p x(t,1) = b p(t,1) in (0,T)

as well as

0T p(t,1) + αu(t)v(t) u(t)dt 0for all v L2(0,T) with |v(t)| 1 a.e.

In case α > 0, this variational inequality is equivalent to the projection formula

u(t) = proj[1,1] p(t,1) α .

Supplementary Material

There is no analytic solution known. An interesting modification is to take a desired state

yd(x) = 1 2(1 x).

Then the optimal control oscillates when approaching the final time T.

References

   K. Eppler and F. Tröltzsch. On switching points of optimal controls for coercive parabolic boundary control problems. Optimization. A Journal of Mathematical Programming and Operations Research, 17(1):93–101, 1986. ISSN 0233-1934. doi: 10.1080/02331938608843105.

   K. Schittkowski. Numerical solution of a time-optimal parabolic boundary value control problem. Journal of Optimization Theory and Applications, 27(2):271–290, 1979. ISSN 0022-3239. doi: 10.1007/BF00933231.

   F. Tröltzsch. The generalized bang-bang-principle and the numerical solution of a parabolic boundary-control problem with constraints on the control and the state. Zeitschrift für Angewandte Mathematik und Mechanik, 64(12):551–556, 1984. ISSN 0044-2267. doi: 10.1002/zamm.19840641218.