scdist1 details:

Keywords: 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 2012-08-21 by Roland Herzog. Published on 2012-08-21


scdist1 description:


Introduction

This problem is a standard linear-quadratic optimal control problem with pointwise state constraints, but no constraints on the control variable. It appears in Meyer et al. [2005] as a test example for the so-called Lavrentiev regularization approach. By the latter, optimal control problems with state constraints of the form y(x) yc(x) are approximated by problems involving mixed control-state constraints y(x) + 𝜀u(x) yc(x). While problems with pointwise state constraints usually involve a measure-valued Lagrange multiplier, see Casas [1986], the corresponding quantity in the regularized problem is a function.

The following state-constrained problem and its solution appear in [Meyer et al.2005, Section 7, Example 2] and it features a line measure as the state constraint multiplier.

Variables & Notation

Unknowns

u L2(Ω) control variable y H1(Ω)state variable

Given Data

The given data is chosen in a way which admits an analytic solution.

Ω = (0,1)2 computational domain Ω1 = {x = (x1,x2) Ω : x1 < 0.5} subdomain of Ω Ω2 = {x = (x1,x2) Ω : x1 > 0.5} subdomain of Ω Γ̂ = {x = (x1,x2) Ω : x1 = 0.5} line inside Ω yd = 0.5 + x12,x1 < 0.5 0.75, x1 0.5 desired state (discontinuous) ud = 2.5 x12,x1 < 0.5 2.25, x1 0.5 desired control (continuous) yc = 2x1 + 1,x1 < 0.5 2, x1 0.5 state constraint (continuous)

Note that there is a typo in the specification of yd in [Meyer et al.2005, Section 7, Example 2].

Problem Description

Minimize1 2y ydL2(Ω)2 + 1 2u udL2(Ω)2 s.t. y + y = uin Ω y n = 0on Ω and y ycin Ω¯.

Optimality System

The following optimality system for the state y H1(Ω), the control u L2(Ω), the adjoint state p H1(Ω) and the state constraint multiplier μ M(Ω¯), given in the strong form, characterizes the unique minimizer.

y + y = u in Ω y n = 0 on Ω p + p = y yd μΩin Ω p n = μΩ on Ω u ud + p = 0 in Ω y yc 0 in Ω¯ μ 0 in M(Ω¯) μ,y ycM(Ω¯),C(Ω¯) = 0

The space M(Ω¯) is the dual of C(Ω¯) and it consists of all signed, real, regular Borel measures on Ω¯. Here, μΩ and μΩ denote the restrictions of the measure μ to Ω and its boundary, respectively.

Supplementary Material

The optimal state, adjoint state, control and state constraint multiplier are known analytically:

y = 2 p = 0.5 x12,x1 < 0.5 0.25, x1 0.5 u = 2 μ,yM(Ω¯),C(Ω¯) =Ω2y(x)dx +Γ ̂y(s)dsfor y C(Ω¯).

Note that μ consists of a regular part (the characteristic function of Ω2) plus a singular part (the line measure concentrated on Γ̂). In particular, the boundary part μΩ vanishes.

Revision History

References

   E. Casas. Control of an elliptic problem with pointwise state constraints. SIAM Journal on Control and Optimization, 24(6):1309–1318, 1986. doi: 10.1137/0324078.

   C. Meyer, A. Rösch, and F. Tröltzsch. Optimal control of PDEs with regularized pointwise state constraints. Computational Optimization and Applications, 33(2–3): 209–228, 2005. doi: 10.1007/s10589-005-3056-1.