Departmentof Mathematics

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 $y\left(x\right)\ge y_c\left(x\right)$ are approximated by problems involving mixed control-state constraints $y\left(x\right)+𝜀u\left(x\right)\ge y_c\left(x\right)$. 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

$$\begin{array}{ccc}\hfill u\in L^2\left(\Omega \right)& \text{control variable}\hfill & \hfill \\ \hfill y\in H^1\left(\Omega \right)& \text{state variable}\hfill \end{array}$$

Given Data

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

$$\begin{array}{ccccc}\hfill \Omega & ={\left(0,1\right)}^2\hfill & \hfill & \text{computational domain}\hfill & \hfill \\ \hfill {\Omega }_1& =\left\{x=\left(x_1,x_2\right)\in \Omega :x_1<0.5\right\}\hfill & \hfill & \text{subdomain of }\Omega \hfill \\ \hfill {\Omega }_2& =\left\{x=\left(x_1,x_2\right)\in \Omega :x_1>0.5\right\}\hfill & \hfill & \text{subdomain of }\Omega \hfill \\ \hfill \hat{\Gamma }& =\left\{x=\left(x_1,x_2\right)\in \Omega :x_1=0.5\right\}\hfill & \hfill & \text{line inside }\Omega \hfill \\ \hfill y_d& =\left\{\begin{array}{cc}-0.5+x_1^2,\hfill & x_1<0.5\hfill \\ 0.75,\hfill & x_1\ge 0.5\hfill \end{array}\right.\hfill & \hfill & \text{desired state (discontinuous)}\hfill \\ \hfill u_d& =\left\{\begin{array}{cc}2.5-x_1^2,\hfill & x_1<0.5\hfill \\ 2.25,\hfill & x_1\ge 0.5\hfill \end{array}\right.\hfill & \hfill & \text{desired control (continuous)}\hfill \\ \hfill y_c& =\left\{\begin{array}{cc}2x_1+1,\hfill & x_1<0.5\hfill \\ 2,\hfill & x_1\ge 0.5\hfill \end{array}\right.\hfill & \hfill & \text{state constraint (continuous)}\hfill \end{array}$$

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

Problem Description

$$\begin{array}{ccc}\hfill \text{Minimize}& \frac{1}{2}\parallel y-y_d{\parallel }_{L^2\left(\Omega \right)}^2+\frac{1}{2}\parallel u-u_d{\parallel }_{L^2\left(\Omega \right)}^2\hfill & \hfill \\ \hfill \text{s.t.}& \left\{\begin{array}{ccccc}\hfill -△y+y& =u\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill \frac{\partial y}{\partial n}& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill \end{array}\right.\hfill \\ \hfill \text{and}& y\ge y_c\text{in }\overline{\Omega }.\hfill \end{array}$$

Optimality System

The following optimality system for the state $y\in H^1\left(\Omega \right)$, the control $u\in L^2\left(\Omega \right)$, the adjoint state $p\in H^1\left(\Omega \right)$ and the state constraint multiplier $\mu \in M\left(\overline{\Omega }\right)$, given in the strong form, characterizes the unique minimizer.

$$\begin{array}{ccccc}\hfill -△y+y& =u\hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill \frac{\partial y}{\partial n}& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill -△p+p& =y-y_d-{\mu }_{\Omega }\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill \frac{\partial p}{\partial n}& =-{\mu }_{\partial \Omega }\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill u-u_d+p& =0\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill y-y_c& \ge 0\hfill & \hfill & \text{in }\overline{\Omega }\hfill \\ \hfill \mu & \ge 0\hfill & \hfill & \text{in }M\left(\overline{\Omega }\right)\hfill \\ \hfill {⟨\mu ,y-y_c⟩}_{M\left(\overline{\Omega }\right),C\left(\overline{\Omega }\right)}& =0\hfill \end{array}$$

The space $M\left(\overline{\Omega }\right)$ is the dual of $C\left(\overline{\Omega }\right)$ and it consists of all signed, real, regular Borel measures on $\overline{\Omega }$. Here, ${\mu }_{\Omega }$ and ${\mu }_{\partial \Omega }$ denote the restrictions of the measure $\mu$ to $\Omega$ and its boundary, respectively.

Supplementary Material

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

Note that $\mu$ consists of a regular part (the characteristic function of ${\Omega }_2$) plus a singular part (the line measure concentrated on $\hat{\Gamma }$). In particular, the boundary part ${\mu }_{\partial \Omega }$ vanishes.

Revision History

• 2014–11–14: added missing term $u_d$ to the objective (thanks to Stefan Takacs)
• 2012–11–14: problem added to the collection

References