Optimality conditions for some nonqualified problems of distributed control.

*(English)*Zbl 0681.49025This paper deals with the problem of deriving necessary and sufficient conditions of optimality for elliptic partial differential equation optimal control problems with state constraints. Namely the following control problem is considered. \(-\Delta z+z=v\quad in\quad \Omega \subset {\mathbb{R}}^ n;\quad z=0\quad on\quad \partial \Omega\) \(v\in L^ 2(\Omega);\quad z\in H^ 2(\Omega)\cap H^ 1_ 0(\Omega)=X.\) Minimize:
\[
J(v)=\frac{1}{2\eta}\int_{\Omega}v^ 2dx+\int_{\Omega}| z(v)- z_ d|^ 2dx
\]
subject to either the bilateral constraint: \(| z| \leq \alpha\) almost everywhere in \(\Omega\), or the unilateral constraint: \(z\leq \alpha\) almost everywhre in \(\Omega\).

The authors use the Fenchel duality method of conex analysis to define a dual problem (P*) with respect to the adjoint state p. This problem is not coercive and so has no solution in X. This difficulty is overcome by extending the definition space for the adjoint state in (P*). In the bilateral case the extension for X is the space: \[ BL_ 0(\Omega)=\{p\in L^ 2(\Omega),\quad (-\Delta p+p)\in M_ 1(\Omega),\quad p=0\quad on\quad \partial \Omega \} \] where \(M_ 1(\Omega)\) is the space of bounded measures on \(\Omega\). The derivation of optimality conditions is obtained through the definition of \(\psi\) (\(\mu)\), where \(\psi\) is a convex function with linear growth and \(\mu\) a bounded measure and the extension to \(BL_ 0(\Omega)\) of Green formula. In the unilateral case it turns out that the problem can be separated in a problem without constraint for \(z<0\) and a problem similar to the bilateral case for \(z>0\). Optimality conditions are also derived.

The authors use the Fenchel duality method of conex analysis to define a dual problem (P*) with respect to the adjoint state p. This problem is not coercive and so has no solution in X. This difficulty is overcome by extending the definition space for the adjoint state in (P*). In the bilateral case the extension for X is the space: \[ BL_ 0(\Omega)=\{p\in L^ 2(\Omega),\quad (-\Delta p+p)\in M_ 1(\Omega),\quad p=0\quad on\quad \partial \Omega \} \] where \(M_ 1(\Omega)\) is the space of bounded measures on \(\Omega\). The derivation of optimality conditions is obtained through the definition of \(\psi\) (\(\mu)\), where \(\psi\) is a convex function with linear growth and \(\mu\) a bounded measure and the extension to \(BL_ 0(\Omega)\) of Green formula. In the unilateral case it turns out that the problem can be separated in a problem without constraint for \(z<0\) and a problem similar to the bilateral case for \(z>0\). Optimality conditions are also derived.

Reviewer: J.Henry

##### MSC:

49K20 | Optimality conditions for problems involving partial differential equations |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49N15 | Duality theory (optimization) |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |