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Contrôle de systèmes elliptiques semilinéaires comportant des contraintes sur l’état. (Control of semilinear elliptic systems with state constraints). (French) Zbl 0656.49011
Nonlinear partial differential equations and their applications, Coll. de France Semin., Vol. VIII, Paris/Fr. 1984-85, Pitman Res. Notes Math. Ser. 166, 69-86 (1988).
[For the entire collection see Zbl 0643.00014.]
The paper deals with the optimal control problem \[ -\Delta y+f(y)=u\quad on\quad \Omega \subset R^ n,\quad n\leq 3;\quad y=0\quad on\quad \partial \Omega;\quad \min J(u),\quad u\in K,\quad y_ u\in K_ Y;\quad where\quad \] \[ J(u)=(N/2)\int_{\Omega}u^ 2dx+(1/2)\int_{\Omega}(y_ u-y_ d)^ 2dx,\quad N>0. \] K and \(K_ Y\) are closed convex subsets of \(L_ 2(\Omega)\) and \(H^ 2(\Omega)\cap H^ 1_ 0(\Omega)\) respectively. The cases \[ K_ Y=\{y\in H^ 2(\Omega)\cap H^ 1_ 0(\Omega),\quad \int_{\Omega}| y| dx\leq \delta \} \] and \[ K_ Y=\{y\in H^ 2(\Omega)\cap H^ 1_ 0(\Omega),\quad | y(x)| \leq \delta x\in \Omega \}, \] \(\delta\) \(>0\), are investigated. The optimality conditions are obtained in a non- qualified form. Some qualification results for specific cases are given.
Reviewer: I.Bock

MSC:
49K20 Optimality conditions for problems involving partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
93C20 Control/observation systems governed by partial differential equations
35B37 PDE in connection with control problems (MSC2000)
93C10 Nonlinear systems in control theory