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Shape sensitivity analysis of state constrained optimal control problems for distributed parameter systems. (English) Zbl 0702.49015
Control of partial differential equations, Proc. IFIP Work. Conf., Santiago de Compostela/Spain 1987, Lect. Notes Control Inf. Sci. 114, 236-245 (1989).
[For the entire collection see Zbl 0668.00021.]
The differentiable stability with respect to \(\epsilon\) for the optimal control problem \[ \min (1/2)\int_{\Omega_{\epsilon}}(y-z_ d)^ 2dx\quad +\quad (\alpha /2)\int_{\Omega_{\epsilon}}u^ 2dx\quad over\quad L^ 2(\Omega_{\epsilon}), \]
\[ -\Delta y=u\text{ in } \Omega_{\epsilon},\quad y=0\text{ on } \partial \Omega_{\epsilon} \] is considered. By finding an expression for the closure of the tangent cone to the set \[ K=\{\phi \in H^ 2(\Omega_{\epsilon})\cap H^ 1_ 0(\Omega_{\epsilon})| \quad \phi \geq a\quad q.e.\text{ in } \Omega_{\epsilon}\} \] the problem is reduced to the solution of a related optimal control problem over a certain cone.
Reviewer: S.P.Banks

49K40 Sensitivity, stability, well-posedness
49J20 Existence theories for optimal control problems involving partial differential equations