×

Study of a non-classical perturbed optimal control system. (Étude d’un système de contrôle optimal perturbé non classique.) (French) Zbl 0899.49002

The paper deals with a perturbed optimal control problem arising in aerodynamics. The state equation is \[ \begin{cases} -\Delta y_\varepsilon(v)=0 & \text{in \(\Omega\);}\\ \partial y_\varepsilon(v)\over\partial\nu+\varepsilon y_\varepsilon(v)=0 & \text{on\;\(\Gamma\)},\qquad\qquad y_\varepsilon(v)\in H^1(\Omega),\end{cases} \] where \(\Gamma=\partial\Omega\) is smooth, and \(\Omega\) locally lies on one side of \(\Gamma\). Denoting by \(T_\varepsilon:L^2(\Gamma)\to L^2(\Gamma)\) the map \(v\mapsto {y_\varepsilon(v)}_\Gamma\), the cost functional is \(J_\varepsilon(v)=\| T_\varepsilon(v)-z\|^2_2\), with \(z\) a given non zero vector in \(L^2(\Gamma)\), and \(v\) varies in some subspace of admissible controls. The paper deals with existence and uniqueness of optimal controls \(v_\varepsilon\), and with their behaviour as \(\varepsilon\to 0\).
Reviewer: L.Ambrosio (Pavia)

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
93C73 Perturbations in control/observation systems
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Benkiran, T., On a non Classical Perturbed Boundary Optimal Control System, Libertas Mathematica, Vol.XVI (1996). · Zbl 0871.49007
[2] Benkiran, T., Etude d’un problème de perturbation singulière non classique, Bolletino U.M.I. (7)11-A (1997), 93-103. · Zbl 0902.35011
[3] Brezis, H., Analyse fonctionnelle, Théorie et Applications, Masson, Paris1983. · Zbl 0511.46001
[4] Lions, J.L., Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Dunod Gauthier-Villars, Paris1968. · Zbl 0179.41801
[5] Lions, J.L., Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Springer-Verlag, Berlin1973. · Zbl 0268.49001
[6] Lions, J.L. and Magenes, E., Problèmes aux limites non homogènes et applications, Vol.1, 2, Dunod, Paris1969. · Zbl 0165.10801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.