Sensitivity of Dynamic Structures, Case of a Smart Beam

B. Rousselet

Journal of convex analysis, Tome 9 (2002) no. 2, pp. 649-663

Résumé

Continuity of the solution with respect to data is a classical theme; for evolution equation this is usually restricted to continuity with respect to initial data and right hand side. Here we are interested in continuity with respect to coefficients appearing in bilinear forms (or in the coefficient of the operators involved in partial differential equations) or in the shape of the domain in which the system is posed. From a mathematical point of view, as noted in previous work of the author ["Quelques resultats en optimisation de domaine", These d'Etat, Univ. de Nice Sophia Antipolis (1982)], the classical implicit function theorem is not applicable for equations of Petrowsky type. We consider the sensitivity of a classical abstract optimal control problem to present the approach. As we consider beams of variable thickness, the partial differential equation involves variable coefficients; in the situation of exact control, the controlability issue seems still to be open. In this paper, we assume that controlability is satisfied. In the situation of classical optimal control, the existence of an optimal control may be obtained classically by lower semi continuity.