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Optimality conditions and numerical approximations for some optimal design problems. (English) Zbl 0731.49010
Summary: In structural optimization one often encounters the problem of minimizing the weight of a plate or a beam under some geometric constraints or bounds on the deflection or stress. Sometimes these problems have been studied as state constrained optimal control problems governed by elliptic differential equations, the control being a parameter that appears in the coefficients of the corresponding differential operator. In this paper we consider three aspects of these problems. Firstly, we apply sensitivity analysis and prove the existence of a solution. Next we derive the optimality system from an abstract theorem of existence of a Lagrange multiplier. And finally, we perform the numerical discretization of the control problem and prove the convergence of approximate solutions. In order to derive the optimality conditions and to prove the convergence of the numerical approximations we make a stability hypothesis of the Slater type, which avoids the necessity of enlarging the set of admissible states in the discretization. This approach is interesting since it is well known that this enlargement of the admissible state set diminishes the order of convergence.

49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
74M05 Control, switches and devices (“smart materials”) in solid mechanics