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Approximation of Young measures by functions and application to a problem of optimal design for plates with variable thickness. (English) Zbl 0824.49006

The aim of this paper is to present an approximation of parametrized measures (in the sense of L. C. Young) by a dense family of continuous functions in one or several dimensions. The approximation functions form a dense set of linearly independent, bounded functions with positive values which separate the Dirac measure.
In one dimension, the main tool is the method of orthogonal polynomials while in several dimensions, the method of moments together with the Riesz-Carathéodory theorem is used.
As application of the main result, the weak-\(*\)-compactness of some subsets of \(L^ \infty(\Omega)\), \(\Omega\subset \mathbb{R}^ n\), is proved. Next, this result is applied to the prove of the existence of optimal solution in some optimal control problem for plates with variable thickness. Namely, some energy functional is minimized over the set of all solutions of some second-order partial differential equation which describes the deflection of a symmetric plate with midplane (the controls, i.e., half-thickness, are in the coefficients of this equation). Introducing the generalized thickness and the corresponding relaxation of the state equation, the existence of the generalized optimal control is proved.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49J40 Variational inequalities
74M05 Control, switches and devices (“smart materials”) in solid mechanics
35B37 PDE in connection with control problems (MSC2000)
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