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Optimal design problems for a dynamic viscoelastic plate. I: Short memory material. (English) Zbl 0845.49001
Several optimal control problems with the same state problem – a pseudohyperbolic variational inequality with a linear operator – are considered. The control parameters appearing in the coefficients of the variational inequality as well as on the right-hand side are analyzed. The thickness of the dynamic viscoelastic plate with velocity constraints plays the role of control variable. Cost functionals correspond with adjusting the deflection (or the field moments) to a prescribed function $$z_d$$ (with minimal cost). Existence of an optimal control is proven on the abstract level. Using the method of penalization an existence and uniqueness theorem for a solution of an initial-boundary value problem for a pseudohyperbolic variational inequality is proved.
##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations 49J40 Variational inequalities 35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators 74Hxx Dynamical problems in solid mechanics
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