An optimal control problem for a pseudoparabolic variational inequality
The paper deals with an optimal control problem described by: \(u(t,e)\in K(e)\) for a.e. \(t\in [0,T]\), for a.e. \(t\in [0,T]: \langle A(t,e)u'(t,e)+B(t,e)u(t,e)\), \(v-u(t,e)\rangle\geq \langle f(t,e)\), \(v- u(t,e)\rangle\) for all \(v\in K(e)\), \(u(0,e)=u_ 0(e)\in K(e)\), \(j(u(\bar e),\bar e)=\min_{e\in U_{ad}} j(u(e),e)\), where \(A(t,e)\), \(B(t,e)\) are families of elliptic operators operating from the Hilbert space \(V\) into its dual \(V^*\), \(U_{ad}\) is a compact subset of a Banach space \(U\), \(K(e)\) is closed and convex subset of the space \(V\) for every \(e\in U_{ad}\). An existence and uniqueness theorem for a state pseudoparabolic inequality is proved using the penalization method. The existence of an optimal control \(\bar e\) is verified. The theory is applied to the optimal design problem for a viscoelastic plate with an obstacle, where a variable thickness of the plate plays the role of a control parameter.
scientific article; zbMATH DE number 58781
scientific article