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The optimal control problem in coefficients for the pseudoparabolic variational inequality. (English) Zbl 0713.49015

Differential equations and their applications, Proc. 7th Conf., Equadiff 7, Prague/Czech. 1989, Teubner-Texte Math. 118, 232-235 (1990).
[For the entire collection see Zbl 0704.00019.]
The paper deals with the optimal control problem \[ J(u(\bar e),\bar e)=\min_{e\in U_{ad}}J(u(e),e),\text{ subject to } u(e)=u(\cdot,e)\in W^ 1_ 2([0,T],V), \]
\[ u(t,e)\in K(e),\quad t\in [0,T], \]
\[ <A_ 1(e)u_ t'(t,e)+A_ 0(e)u(t,e), v-u(t,e)>\geq <f(t)+B(e), v-u(t,e)> \]
\[ for\quad all\quad v\in K(e),\quad t\in [0,T],\quad u(0,e)=u_ 0(e), \] where \(U_{ad}\) is a compact set of admissible controls in a Hilbert space U, K(e) are closed convex subsets of a Hilbert space V, \(A_ i(t,e):\) \(V\to V^*\) are elliptic operators, \(f(t)\in V^*\), B: \(U\to V^*\). Using the penalty function method the existence of an optimal control \(\bar e\) is proved and generalized optimality conditions via the results of V. Barbu [see “Optimal control of variational inequalities” (1984; Zbl 0574.49005)] are deduced.
Reviewer: I.Bock

MSC:

49J40 Variational inequalities
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.