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Minimax control of nonlinear evolution equations. (English) Zbl 0823.49006
In this paper the author proves the existence of an optimal control for a minimax problem. The control system is \[ -\dot x(t)\in \partial\varphi(t, x(t), \lambda)+ f_ 1(t, x(t), \lambda)+ f_ 2(t, x(t), \lambda) u(t)\quad\text{a.e.} \]
\[ x(0)= x_ 0(\lambda),\quad u(t)\in U(t)\quad\text{a.e.}, \] where \(\lambda\) models noise, disturbances and inaccuracy of measurement. The cost functional is given by \[ J(\lambda, u)= \int^ b_ 0 L(t, x(t, \lambda), \lambda, u(t)) dt. \] Since \(\lambda\) is not a priori known, the system analyst takes a pessimistic approach and minimizes the maximum cost: \[ \inf_ u\;\sup_ \lambda J(\lambda, u). \] The abstract result of existence is applied to obstacle problems, semilinear systems, and differential variational inequalities. The contents of this article are the same as those presented by the author in Appl. Math. Comput. 68, No. 2-3, 217-236 (1995; cited below).

49J35 Existence of solutions for minimax problems
49J20 Existence theories for optimal control problems involving partial differential equations
49J40 Variational inequalities
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