Differential
Inclusions, Control and Optimization 21 (2001) 5-50

doi: 10.7151/dmdico.1016

Nikolaos S. Papageorgiou and Nikolaos Yannakakis

*National Technical University, Department of
Mathematics
Zografou Campus, Athens 157 80, Greece*

In this paper, first we consider parametric control systems driven by nonlinear evolution equations defined on an evolution triple of spaces. The parametres are time-varying probability measures (Young measures) defined on a compact metric space. The appropriate optimization problem is a minimax control problem, in which the system analyst minimizes the maximum cost (risk). Under general hypotheses on the data we establish the existence of optimal controls.

Then we pass to nonparametric systems, which are governed by nonlinear evolution equations with nonmonotone operators. We prove two existence results for such evolution inclusions, which are of independent interest and extend significantly the results existing in the literature. Then we solve time-optimal and Meyer-type optimization problems. In Section 5, we derive necessary conditions for saddle point optimality in the minimax control problem. We conclude the paper with three examples of distributed parameter control systems.

**Keywords and phrases:** evolution triple, compact embedding, monotone operator,
pseudomonotone operator, L-generalized pseudomonotonicity, integration by parts, evolution
inclusion, saddle point, necessary conditions, adjoint equation, distributed parameter
systems.

**2000 Mathematics Subject Classification:** 49J35, 49J27, 49K27, 34G20.

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Received 17 February 2000