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A continuous version of the relaxation theorem for nonlinear evolution inclusions. (English) Zbl 0864.34057

Summary: We consider parametric nonlinear evolution inclusions defined on an evolution triple of spaces. First, we prove some continuous dependence results for the solution sets of both the convex and nonconvex problem and for the set of solution-selector pairs of the convex problem. Subsequently, we derive a parametrized version of the Filippov-Gronwall estimate in which the parameter varies in a continuous fashion. Using that estimate, we prove a continuous version of the nonlinear relaxation theorem. An example of a nonlinear parabolic control system is worked out in detail.

MSC:

34G20 Nonlinear differential equations in abstract spaces
35K55 Nonlinear parabolic equations
34H05 Control problems involving ordinary differential equations
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