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On parametric evolution inclusions of the subdifferential type with applications to optimal control problems. (English) Zbl 0818.34010

This article deals with parametric nonlinear evolution inclusions of the form

-x ˙(t)ϕ(t,x(t),λ)+F(t,x(t),λ)a.e.inT,x(0)=x 0 (λ),

defined on a separable Hilbert space H, where T=[0,b], ϕ(t,·,λ) denotes the subdifferential of a proper, lower semicontinuous convex function, and λE is a parameter taking values in a complete metric space E.

Several continuous dependence results for the above problem are provided under mild assumptions on the initial data. In particular, if S(λ) denotes the set of strong solutions, it is shown that the multifunction λS(λ) has a closed graph in E×C(T,H) (λ n λ in Elim ¯S(λ n )S(λ)), and it is lower semicontinuous (λ n λ in ES(λ)lim ̲S(λ n )). Therefore, the multifunction is K-continuous in the classical Kuratowski sense. Under certain stronger hypotheses, it is shown that it is also Vietoris and Hausdorff continuous.

These results lead to a sensitivity (variational) analysis of a class of nonlinear, infinite-dimensional optimal control problems, namely, that of minimizing the functional T L(t,x(t),u(t),λ)dt subject to -x ˙(t)ϕ(t,x(t),λ)+g(t,x(t),λ)+B(t,λ)u(t) a.e. in T, x(0)=x 0 (λ), u(t)U(t,λ) a.e. in T. Some examples illustrate the applicability of the results obtained, including a parametrized family of nonlinear parabolic variational inequalities with unilateral constraints (obstacle problems), a sequence of optimal control problems with rapidly oscillating coefficients in their dynamics, differential variational inequalities, and a certain class of multivalued parabolic partial differential equations related to the study of free boundary problems.

MSC:
34A60Differential inclusions
35K55Nonlinear parabolic equations
35L85Linear hyperbolic unilateral problems; linear hyperbolic variational inequalities
34G20Nonlinear ODE in abstract spaces
49J24Optimal control problems with differential inclusions (existence) (MSC2000)