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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 $$-\dot x(t)\in \partial\varphi(t, x(t), \lambda)+ F(t, x(t), \lambda)\quad\text{a.e. in }T,\quad x(0)= x\sb 0(\lambda),$$ defined on a separable Hilbert space $H$, where $T= [0,b]$, $\partial\varphi(t,\cdot, \lambda)$ denotes the subdifferential of a proper, lower semicontinuous convex function, and $\lambda\in E$ 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 $S(\lambda)$ denotes the set of strong solutions, it is shown that the multifunction $\lambda\mapsto S(\lambda)$ has a closed graph in $E\times C(T, H)$ ($\lambda\sb n\to \lambda$ in $E\Rightarrow \varlimsup S(\lambda\sb n)\subset S(\lambda)$), and it is lower semicontinuous ($\lambda\sb n\to \lambda$ in $E\Rightarrow S(\lambda) \subset \varliminf S(\lambda\sb n)$). 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 $\int\sb T L(t, x(t), u(t), \lambda) dt$ subject to $-\dot x(t)\in \partial\varphi(t, x(t), \lambda)+ g(t, x(t), \lambda)+ B(t, \lambda) u(t)$ a.e. in $T$, $x(0)= x\sb 0(\lambda)$, $u(t)\in U(t, \lambda)$ 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.

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)
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