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

$-\stackrel{˙}{x}\left(t\right)\in \partial \varphi \left(t,x\left(t\right),\lambda \right)+F\left(t,x\left(t\right),\lambda \right)\phantom{\rule{1.em}{0ex}}\text{a.e.}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}T,\phantom{\rule{1.em}{0ex}}x\left(0\right)={x}_{0}\left(\lambda \right),$
defined on a separable Hilbert space $H$, where $T=\left[0,b\right]$, $\partial \varphi \left(t,·,\lambda \right)$ 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 $S\left(\lambda \right)$ denotes the set of strong solutions, it is shown that the multifunction $\lambda ↦S\left(\lambda \right)$ has a closed graph in $E×C\left(T,H\right)$ (${\lambda }_{n}\to \lambda$ in $E⇒\overline{lim}S\left({\lambda }_{n}\right)\subset S\left(\lambda \right)$), and it is lower semicontinuous (${\lambda }_{n}\to \lambda$ in $E⇒S\left(\lambda \right)\subset \underline{lim}S\left({\lambda }_{n}\right)$). 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 ${\int }_{T}L\left(t,x\left(t\right),u\left(t\right),\lambda \right)dt$ subject to $-\stackrel{˙}{x}\left(t\right)\in \partial \varphi \left(t,x\left(t\right),\lambda \right)+g\left(t,x\left(t\right),\lambda \right)+B\left(t,\lambda \right)u\left(t\right)$ a.e. in $T$, $x\left(0\right)={x}_{0}\left(\lambda \right)$, $u\left(t\right)\in U\left(t,\lambda \right)$ 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.