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Time-dependent subdifferential evolution inclusions and optimal control. (English) Zbl 0909.49005

The present monograph concerns the study of an evolution inclusion of subdifferential type:

$-\stackrel{˙}{x}\left(t\right)\in \partial g\left(t,x\left(t\right)\right)+F\left(t,x\left(t\right)\right)\phantom{\rule{1.em}{0ex}}\text{a.e.}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\left[0,b\right],\phantom{\rule{1.em}{0ex}}x\left(0\right)=a·\phantom{\rule{2.em}{0ex}}\left(\mathrm{E}\right)$

The function $g\left(t,·\right)$ is assumed to be convex, and the symbol “$\partial$” is understood as the subdifferential operator in the sense of convex analysis. Here $F\left(t,·\right):H\to H$ is a nonmonotone set-valued perturbation with a time varying domain which satisfies a certain growth condition, and $H$ is a separable Hilbert space. The authors discuss several issues related to the above evolution system: existence of solutions, relaxation, dependence of the solution set on external parameters, path-connectedness of the solution set. In a second part, the authors discuss an abstract optimal control problem which consists in minimizing the cost functional

$J\left(x,u\right)={\int }_{0}^{b}L\left(t,x\left(t\right),u\left(t\right)\right)dt,$

among all trajectories $x$ satisfying the evolution inclusion (E), and all measurable controls $u$ satisfying the feedback inclusion

$u\left(t\right)\in U\left(t,x\left(t\right)\right)\phantom{\rule{1.em}{0ex}}\text{a.e.}$

Special attention is paid to existence results, relaxability, and well-posedness.

##### MSC:
 49J24 Optimal control problems with differential inclusions (existence) (MSC2000) 34A60 Differential inclusions 49J52 Nonsmooth analysis (other weak concepts of optimality) 34G20 Nonlinear ODE in abstract spaces 35G10 Initial value problems for linear higher-order PDE 35K25 Higher order parabolic equations, general 49J27 Optimal control problems in abstract spaces (existence) 49J40 Variational methods including variational inequalities 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 35R70 PDE with multivalued right-hand sides