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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: $$-\dot x(t)\in\partial g(t, x(t))+ F(t, x(t))\quad\text{a.e. on }[0,b],\quad x(0)= a.\tag E$$ The function $g(t,\cdot)$ is assumed to be convex, and the symbol “$\partial$” is understood as the subdifferential operator in the sense of convex analysis. Here $F(t,\cdot): H\to H$ 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 $$J(x, u)= \int^b_0 L(t,x(t), u(t))dt,$$ among all trajectories $x$ satisfying the evolution inclusion (E), and all measurable controls $u$ satisfying the feedback inclusion $$u(t)\in U(t, x(t))\quad\text{a.e.}$$ Special attention is paid to existence results, relaxability, and well-posedness.

49J24Optimal control problems with differential inclusions (existence) (MSC2000)
34A60Differential inclusions
49J52Nonsmooth analysis (other weak concepts of optimality)
34G20Nonlinear ODE in abstract spaces
35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general
49J27Optimal control problems in abstract spaces (existence)
49J40Variational methods including variational inequalities
49J45Optimal control problems involving semicontinuity and convergence; relaxation
35R70PDE with multivalued right-hand sides