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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:

-x ˙(t)g(t,x(t))+F(t,x(t))a.e.on[0,b],x(0)=a·(E)

The function g(t,·) is assumed to be convex, and the symbol “” is understood as the subdifferential operator in the sense of convex analysis. Here F(t,·):HH 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)= 0 b 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)U(t,x(t))a.e.

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

MSC:
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