*(English)*Zbl 0909.49005

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

The function $g(t,\xb7)$ is assumed to be convex, and the symbol “$\partial $” is understood as the subdifferential operator in the sense of convex analysis. Here $F(t,\xb7):H\to H$ 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

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

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 |