Discussiones Mathematicae Differential Inclusions 16(1) (1996) 53-74

SUBDIFFERENTIAL EVOLUTION INCLUSIONS

Tiziana Cardinali and Simona Pieri

*University of Perugia, Department of Mathematics
Via Vanvitelli, 1 Perugia, 06123, Italy*

In this paper we study Cauchy problems for retarded evolution inclusions, where the
Fréchet subdifferential of a function F:Ω→**R**∪{+∞} (Ω is
an open subset of a real separable Hilbert space) having a φ-monotone
subdifferential of oder two is present. First we establish the existence of extremal
trajectories and we show that the set of these trajectories is dense in the solution set
of the original convex problem for the norm topology of the Banach space C([-r, T_{0}],
Ω) ("strong relaxation theorem"). Then we prove
that this density result allows one to establish a nonlinear "bang-bang
principle" for a class of control systems with dealy on a nonconvex constraint. We
want to observe that the results here extend those of [11] and [13].

**Keywords: **Fréchet subdifferential, subdifferential evolution inclusion, dealy,
strong solution, extremal trajectory, solution set, relaxation theorem, bang-bang
principle, control system.

**1991 Mathematics Subject Classification:** 34G20.

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