The bang-bang principle for controlled systems of subdifferential type

In a separable Hilbert space, a nonlinear evolutionary controlled system with evolutionary operators which are subdifferentials of a proper, convex, and lower semicontinuous function depending on time is considered. The control is subjected to a constraint which is a convex closed bounded set from a separable reflexive Banach space. It is shown that any trajectory of the controlled system corresponding to a measurable control can be approximated, to any degree of accuracy and uniformly in time, by trajectories corresponding to step controls whose values are extreme points of the constraint on control. The obtained results are illustrated by an example of a controlled system with a discontinuous nonlinearity.