Properties of Relaxed Trajectories for a Class of Nonlinear Evolution Equations on a Banach Space
Naveed Ahmed · SIAM Journal on Control and Optimization · 1983
In this paper, we consider a class of nonconvex control problems in a Banach space, which is convexified by introducing measure-valued controls. We study some topological properties of the set of trajectories of the original system and that of the relaxed system. It is shown that, under certain reasonable assumptions, the set of trajectories of the original system is dense (in an appropriate topology) in the set of relaxed trajectories. As a corollary of this result it follows that the attainable set of the original system is dense in that of the relaxed system. These results are directly useful in terminal optimization problems and time-optimal control problems. For illustration, we present two examples in § 6.