Properties of the Relaxed Trajectories of Evolution Equations and Optimal Control
Nikolaos S. Papageorgiou · SIAM Journal on Control and Optimization · 1989
This paper considers a class of nonconvex control problems in a Banach space, governed by a semilinear evolution equation and establishes the existence of admissible pairs. Then the introduction of measure-valued controls convexifies the system, and under general assumptions it is shown that the set of trajectories of the original system is dense (in an appropriate topology) in the set of relaxed trajectories. Some useful topological properties of the set of relaxed trajectories are also determined. Furthermore, some optimization problems are solved and in many cases the values of the original and relaxed problems are shown to be equal. Finally, another relaxation result is proved for a different class of systems, described by a general nonlinear evolution equation.