Relaxed controls and measure controls

The concept of measure differential equation was recently introduced in [10]. Such a concept allows deterministic modeling of uncertainty, finite-speed diffusion, concentration, and other phenomena. Moreover, it represents a natural generalization of Ordinary Differential Equations to measures. In this paper, we introduce a new idea to generalize the concept of relaxed control to the new framework of measure differential equations. A relaxed control is defined as probability measure on the space of controls, and, similarly, a measure control is a feedback relaxed control which depend on the measure distribution on the state space representing the state of the system. We establish regularity properties of measure controls to ensure existence and uniqueness of trajectories.