Well-posedness and Bilinear Controllability of a Repairable System with Degraded State

In this work, we consider the dynamics of repairable systems characterized by three distinct states: one signifying normal operational states, another representing degraded conditions and a third denoting failed conditions. These systems are characterized by their ability to be repaired when failures and/or degradation occur. Typically described by transport equations, these systems exhibit a coupled nature, interlinked through integro-differential equations and integral boundary conditions that dictate the transitions among all the states. In this paper, we address two less-explored facets: 1) the well-posedness and the asymptotic behavior of such systems with maximum repair time being finite; and 2) the bilinear controllability of the system via repair actions. In particular, we focus on the case where only one degraded and one failed states exist. We first discuss part 1) for given time-independent repair rates and then design the space-time dependent repair strategies that can manipulate system dynamics to achieve the desired level over a finite horizon. Our objective is to enhance the system availability -- the probability of being operational when needed over a fixed period of time. We present rigorous analysis and develop control strategies that leverage the bilinear structure of the system model.