Minimum time problem for the double integrator with a loss-of-control region

In this paper we address the minimum time problem to reach the origin for the double integrator but, in contrast with the classical version of this problem, the control is constrained to be frozen as long as the corresponding state belongs to a given region of the state space called loss control region . This situation prevents switches to occur in the loss control region and, therefore, a new analysis has to be performed. To this aim we prove a Pontryagin maximum principle adapted to this setting. The necessary conditions involve a costate that admits discontinuity jumps at the interface between the loss control region and its complement, and an averaged Hamiltonian gradient condition to determine the optimal constant value of the control in the loss control region. The purpose of this paper is to highlight the use of this principle and also the new behaviors that are observed in this new setting, such as the lacks of dynamical programming principle, feedback expression and saturation of the control constraint set.