Time-optimal control for the hybrid double integrator with state-driven jumps

The problem of steering the state of a double integrator from a given initial condition to the origin in minimum time and in the presence of point-wise constraint on the control input has been thoroughly characterized in the case of purely continuous-time systems. The objective of this paper consists in extending the theory to the hybrid setting, namely with the double integrator dynamics potentially undergoing state-driven jumps. In particular, the optimal solution is characterized and compared with the purely continuous one, namely assessing potential advantages of a hybrid scheme in controlling the state of a double integrator to the origin. Moreover, we first construct the optimal solution in the case of a prescribed number of jumps before reaching the origin and then we provide sufficient conditions ensuring that the overall minimum-time solution undergoes a finite number of jumps.