Optimal control for a class of impulsive switched systems

In this paper, an optimal switch-time control problem is solved for a class of impulsive switched autonomous systems. The considered systems jump at the switching times, and the sequence of active subsystems is pre-specified. The control variables consist of the impulse times and a set of scalars which determine the jump amplitudes. Moreover, the subsystems do not require a refractory period, which can bring more generality. Using the calculus of variation, the partial derivatives of the cost with respect to the control variables are derived, based on which the optimality conditions are given. Meanwhile, the obtained formulas can be used in some gradient descent algorithms to locate the optimal control variables. Finally, the viability of the proposed method is illustrated through two numerical examples.