Variation based shooting method for the variable time impulsive system ontimizntion

This paper proposes a variation based shooting method to solve the variable time impulsive system optimization problem. For the variable time impulsive system optimization, the moments when the impulsive control happens (impulsive moments/impulsive time) are not fixed but a function of the states. Therefore, the impulsive time also needs to be optimized besides the controls. Through the study of the necessary conditions based on the variational calculus, a series of differential equations and a series of impulsive equations are to be satisfied with the boundary constraints, and the impulsive time can be implicitly decided using so called jump condition. This optimization problem is then converted to a two point boundary value(TPBV) problem. Considering both the differential equations and impulsive equations are involved in this TPBV problem, a variation based shooting method is proposed for numerical solutions. By mapping the variations of the final constraints into that of the initial constraints, the initial states and costates can be properly adjusted and the TPBV problem can be iteratively solved. Simulation examples are also presented for illustrative purpose.