Optimal Impulsive Linear Systems: Sufficient Conditions and Maximum Number of Impulses

The optimal control of a linear system with impulsive force control inputs is considered. The cost functional to be minimized is the integral over time of the magnitude of the control force per unit mass, which is equivalent to the sum of the magnitudes of the discontinuities in the velocity vector caused by the force impulses. Previously derived necessary conditions for an optimal solution are shown to also be sufficient conditions for a global minimum. A proof is also given that there exists a maximum number q of impulses required for any solution that satisfies the specified boundary conditions. The value of q is equal to the number of specified final state variables and thus the optimal solution requires at most q impulses. In addition, a procedure is derived and illustrated whereby a solution using more than q impulses can be reduced to a q-impulse solution of equal or lower cost.