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.