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Abstract
Linear time-invariant systems of the form are considered. The problem is to find a control vector u(t) that will drive the state of this system from a given z(0) to some (not necessarily fixed) final state z(t
f
) in some (not necessarily fixed) time t
f
while minimizing a cost functional of the form
The original system is re-written in the form of a descriptor-variable system consisting of the equations diag
It is shown that there is a matrix
such that
thus the original optimal control problem is equivalent to the following problem. Find a control vector u(t) that drives the descriptor vector x(t) from a given x(0) to some x(t
f
) while minimizing
Using the results of an earlier paper on optimal control of descriptor systems (Ibrahim et al. 1988), necessary conditions are derived for the existence of minima of
; the problem of finding sufficient conditions for the existence of minima of
is not considered. Various formulae are obtained for designing open-loop controls corresponding to three different terminal boundary conditions (t
f
fixed, z(t
f
)=0; t
f
fixed, z(t
f
) free; t
f
free, z(t
f
)=0). The problem of choosing the matrix Q of the original cost functional J is investigated in some detail.