Optimum control of linear systems with a "Modified" energy constraint

Linear control processes are considered under the following optimization criteria: 1) minimization of the terminal error and 2) minimization of the required time (
T
) to reach a desirable state. The constraint on the control vector
(u(t))
is considered to be
\int\liminf{t_{j}} \limsup{t_{j+1}} u'(t)Qu(t)dt \leq c_{j} j = 1, 2, ... m
where
Q
is a positive definite matrix, and
t_{j} < t <t_{j+1}
represents any given interval contained in the interval
0<t<T
, and c
j
can be considered as the available energy during that interval. A condition of optimality has been obtained which can be used analytically. Furthermore, a numerical procedure is developed for determination of the optimum control vector.