A PROPER RELAXATION OF CONTROLS WITH VARIABLE SHIFTS

In a previous paper written jointly with Q. J. Zhu (1992, J. Math. Ana l. Appl. 169, 546-561) we studied optimal control problems defined by functional-integral equations (and, in particular, ordinary differenti al equations) with shifts in the controls and with the shifted control s not necessarily separated (i.e., either additively or nonadditively coupled). In that paper it was assumed that the domain of the state an d control functions is a cartesian product of an interval with a compa ct metric space and that each shift h(j), j = 1,..., k, has a one-dime nsional component of the form t(1) - d(j), where d(1),..., d(k) are co nstant, possibly noncommensurate, delays and advances. In the present note we extend those results to the case where each di is replaced by a function d(j)(.) that may vary and take on, at different times, both positive and nonpositive values. (C) 1995 Academic Press, Inc.