A Proper Relaxation of Controls with Variable Shifts

In a previous paper written jointly with Q. J. Zhu (1992, J. Math. Anal. Appl.169, 546-561) we studied optimal control problems defined by functional-integral equations (and, in particular, ordinary differential equations) with shifts in the controls and with the shifted controls not necessarily separated (i.e., either additively or nonadditively coupled). In that paper it was assumed that the domain of the state and control functions is a cartesian product of an interval with a compact metric space and that each shift hj, j = 1, ..., k, has a one-dimensional component of the form t1 − dj, where d1, ..., dk are constant, possibly noncommensurate, delays and advances. In the present note we extend those results to the case where each dj is replaced by a function dj(•) that may vary and take on, at different times, both positive and nonpositive