Abstract
The optimal control of a system whose states are governed by a nonlinear autonomous Volterra integrodifferential equation with unbounded time interval is considered. Specifically, it is assumed that the delay occurs only in the state variable. The results obtained extend those of Brock and Haurie [Math. Oper. Res., 1 (1976), pp. 337–346] and Leizarowitz [Math. Oper. Res., 10 (1985), pp. 450–461]. In particular, it is shown that (under appropriate hypotheses) catching-up optimal solutions asymptotically approach a unique optimal steady state, and thus enjoy the so-called “turnpike” property found in the economics literature. By combining this result with an associated optimal control problem, the desired existence result is obtained. Furthermore, it is remarked that, in addition to extending these earlier works to the time-delay case, the results presented below utilize convexity, seminormality, and growth hypotheses that in some cases are weaker than those encountered in the above-mentioned papers.
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