0, u_0\in(\alpha,1). $$ Therefore, there exists a point $u_*\in(\alpha,1)$ where the maximum value is achieved: $J(u_*)=\max\{J(u), u\in[0,1]\}$. Then $u(t)\equiv u_*$ is an optimal control. Note that in general $u_*$ does not have to be unique (appropriate non-uniqueness examples are readily constructed). Suppose next that $F$ and $G$ are monotone and satisfy (H4). We claim that the above optimal control $u(t)\equiv u_*$ is unique then. Indeed, the value of the functional $J$ with the constant control $u$ is $$ J(x(\cdot),u)=(1-u)F(x_u)=\big[1-\frac{G(x_u)}{F(x_u)}\big]\cdot F(x_u)=F(x_u)-G(x_u), $$ which assumes the unique maximum value at $x_{u_*}$ when $u=u_*$. \end{proof} \begin{example}\label{Ram} \rm Controlled Ramsey model with delay. \end{example} The differential equation \begin{equation}\label{Ramsey1} \frac{dK(t)}{dt}=B K^p(t-\tau)-bK(t)\,. \end{equation} was proposed as a modified Ramsey economic model with delay \cite{IvaSwi08,Ram28}. Consider here the respective control problem \eqref{Ramsey2}-\eqref{consumption}-\eqref{functional} $$ \frac{dK(t)}{dt}=u(t)B K^p(t-\tau)-bK(t)\,, $$ where $ B>0$, $b>0$, $0 < p < 1$ and $u(t) \equiv u\in[0,1]$ is a constant control. It is easy to check that assumptions (H3$\;'$) and (H4) hold with $\alpha = 0$. One readily finds the steady state $x_u$ and the respective value of the functional $J(x_u)$ as $$ x_u=\Big(\frac{B}{b}\Big)^{1/(1-p)}\cdot u^{1/(1-p)}, \quad J(u)=\Big(\frac{B}{b}\Big)^{p/(1-p)}\,\cdot (1-u)\cdot u^{p/(1-p)}. $$ The unique maximum value of $J(u)$ is achieved when $u=p$. \subsection*{Acknowledgments} This research was supported in part by the NSF (USA) and the ARC (Australia). This work was done during the first author's visit and stay at the CIAO/GSITMS of the University of Ballarat, Australia. He is thankful for the hospitality and support extended to him during this visit. \begin{thebibliography}{00} \bibitem{ChoHal78} S. N. Chow, J. K. 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