The Existence of Optimal Controls for Problems Defined on Time Scales

Abstract

In this paper we investigate the existence of optimal solutions for dynamic optimization problems defined on time scales. We use the classical convexity and seminormality conditions originating in the works of L. Tonelli and E. J. McShane for problems in the calculus of variations and in the works of L. Cesari, C. Olech, R. T. Rockafellar and others for problems in optimal control theory, thus extending these classical results to optimal control problems whose states satisfy a dynamic equation on an arbitrary time scale. As applications of our results we focus on three examples of time scales—the real line, the integers and a monotone sequence of points converging to one.

Appendix 1: Proof of Growth Condition (\(\mathbf {\mathrm {g}_2}\)) for the Economic Growth Model

We first observe that since $F'(\cdot )$ decreases from $\infty$ to $0$ there exists a $x_\delta >0$ such that $F'(x_\delta )=\delta$. Since $F'' <0$ we have $F(x)\le F(x_\delta )+F'(x_\delta )(x-x_\delta )$ for all $x\ge 0$. Multiplying through by $1/\mu (t) >0$ and rearranging the terms gives us

$$\begin{aligned} \frac{1}{\mu (t)}[F(x)-\delta x] \le \frac{B}{\mu (t)} \end{aligned}$$

for all $t\in [0,1[_\mathbb T$ and all $x\ge 0$, where $B=F(x_\delta )-\delta x_\delta$. Similarly, for any $\varepsilon >0$ there exists $c_\varepsilon >0$ so that $V'(c_\varepsilon )=(1-\beta )/\varepsilon$ and $V'' <0$ gives us

$$\begin{aligned} \frac{1}{1-\beta }V(c)\le \frac{1}{1-\beta }V(c_\varepsilon )+\frac{1}{\varepsilon }(c-c_\varepsilon ) \end{aligned}$$

for all $c\ge 0$ or, equivalently that

$$\begin{aligned} -c\le C_\varepsilon -\frac{\varepsilon }{1-\beta } V(c) \end{aligned}$$

for all $c >0$, where $C_\varepsilon = (\varepsilon /(1-\beta )) V(c_\varepsilon ) -c_\varepsilon$. Further since $\mu (t) <1$ it follows that $-c/\mu (t) \le -c$ so that we have

$$\begin{aligned} \frac{-c}{\mu (t)}\le -c \le C_\varepsilon -\frac{\varepsilon }{1-\beta } V(c). \end{aligned}$$

Combining the above gives us

$$\begin{aligned} \left| \frac{1}{\mu (t)}[F(x)-\delta x -c]\right| \le \left[ \frac{B}{\mu (t)} +C_\varepsilon \right] -\frac{\varepsilon }{1-\beta } V(c)=\psi _\varepsilon (t)-\frac{\varepsilon }{1-\beta } V(c), \end{aligned}$$

for all $(t,x,u)\in M=\{(s,y,v):\ s\in [0,1[_\mathbb T,\ y\ge 0,\ v\in U(t,x)\}$, where $\psi _\varepsilon (t)=(B/\mu (t))+C_\varepsilon$ is locally $\Delta$-Lebesgue integrable, which is precisely growth condition $(\mathrm {g}_2)$.