Optimality necessary conditions for an optimal control problem on time scales

1.   Introduction

In recent years, the calculus of variations and optimal control problems on time scales have attracted much attention. For example, the calculus of variations on time scales was discussed in ^[1,2,3,4], some maximum principles on time scales were studied in ^[5,6,7,8,9], while the existence of optimal solutions or the necessary conditions of optimality for some optimal control problems on time scales were investigated in ^{[10,11,12,13,14,15]}. In particular, Peng et al. ^[11] presented the necessary conditions of optimality for the Lagrange problem of systems governed by linear dynamic equations on time scales with quadratic cost functional. It is necessary to point out that the controlled state variable in ^[11] satisfies the initial value condition.

Throughout this paper, we always assume that $\mathbb{T}$ is a time scale, that is, $\mathbb{T}$ is an arbitrary nonempty closed subset of the real numbers ^[16], $T > 0$ is fixed, $0, \ T\in\mathbb{T}$ and $\sigma(T) = T$. For each interval $\bf{I}$ of $\mathbb{R}$, we denote by $\bf{I}_{\mathbb{T}} = \bf{I}\cap\mathbb{T}$. The notation $\sigma$, which is standard in the study of time scales will be recalled in section 2 as well as the related tools required to follow the paper.

Let $U_{ad}$ be the admissible control set. For any given control policy $u\in U_{ad}$, it is assumed that the change in the controlled state variable $x(t)$ can be described by the following dynamic equation

At the same time, we assume that $x(t)$ satisfies the following loop condition

Suppose that $x_{u}$ is the solution of the controlled system (1.1)–(1.2) corresponding to the control policy $u$ and $x_{d}$ is the desired value. In this paper, we will study optimality necessary conditions for the optimal control problem $(P)$: Find a $u_{0}\in U_{ad}$ such that

where

is the quadratic cost functional. By using the integration by parts on time scales, we obtain some optimality necessary conditions for the problem $(P)$.

2.   Preliminaries

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his PhD thesis ^[17] in 1988 in order to unify continuous and discrete analysis. For more details, one can see ^[16,18,19]. In this section, we will recall some foundational definitions and results from the calculus on time scales which will be used in the paper.

Definition 2.1. The forward jump operator $\sigma:\mathbb{T}\rightarrow\mathbb{T}$ is defined by

while the backward jump operator $\rho:\mathbb{T}\rightarrow\mathbb{T}$ is defined by

In this definition we put $\inf\emptyset = \sup\mathbb{T}$ and $\sup\emptyset = \inf\mathbb{T}$, where $\emptyset$ denotes the empty set. If $\sigma(t) > t$, then $t$ is called right-scattered, while if $\rho(t) < t$, then $t$ is called left-scattered. Also, if $t < \sup\mathbb{T}$ and $\sigma(t) = t$, then $t$ is called right-dense, and if $t > \inf\mathbb{T}$ and $\rho(t) = t$, then $t$ is called left-dense. If $\mathbb{T}$ has a left-scattered maximum $m$, then we define $\mathbb{T}^{k} = \mathbb{T}-\{m\}$, otherwise $\mathbb{T}^{k} = \mathbb{T}$. Finally, the graininess function $\mu:\mathbb{T}\rightarrow[0, +\infty)$ is defined by

Definition 2.2. Assume $f:\mathbb{T}\rightarrow \mathbb{R}$ is a function and let $t\in \mathbb{T}^{k}$. Then $f^{\Delta}(t)$ is defined to be the number (provided it exists) with the property that given any $\epsilon > 0$, there is a neighborhood $U$ of $t \; (i.e., U = (t-\delta, t+\delta)_{\mathbb{T}} \ for\ some\ \delta > 0)$ such that

In this case, $f^{\Delta}(t)$ is called the delta derivative of $f$ at $t$.

Moreover, $f$ is called delta differentiable on $\mathbb{T}^{k}$ provided $f^{\Delta}(t)$ exists for all $t\in\mathbb{T}^{k}$. The function $f^{\Delta}:\mathbb{T}^{k}\rightarrow \mathbb{R}$ is called the delta derivative of $f$ on $\mathbb{T}^{k}$. A function $F:\mathbb{T}\rightarrow \mathbb{R}$ is called an antiderivative of $f:\mathbb{T}\rightarrow \mathbb{R}$ provided

If $F:\mathbb{T}\rightarrow \mathbb{R}$ is an antiderivative of $f:\mathbb{T}\rightarrow \mathbb{R}$, then the Cauchy integral is defined by

Definition 2.3. A function $f:\mathbb{T}\rightarrow \mathbb{R}$ is called rd-continuous provided it is continuous at right-dense points in $\mathbb{T}$ and its left-sided limits exist (finite) at left-dense points in $\mathbb{T}$.

In the following we will provide some important properties of the exponential function which is specific to time scales. Their proofs can be found in ^[16].

Definition 2.4. A function $p:\mathbb{T}\rightarrow \mathbb{R}$ is called regressive provided

holds. The set of all regressive and rd-continuous functions will be denoted by $\mathcal{R}$. The set of positively regressive functions $\mathcal{R}^{+}$ is defined as the set consisting of those $p\in \mathcal{R}$ satisfying

Lemma 2.1. ^[16] Let $p\in\mathcal{R}$, $t_{0}, s\in \mathbb{T}$ and $e_p(\cdot, t_{0})$ be the exponential function on $\mathbb{T}$. Then

(ⅰ)$\ \ e_p(t, t)\equiv1 \ for\ all\ t\in \mathbb{T};$

(ⅱ)$\ \ e_p^{\Delta }(t, t_0) = p(t)e_p(t, t_0) \ for\ all\ t\in \mathbb{T}^{k}$;

(ⅲ)$\ \ \ e_p(t, t_0) = \frac{1}{e_p(t_0, t)} \ for\ all\ t\in \mathbb{T}$;

(ⅳ)$\ \ \ e_p(t, s)e_p(s, t_0) = e_p(t, t_0) \ for\ all\ t\in \mathbb{T}$;

(ⅴ)$\ \ \ \left(\frac{1}{e_p(t, t_0)}\right)^{\Delta } = -\frac{p(t)}{e_p({\sigma}(t), t_0)} \ for\ all\ t\in \mathbb{T}^{k}$.

Moreover, if $p\in\mathcal{R}^{+}$, then

Lemma 2.2. ^[16] Assume $f, g:\mathbb{T}\rightarrow \mathbb{R}$ are differentiable at $t\in \mathbb{T}^{k}$. Then the product $fg:\mathbb{T}\rightarrow \mathbb{R}$ is differentiable at $t$ with

Lemma 2.3. ^[16] If $a, b\in \mathbb{T}$ and $f, g:\mathbb{T}\rightarrow \mathbb{R}$ are rd-continuous functions, then

In the remainder of this paper, we always assume that Banach space

is equipped with the norm $\left\| x\right\| = \max\limits_{t\in \left[ 0, T\right]_\mathbb{T}}\left| x\left(t\right) \right|$, $p:[0, T]_{\mathbb{T}}\rightarrow(0, +\infty)$ is rd-continuous and denote

Lemma 2.4. For any $g\in C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})$, the following first-order linear periodic boundary value problem (PBVP for short)

has a unique solution

Proof. Since $g$ and $e_{p}$ are rd-continuous, we know that the right side of (2.2) is well defined. By the equation in (2.1), Lemma 2.1 and Lemma 2.2, we get

So,

It follows from (2.3) and the boundary condition in (2.1) that

And so,

Lemma 2.5. ^[20] For any $h\in C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})$, the following first-order linear PBVP

has a unique solution

3.   Main results

From now on, we always suppose that the control space is $C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})$ and the admissible control set $U_{ad}$ is a nonempty convex subset of $C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})$.

Theorem 3.1. Assume that $f, q\in C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})$. Let $(x_{u_{0}}, u_{0})\in C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})\times U_{ad}$ be an optimal pair of the problem $(P)$. Then

and there exists a function $\varphi\in C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})$ such that

Proof. Since $(x_{u_{0}}, u_{0})\in C_{rd}([0, T] _{\mathbb{T}}, \mathbb{R})\times U_{ad}$ is an optimal pair of the problem $(P)$, it must satisfy (3.1).

According to Lemma 2.4, we know that the following PBVP

has a unique solution $\varphi$.

In what follows, we shall show that

For any fixed $u\in U_{ad}$, we first consider the following PBVP

By Lemma 2.5, we know that the PBVP (3.4) has a unique solution $z$.

Next, for $\epsilon\in[0, 1]$, we denote

Then, the hypothesis that $U_{ad}$ is a nonempty convex set yields $u_{\epsilon}\in U_{ad}$ for $\epsilon\in[0, 1]$ and moreover from Lemma 2.5 we obtain

In view of (3.2), (3.4) and Lemma 2.3, we have

which together with (3.5) and (3.6) indicates that for any $\epsilon\in[0, 1]$,

Since $u_{0}$ is an optimal solution of the problem $(P)$, for any $\epsilon\in[0, 1]$, we get

which implies that for any $u\in U_{ad}$,

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11661049).

Conflict of interest

The authors declare that there are no conflict of interest regarding the publication of this paper.