# Vartiational Optimal-Control Problems with Delayed Arguments on Time Scales

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## Abstract

This paper deals with variational optimal-control problems on time scales in the presence of delay in the state variables. The problem is considered on a time scale unifying the discrete, the continuous, and the quantum cases. Two examples in the discrete and quantum cases are analyzed to illustrate our results.

### Keywords

Optimal Control Problem Performance Index Quantum Case Jump Operator Order Linear Differential Equation## 1. Introduction

The calculus of variations interacts deeply with some branches of sciences and engineering, for example, geometry, economics, electrical engineering, and so on [1]. Optimal control problems appear in various disciplines of sciences and engineering as well [2].

Time-scale calculus was initiated by Hilger (see [3] and the references therein) being in mind to unify two existing approaches of dynamic models difference and differential equations into a general framework. This kind of calculus can be used to model dynamic processes whose time domains are more complex than the set of integers or real numbers [4]. Several potential applications for this new theory were reported (see, e.g., [4, 5, 6] and the references therein). Many researchers studied calculus of variations on time scales. Some of them followed the delta approach and some others followed the nabla approach (see, e.g., [7, 8, 9, 10, 11, 12]).

It is well known that the presence of delay is of great importance in applications. For example, its appearance in dynamic equations, variational problems, and optimal control problems may affect the stability of solutions. Very recently, some authors payed the attention to the importance of imposing the delay in fractional variational problems [13]. The nonlocality of the fractional operators and the presence of delay as well may give better results for problems involving the dynamics of complex systems. To the best of our knowledge, there is no work in the direction of variational optimal-control problems with delayed arguments on time scales.

Our aim in this paper is to obtain the Euler-Lagrange equations for a functional, where the state variables of its Lagrangian are defined on a time scale whose backward jumping operator is Open image in new window , Open image in new window , Open image in new window . This time scale, of course, absorbs the discrete, the continuous and the quantum cases. The state variables of this Lagrangian allow the presence of delay as well. Then, we generalize the results to the Open image in new window -dimensional case. Dealing with such a very general problem enables us to recover many previously obtained results [14, 15, 16, 17].

The structure of the paper is as follows. In Section 2 basic definitions and preliminary concepts about time scale are presented. The nabla time-scale derivative analysis is followed there. In Section 3 the Euler-Lagrange equations into one unknown function and then in the Open image in new window -dimensional case are obtained. In Section 4 the variational optimal control problem is proposed and solved. In Section 5 the results obtained in the previous sections are particulary studied in the discrete and quantum cases, where two examples are analyzed in details. Finally, Section 6 contains our conclusions.

## 2. Preliminaries

In order to define the backward time-scale derivative down, we need the set Open image in new window which is derived from the time scale Open image in new window as follows. If Open image in new window has a right-scattered minimum Open image in new window , then Open image in new window . Otherwise, Open image in new window .

Definition 2.1 (see [18]).

Moreover, we say that Open image in new window is (nabla) differentiable on Open image in new window provided that Open image in new window exists for all Open image in new window .

The following theorem is Theorem Open image in new window in [19] and an analogue to Theorem Open image in new window in [4].

Theorem 2.2 (see [18]).

- (i)
If Open image in new window is differentiable at Open image in new window then Open image in new window is continuous at Open image in new window .

- (ii)If Open image in new window is continuous at Open image in new window and Open image in new window is left-scattered, then Open image in new window is differentiable at Open image in new window with
- (iii)If Open image in new window is left-dense, then f is differentiable at Open image in new window if and only if the limitexists as a finite number. In this case
- (iv)If Open image in new window is Open image in new window -differentiable at Open image in new window , then

- (i)
Open image in new window or any any closed interval (the continuous case) Open image in new window , Open image in new window and Open image in new window .

- (ii)
Open image in new window , Open image in new window or any subset of it (the difference calculus, a discrete case) Open image in new window , Open image in new window , Open image in new window , and Open image in new window .

- (iii)
Open image in new window , Open image in new window , (quantum calculus) Open image in new window , Open image in new window , Open image in new window , and Open image in new window .

- (iv)
Open image in new window , Open image in new window , Open image in new window (unifying the difference calculus and quantum calculus). There are Open image in new window , Open image in new window , Open image in new window , and Open image in new window . If Open image in new window then Open image in new window and so Open image in new window . Note that in this example the backward operator is of the form Open image in new window and hence Open image in new window is an element of the class Open image in new window of time scales that contains the discrete, the usual, and the quantum calculus (see [17]).

Theorem 2.4.

- (1)
the sum Open image in new window is nabla differentiable at Open image in new window and Open image in new window

- (2)
for any Open image in new window , the function Open image in new window is nabla differentiable at Open image in new window and Open image in new window ;

- (3)the product Open image in new window is nabla differentiable at Open image in new window and
For the proof of the following lemma we refer to [20].

Lemma 2.5.

Throughout this paper we use for the time-scale derivatives and integrals the symbol Open image in new window which is inherited from the time scale Open image in new window . However, our results are true also for the Open image in new window -time scales (those time scales whose jumping operators have the form Open image in new window ). The time scale Open image in new window is a natural example of an Open image in new window -time scale.

Definition 2.6.

The following lemma which extends the fundamental lemma of variational analysis on time scales with nabla derivative is crucial in proving the main results.

Lemma 2.7.

The proof can be achieved by following as in the proof of Lemma Open image in new window in [9] (see also [17]).

## 3. First-Order Euler-Lagrange Equation with Delay

where we have used the fact that Open image in new window on Open image in new window

In (3.8), once choose Open image in new window such that Open image in new window and Open image in new window on Open image in new window and in another case choose Open image in new window such that Open image in new window and Open image in new window on Open image in new window , and then make use of Lemma 2.7 to arrive at the following theorem.

Theorem 3.1.

holds along Open image in new window for all admissible variations Open image in new window satisfying Open image in new window , Open image in new window .

The necessary condition represented by (3.12) is obtained by applying integration by parts in (3.7) and then substituting (3.11) in the resulting integrals. The above theorem can be generalized as follows.

Theorem 3.2.

## 4. The Optimal-Control Problem

where Open image in new window is a Lagrange multiplier or an adjoint variable.

Note that condition (4.6) disappears when the Lagrangian Open image in new window is free of the delayed time scale derivative of Open image in new window .

## 5. The Discrete and Quantum Cases

- (i)
The Discrete Case

holds along Open image in new window for all admissible variations Open image in new window satisfying Open image in new window , Open image in new window .

In this case the Open image in new window -optimal-control problem would read as follows.

Note that condition (5.9) disappears when the Lagrangian Open image in new window is independent of the delayed Open image in new window derivative of Open image in new window .

Example 5.1.

- (ii)
The Quantum Case

holds along Open image in new window for all admissible variations Open image in new window satisfying Open image in new window , Open image in new window .

In this case the Open image in new window -optimal-control problem would read as follows.

where Open image in new window is a constant and Open image in new window and Open image in new window are functions with continuous first and second partial derivatives with respect to all of their arguments.

Note that condition (5.25) disappears when the Lagrangian Open image in new window is independent of the delayed Open image in new window derivative of Open image in new window .

Example 5.2.

Clearly the solutions for this equation are Open image in new window and Open image in new window . For details see [16].

## 6. Conclusion

In this paper we have developed an optimal variational problem in the presence of delay on time scales whose backward jumping operators are of the form Open image in new window , Open image in new window , Open image in new window , called Open image in new window -time scales. Such kinds of time scales unify the discrete, the quantum, and the continuous cases, and hence the obtained results generalized many previously obtained results either in the presence of delay or without. To formulate the necessary conditions for this optimal control problem, we first obtained the Euler-Lagrange equations for one unknown function then generalized to the n-dimensional case. The state variables of the Lagrangian in this case are defined on the Open image in new window -time scale and contain some delays. When Open image in new window and Open image in new window with the existence of delay some of the results in [14] are recovered. When Open image in new window and Open image in new window and the delay is absent most of the results in [16] can be reobtained. When Open image in new window and the delay is absent some of the results in [15] are reobtained. When the delay is absent and the time scale is free somehow, some of the results in [17] can be recovered as well.

Finally, we would like to mention that we followed the line of nabla time-scale derivatives in this paper, analogous results can be originated if the delta time-scale derivative approach is followed.

## Notes

### Acknowledgment

This work is partially supported by the Scientific and Technical Research Council of Turkey.

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