Advances in Difference Equations

, 2009:840386 | Cite as

Vartiational Optimal-Control Problems with Delayed Arguments on Time Scales

  • Thabet Abdeljawad (Maraaba)
  • Fahd Jarad
  • Dumitru Baleanu
Open Access
Research Article

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

A time scale is an arbitrary closed subset of the real line Open image in new window . Thus the real numbers and the natural numbers, Open image in new window , are examples of a time scale. Throughout this paper, and following [4], the time scale will be denoted by Open image in new window . The forward jump operator Open image in new window is defined by
while the backward jump operator Open image in new window is defined by

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]).

Assume that Open image in new window is a function and Open image in new window . Then the backward time-scale derivative Open image in new window is the number (provided that it exists) with the property that given any Open image in new window there exists a neighborhood Open image in new window of Open image in new window (i.e., Open image in new window for some Open image in new window ) such that

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]).

Assume that Open image in new window is a function and Open image in new window , then one has the following.
  1. (i)
     
  2. (ii)
     
  3. (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 limit
    exists as a finite number. In this case
     
  4. (iv)
     
Example 2.3.
  1. (i)
     
  2. (ii)
     
  3. (iii)
     
  4. (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.

Suppose that Open image in new window are nabla differentiable at Open image in new window , then,
  1. (1)
     
  2. (2)
     
  3. (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.

A function Open image in new window is called a nabla antiderivative of Open image in new window provided Open image in new window for all Open image in new window . In this case, for Open image in new window , we write

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.

holds if and only if

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

We will shortly write
We calculate the first variation of the functional Open image in new window on the linear manifold Open image in new window . Let Open image in new window , then
and where Lemma 2.5 and that Open image in new window are used. If we use the change of variable Open image in new window , which is a linear function, and make use of Theorem Open image in new window in [4] and Lemma 2.5 we then obtain

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

Splitting the first integral in (3.6) and rearranging will lead to
If we make use of part ( Open image in new window ) of Theorem 2.4 then we reach

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.

Then the necessary condition for Open image in new window to possess an extremum for a given function Open image in new window is that Open image in new window satisfies the following Euler-Lagrange equations
Furthermore, the equation:

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.

Then a necessary condition for Open image in new window to possess an extremum for a given function Open image in new window is that Open image in new window satisfies the following Euler-Lagrange equations:
Furthermore, the equations
hold along Open image in new window for all admissible variations Open image in new window satisfying

4. The Optimal-Control Problem

Our aim in this section is to find the optimal control variable Open image in new window defined on the Open image in new window -time scale, which minimizes the performance index
subject to the constraint
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. To find the optimal control, we define a modified performance index as

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

Using (3.11) and (3.12) of Theorem 3.2 with Open image in new window ( Open image in new window , Open image in new window , Open image in new window ), the necessary conditions for our optimal control are (we remark that as there is no any time-scale derivative of Open image in new window , no boundary constraints for it are needed)

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

We recall that the results in the previous sections are valid for time scales whose backward jump operator Open image in new window has the form Open image in new window , in particular for the time scale Open image in new window .
  1. (i)

    The Discrete Case

     
If Open image in new window and Open image in new window (of special interest the case when Open image in new window ), then our work becomes on the discrete time scale Open image in new window . In this case the functional under optimization will have the form
The necessary condition for Open image in new window to possess an extremum for a given function Open image in new window is that Open image in new window satisfies the following Open image in new window -Euler-Lagrange equations:
Furthermore, the equation

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.

Find the optimal control variable Open image in new window defined on the time scale Open image in new window , which minimizes the Open image in new window -performance index
subject to the constraint
The necessary conditions for this Open image in new window -optimal control are

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.

In order to illustrate our results we analyze an example of physical interest. Namely, let us consider the following discrete action:
subject to the condition
The corresponding Open image in new window -Euler-Lagrange equations are as follows:
We observe that when the delay is removed, that is, Open image in new window , the classical discrete Euler-Lagrange equations are reobtained.
  1. (ii)

    The Quantum Case

     
If Open image in new window and Open image in new window , then our work becomes on the time scale Open image in new window . In this case the functional under optimization will have the form
Using the Open image in new window -integral theory on time scales, the functional Open image in new window in (5.14) turns to be
The necessary condition for Open image in new window to possess an extremum for a given function Open image in new window is that Open image in new window satisfies the following Open image in new window -Euler-Lagrange equations:
Furthermore, the equation

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.

Find the optimal control variable Open image in new window defined on the Open image in new window -time scale, which minimizes the performance index
subject to the constraint
such that

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.

The necessary conditions for this Open image in new window -optimal control are

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.

Suppose that the problem is that of finding a control function Open image in new window defined on the time scale Open image in new window such that the corresponding solution of the controlled system
satisfying the conditions
is an extremum for the Open image in new window -integral functional ( Open image in new window -quadratic delay cost functional):
According to (5.24) and (5.25), the solution of the problem satisfies
and of course
When the delay is absent (i.e., Open image in new window ), it can be shown that the above system is reduced to a second-order Open image in new window -difference equation. Namely, reduced to
If we solve recursively for this equation in terms of an integer power series by using the initial data, then the resulting solution will tend to the solutions of the second order linear differential equation:

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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Copyright information

© Thabet Abdeljawad (Maraaba) et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Thabet Abdeljawad (Maraaba)
    • 1
  • Fahd Jarad
    • 1
  • Dumitru Baleanu
    • 1
    • 2
  1. 1.Department of Mathematics and Computer ScienceÇankaya UniversityAnkaraTurkey
  2. 2.Institute of Space SciencesMagurele-BucharestRomania

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