The control of systems with time delay has been of considerable concern. Delays occur frequently in biological, chemical, electronic and transportation systems. Wu, et al. [14] built up a computational method for solving an optimal control problem which is represented by a switched dynamical system with time delay. Kharatishidi [7] has approached this problem by extending Pontryagin’s maximum principle to time delay systems. The actual solution involves a two-point boundary-value problem in which advances and delays are exhibited. In addition, this solution does not yield a feedback controller. Time-optimal control of delay systems has been considered by Oguztoreli [11] who obtained several results concerning bang-bang controls which parallel those of LaSalle [9] for non delay systems. For a time-invariant system with an infinite upper limit in the performance measure, Krasovskii [8] has developed the forms of the controller and the performance measure. Ross [12] has acquired a set of differential equations for the unknowns in the forms of Krasovskii. However, Ross’s results are not applicable to time-varying systems with a finite limit in the performance measure. In [1], the authors presented an optimal regular for a linear system with multiple state and input delays and a quadratic criterion. The optimal regulator equations were obtained reducing the original problem to the linear-quadratic regulator design for a system without delays (see [3] and [4]). B-splines (where Bezier form is a special case of B-splines), due to numerical stability and arbitrary order of accuracy, have become popular tools for solving differential equations. The use of Bezier curves for solving time-delayed optimal control systems with Páde approximation is a novel idea. The stated technique reduces the CPU time and the computer memory comparing with existing methods such as methods in [2, 10] and at the same time keeps the solution accuracy. Although the stated technique is very easy to utilize and straightforward, the obtained results are satisfactory (see numerical results). In this paper, one may utilize the Bezier polynomials. There are many papers and books deal with the Bezier curves or surface techniques [6]. The organization of this study is arranged as follows: In Section 2, problem transformation is presented. Section 3 is referred to the Bezier curve technique. Some Numerical examples are provided in Section 4. Section 5 is devoted the conclusion.

Consider the time-delayed optimal control problem

where x ∈ ℛⁿ, u ∈ ℛ^m and π ∈ ℛ^p are the state, control and unknown parameter vectors respectively, η ∈ R^q represents the terminal inequality constraints, t_f is end time, and σ is the delay time associated with the state vector x. For the sake of simplicity, our discussion will be confined to the case of a single time delay σ. However, all the results can be extended in a straightforward manner to the case of multiple time delays. The time-delayed optimal control problem (2.1) can be transformed to one without a time-delayed argument utilizing the following approximation scheme. Let X(s) be a two-sided Laplace transform of x(t) (see [10]):

Presently, X(s) is defined over the strip of convergence α < Re(s) < β, where Re(s) denotes the real part of s, and ‘⇔’ denotes a two-sided Laplace transformation. The two-sided Laplace transform is utilized because x(t) = ξ(t) ≠ 0 for t ≤ 0. Given that x(t) ⇔ X(s), the two-sided Laplace transforms of x(t − σ) and ẋ(t) are presented by

where the strip of convergence after the differentiation is assumed to exist and might or might not be the same as that for X(s). To omit variables with a time-delayed argument in (2.1), let us define y(t) ≜ x(t − σ) and y(t) ⇔ Y (s). The two-sided Laplace transforms Y (s) and X(s) are referred by (see (2.6))

Utilizing a first-order Páde approximation, one may have

If an inverse Laplace transformation was performed on the last equation, one may have (see (2.2) and (2.7))

The time-delayed optimal control problem is approximately changed to one of minimizing J(x(t), y(t), u(t), π, t) subject to

with the following conditions

and the terminal condition η(x_f, π) ≥ 0. An alternative approximation scheme (but equivalent) is to state Y (s) in (2.9) as follows:

where Q(s) is defined as

The inverse Laplace transforms of (2.16) and (2.17) are

Utilizing this technique, the original optimization problem is transformed to one of minimizing J(x(t), q(t), u(t), π, t) subject to

with the following conditions

and the terminal condition η(x_f, π) ≥ 0.

To enhance the accuracy of the above-described approximation schemes, the time delay σ can be subdivided into smaller sections. For example, one may define

Again, utilizing a first-order Páde approximation, one may obtain

The time-delayed problem is changed to one of minimizing J(x(t), z(t), u(t), π, t) subject to

with the initial conditions

and the final condition (2.4). Repeated applications of this method will result progressively improved achievement. However, this is at the expense of an increase in the system order, resulting in an increase in the computation time.

Our strategy is utilizing Bezier curves to approximate the solutions x_i(t) and u(t) where x_i(t) are given below. Define the Bezier polynomials of degree n for t ∈ [t₀, t_f ] as follows:

where h = t_f − t₀, and

is the Bernstein polynomial of degree n for t ∈ [t₀, t_f ], ari and b_r are the control points. By substituting x_i(t) and u(t) in (2.27)–(2.30), one may define the problem which can be solved by Maple 16.

Ghomanjani et al. [6] proved the convergence of this method where n → ∞ when the optimal control system solved by Bezier curve method (for more explanation, see [5]). For the convergence of the time-delayed optimal control problem, one may use Páde approximation, then the problem is converted to an optimal control problem (OCP), where the convergence of OCP is in [6].

Termination criterions are ||x_i(t) − x_i,exact(t)||_∞ < ε, and ||u(t) − u_exact(t)|| < ε

In this section, some numerical examples are presented for illustrating the proposed technique.

Example 4.1

Consider the following time-delay system:

For this example the exact solution is given by [10] as follows:

where δ = 2.5599. Utilizing the proposed method, one may have

The delayed differential equations (4.1) then become

by using this method, we have

The optimal value of J in proposed method is 3.04246518971317. This value compares well with that given in [10] (J = 3.256613). In proposed method, one may obtain

the graphs of approximated and exact solution u(t) and x_i(t) for i = 1, 2, 3,4 are respectively plotted in Figs. 1, 2, 3, 4 and 5.

Example 4.2

Consider the following time-delay system (see [10]):

where the exact solution is

Utilizing this method, one may achieve

presently, one may have

The optimal value of J in proposed method is 0.0671730978076868. This value compares well with that given in [10] (J = 0.1967). The graphs of approximated and exact solution u(t) are plotted in Fig. 6.

Example 4.3

Consider the following time-delay system (see [10]):

now, one may have

utilizing this method, one may achieve

using Bezier curve, one may have

then one may substitute (4.6) in (4.5), and solve this system by using Maple 16 software. The optimal value of J in proposed method is 161.712666656000, when the values of J in [10], Banks [2], and Wong et al. [13] are respectively J = 161.88, J = 162.019, and J = 162.104. The graphs of approximated solution u(t) and x₁(t) are plotted in Figs. 7 and 8.

Example 4.4

Consider the following time-delay system (see [10]):

now, one may have

hence

using Bezier curve, one may have

then one may substitute (4.8) in (4.7), and solve this system by using Maple 16 software. The optimal value of J in proposed method is 1.43068782747222, when the value of J in [10] is J = 1.849730. The graphs of approximated solution u(t) and x₁(t) are plotted in Figs. 9 and 10.

Time-delayed optimal control problems with terminal inequality constraints can be approximately solved by a combined parameter. To this end, a Páde approximation is utilized to acquire a corresponding problem without a time-delayed argument. The results obtained by the Bezier curve are in good agreement with the given exact solutions. The study shows that the method is effective technique to solve time-delayed optimal control problems, and the technique is easy to implement and computationally very attractive without sacrificing the accuracy of the solution.

The author would like to thank the anonymous reviewer of this paper for their careful reading, constructive comments and nice suggestions which has improved the paper very much.