An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

Abstract

In this paper, an efficient finite difference method is presented for the solution of time‐delay optimal control problems with time‐varying delay in the state. By using the Pontryagin's maximum principle, the original time‐delay optimal control problem is first transformed into a system of coupled two‐point boundary value problems involving both delay and advance terms. Then the derived system is converted into a system of linear algebraic equations by using a second‐order finite difference formula and a Hermite interpolation polynomial for the first‐order derivatives and delay terms, respectively. The convergence analysis of the proposed approach is provided. The new scheme is also successful for the optimal control of time‐delay systems affected by external persistent disturbances. Numerical examples are included to demonstrate the validity and applicability of the new technique. Some comparative results are included to illustrate the effectiveness of the proposed method.

Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model

“…The terms c 1 E and c 2 I represent the cost of reducing the exposed and infected population respectively, while c 3 u 2 1 is the cost of quarantine and also, c 4 u 2 2 is the cost of monitoring and treatment. The necessary conditions that an optimal control must satisfy come from the Pontryagin's Minimum Principle [34][35][36][37]. This principle converts Eqs.…”

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

“…Example Consider the following example: subject to the time‐delay system in which the analytic solution for U ( t ) is and with δ =−0.3932, J ≃3.1017. This example solved by several numerical techniques such as averaging approximations method, single‐term Walsh series, variational iteration method, and finite difference method with h = 0.01. In Table , these numerical results are compared to the results obtained using the present method for different values of k , M .…”

“…This example solved by several numerical techniques such as averaging approximations method, 50 single-term Walsh series, 51 variational iteration method, 56 and finite difference method 57 with h = 0.01. In Table 6, these numerical results are compared to the results obtained using the present method for different values of k, M. Numerical methods J Averaging approximations method, 50 3.0833 Single-term Walsh series 51 3.0879 Variational iteration method 56 3.1091 Finite difference method 57 3 Example 7. We consider the problem…”

A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

“…[4,20]. [29] 4.79678 Recursive shooting method [18] 4.79682 Line-up competition algorithm [35] 4.7976 Legendre multiwavelets [22] 5.1713 Dynamic programming [28] 6.26775 Walsh series [17] 6.0079 Bernoulli wavelet [32] 2.0481 Example 3. Consider the following TDFOCP…”

“…Furthermore, time-delay fractional optimal control problems (TDFOCPs) are a special type of FOCPs in which the dynamic of system contain some time-delay equations. The control of time-delay systems are frequently used in electronic, age structure, biological, chemical, electronic and transportation systems [15,[23][24][25][26][27][28][29][30][31]. Due to their variety applications in the realistic models of phenomena, TDFOCPs has been investigated by many researchers and considerable attention has been focused on approximate and numerical solution of them.…”