The main objectives of this study are to introduce a new class of optimal control problems governed by a dynamical system of weakly singular variable-order fractional integral equations and to establish a computational method by utilizing the Chebyshev cardinal functions for their numerical solutions. In this way, a new operational matrix of variable-order fractional integration is generated for the Chebyshev cardinal functions. In the established method, first the control and state variables are approximated by the introduced basis functions. Then, the interpolation property of these basis functions together with their mentioned operational matrix is applied to derive an algebraic equation instead of the objective function and an algebraic system of equations instead of the dynamical system. Eventually, the constrained extrema technique is applied by adjoining the constraints generated from the dynamical system to the objective function using a set of Lagrange multipliers. The accuracy of the established approach is examined through several test problems. The obtained results confirm the high accuracy of the presented method.

中文翻译：

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Chebyshev基函数用于一类新的动力奇异系统的新型非线性最优控制问题

这项研究的主要目的是引入一类新的最优控制问题，该问题由弱奇异变量阶分数阶积分方程的动力系统控制，并利用切比雪夫基数函数为其数值解建立计算方法。通过这种方式，为切比雪夫基数函数生成了一个新的可变阶分数积分运算矩阵。在已建立的方法中，首先，控制变量和状态变量通过引入的基函数进行近似。然后，将这些基函数的插值特性及其提到的运算矩阵应用于推导一个代数方程，而不是目标函数和一个代数方程组，而不是动力系统。最终，约束极值技术是通过使用一组拉格朗日乘子将动力学系统生成的约束附加到目标函数而应用的。通过几个测试问题来检验已建立方法的准确性。所得结果证实了所提出方法的高精度。