This paper introduces a highly accurate operational matrix technique based upon the Chebyshev cardinal functions (CCFs) for solving a new category of nonlinear delay optimal control problems (OCPs) involving variable-order (VO) fractional dynamical systems. The VO fractional derivatives are defined in the Caputo type. The main aim is converting such OCPs into systems of algebraic equations. Thus, we first expand the state and control variables in terms of the CCFs with undetermined coefficients. Then, by utilizing the cardinal property of these basis functions, the delay terms in the problem under consideration are expanded in terms of the CCFs. Thereafter, these expansions are substituted into the cost functional, dynamical system and delay conditions. Next, the cardinality of the CCFs together with their operational matrix (OM) of VO fractional derivative are employed to extract a nonlinear algebraic equation from the cost functional, and several algebraic equations from the dynamical system and delay conditions. Eventually, the method of the constrained extremum is employed by coupling the algebraic constraints yielded from the dynamical system and delay conditions with the algebraic equation extracted from the cost functional using a set of Lagrange multipliers. The precision of the method is studied through different types of numerical examples.

中文翻译：

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非线性变阶分数延迟最优控制问题的直接计算方法

本文介绍了一种基于切比雪夫基数函数 (CCF) 的高精度运算矩阵技术，用于解决涉及变阶 (VO) 分数动力学系统的新型非线性延迟最优控制问题 (OCP)。VO 分数导数在 Caputo 类型中定义。主要目标是将此类 OCP 转换为代数方程组。因此，我们首先根据系数不确定的 CCF 扩展状态和控制变量。然后，通过利用这些基函数的基数性质，在 CCF 方面扩展了所考虑问题中的延迟项。此后，这些扩展被代入成本函数、动力系统和延迟条件。下一个，CCF 的基数及其 VO 分数阶导数的运算矩阵 (OM) 被用来从成本函数中提取非线性代数方程，并从动力系统和延迟条件中提取几个代数方程。最后，通过使用一组拉格朗日乘子将从动力系统和延迟条件产生的代数约束与从成本函数中提取的代数方程耦合，采用约束极值的方法。通过不同类型的数值例子研究了该方法的精度。通过使用一组拉格朗日乘子将从动力系统和延迟条件产生的代数约束与从成本函数中提取的代数方程耦合，采用约束极值的方法。通过不同类型的数值例子研究了该方法的精度。通过使用一组拉格朗日乘子将从动力系统和延迟条件产生的代数约束与从成本函数中提取的代数方程耦合，采用约束极值的方法。通过不同类型的数值例子研究了该方法的精度。