论文标题
Bernoulli多项式的应用解决可变级 - 最佳控制型问题
Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems
论文作者
论文摘要
我们提出了两种有效的数值方法，用于解决可变阶分数最佳控制型问题。在Caputo意义上考虑了可变阶分数衍生物，与Riemann-Liouville积分运算符一起在我们的新技术中使用。引入了Bernoulli多项式的可变顺序分数集成的精确操作矩阵。我们的方法进行如下。首先，根据Bernoulli多项式，考虑了状态函数的分化顺序的特定近似值。这样的近似值，以及初始条件，有助于我们获得动态控制型系统中其他现有功能的一些近似值。使用这些近似值和高斯 - legendre集成公式，该问题将降低为非线性代数方程系统。然后给出一些近似最佳状态和控制功能的误差界，这使我们能够获得针对性能索引的近似值绑定的误差。最后，我们解决了一些测试问题，这些问题证明了我们结果的高精度。
We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann-Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss-Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results.
