This paper proposes a new type of smoothing homotopy method for solving general nonlinear optimal control problems via the indirect method, leveraging the Mainardi kernel as the smoothing kernel. The Mainardi kernel is firstly derived from the fundamental solution of the time-fractional diffusion-wave equation, representing a generalized form of the Gaussian kernel. By altering the fractional derivative order, the kernel can seamlessly switch between non-Gaussian and Gaussian forms. Then, parts of the two-point boundary value problems are convolved with the smoothing kernel, and the resulting surrogates are incorporated into the necessary conditions, replacing the terminal state and costate variables. The homotopy process is used for the smoothing parameter: increasing this parameter enhances the smoothing effect, simplifying the homotopy problems, while a parameter value of zero implies no smoothing, reverting to the original problems. Additionally, explicit expressions for the Mainardi kernel at specific derivative orders are derived, avoiding the inefficiencies associated with fractional derivatives. Simulation examples demonstrate that the proposed method offers significant advantages in flexibility and convergence compared to the traditional Gaussian smoothing homotopy method.

中文翻译：

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求解非线性最优控制问题的Mainardi平滑同伦法


本文提出了一种新型平滑同伦方法，利用 Mainardi 核作为平滑核，通过间接方法解决一般非线性最优控制问题。 Mainardi核首先由时间分数阶扩散波方程的基本解推导出来，代表高斯核的广义形式。通过改变分数阶导数阶数，内核可以在非高斯形式和高斯形式之间无缝切换。然后，将部分两点边值问题与平滑核进行卷积，并将所得代理项纳入必要条件，替换最终状态和辅助变量。平滑参数采用同伦过程：增加该参数可以增强平滑效果，简化同伦问题，而参数值为零则意味着不进行平滑，回到原来的问题。此外，导出了特定导数阶数的 Mainardi 核的显式表达式，避免了与分数导数相关的低效率。仿真实例表明，与传统的高斯平滑同伦方法相比，该方法在灵活性和收敛性方面具有显着的优势。