改进的正则化前后扫描迭代算法求解刚性最优控制问题

Improved Regularization Forward-Backward Scan Iterative Algorithm for Solving Rigid Optimal Control Problems

最优控制问题广泛应用于工程学、经济学、生物学等众多领域。由于寻求解析解往往极具挑战性,人们通常采用设计合适的数值算法来求解其数值解。在这些问题中,刚性最优控制问题的数值求解方法尤为关键,这类问题的处理通常面临两难选择:问题的刚性特性容易引起数值方法的不稳定性,而过度追求数值格式的稳定性则可能导致计算成本显著增加,使得算法难以实用。本文专注于一类特定的刚性最优控制问题,通过改进传统的正则化前后扫描迭代算法,既保证了算法的稳定性,同时也显著提高了计算效率。最后通过数值实验验证了上述结论。

Optimal control problems are prevalent in various fields such as engineering, economics, and biology. As finding analytical solutions is often highly challenging, suitable numerical algorithms are typically employed to derive numerical solutions. Among these, the numerical resolution of stiff optimal control problems is particularly crucial, as it often involves a dilemma: the stiffness of the problem can easily lead to instability in numerical methods, while excessively prioritizing the stability of numerical schemes can significantly increase computational costs, rendering the algorithms impractical. This paper focuses on a specific type of stiff optimal control problem and enhances the traditional regularized forward-backward scan iterative algorithm. This improvement not only ensures the stability of the algorithm but also significantly enhances its computational efficiency. Finally, the above conclusions are verified by numerical experiments.