To study optimal control and disturbance attenuation problems, two prominent—and somewhat alternative—strategies have emerged in the last century: dynamic programming (DP) and Pontryagin's minimum principle (PMP). The former characterizes the solution by shaping the dynamics in a closed loop (a priori unknown) via the selection of a feedback input, at the price, however, of the solution to (typically daunting) partial differential equations. The latter, instead, provides (extended) dynamics that must be satisfied by the optimal process, for which boundary conditions (a priori unknown) should be determined. The results discussed in this article combine the two approaches by matching the corresponding trajectories, i.e., combining the underlying sources of information: knowledge of the complete initial condition from DP and of the optimal dynamics from PMP. The proposed approach provides insights for linear as well as nonlinear systems. In the case of linear systems, the derived conditions lead to matrix algebraic equations, similar to the classic algebraic Riccati equations (AREs), although with coefficients defined as polynomial functions of the input gain matrix, with the property that the coefficient of the quadratic term of such equation is sign definite, even if the corresponding coefficient of the original ARE is sign indefinite, as it is typically the case in the $\mathcal {H}_{\infty }$ control problem. This feature is particularly appealing from the computational point of view, since it permits the use of standard minimization techniques for convex functions, such as the gradient algorithm. In the presence of nonlinear dynamics, the strategy leads to algebraic equations that allow us to (locally) construct the optimal feedback by considering the behavior of the closed-loop dynamics at a single point in the state space.

中文翻译：

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将庞特里亚金原理与动态规划结合用于线性和非线性系统

为了研究最优控制和扰动衰减问题，上个世纪出现了两种突出且有些替代的策略：动态规划 (DP) 和庞特里亚金最小原理 (PMP)。前者通过选择反馈输入来塑造闭环中的动态（先验未知）来表征解决方案，但代价是（通常令人生畏的）偏微分方程的解决方案。相反，后者提供了优化过程必须满足的（扩展的）动力学，为此应该确定边界条件（先验未知）。本文讨论的结果通过匹配相应的轨迹将两种方法结合起来，即结合潜在的信息来源：DP 的完整初始条件和 PMP 的最佳动力学知识。所提出的方法为线性和非线性系统提供了见解。在线性系统的情况下，导出的条件导致矩阵代数方程，类似于经典的代数 Riccati 方程 (ARE)，尽管系数定义为输入增益矩阵的多项式函数，具有二次项系数的性质这种方程的符号是定的，即使原始 ARE 的相应系数是符号不定的，正如 $\mathcal {H}_{\infty }$ 控制问题中的典型情况。从计算的角度来看，这个特性特别有吸引力，因为它允许使用标准的最小化技术来处理凸函数，例如梯度算法。