从Hamilton-Jacobi-Bellman到Riccati：通过Carleman线性化的标量情形

中文摘要 AI 辅助

非线性系统的Hamilton-Jacobi-Bellman（HJB）方程是一个非线性偏微分方程，其形式本身难以直接揭示系统性质。本文表明，该方程实际上隐藏着一个Riccati方程，而Carleman线性化能够揭示这一结构。对于具有解析数据的标量输入仿射系统，提升后的HJB方程是一个定义在下三角Toeplitz矩阵上的单一无穷维Riccati方程。该矩阵的首项对应于平衡点处标准线性化的Riccati方程，其余项则包含非线性项。这一过程并非近似。在标准线性化可控且代价可检测的条件下，该方程恰好存在一个具有正对角元的解，且该解可逐项计算。此解即为值函数的导数。我们证明，该解的有限截断是精确的，并在原点附近指数收敛到值函数的导数。标量情形具有闭式解，该闭式解作为基准，用于验证提升方程的所有结果。该结构不依赖于状态维度。我们的方法指向一种一般的最优控制理论，在该理论中，解析非线性系统的HJB方程转化为无穷维Riccati方程。

英文摘要

The Hamilton-Jacobi-Bellman (HJB) equation of a nonlinear system is a nonlinear partial differential equation, and in that form it gives little away. This paper shows that it hides a Riccati equation. Carleman linearization reveals this structure. For scalar input-affine systems with analytic data, the lifted HJB equation is a single infinite-dimensional Riccati equation in a lower-triangular Toeplitz matrix. Its first entry is the Riccati equation of the standard linearization at equilibrium. Its remaining entries hold the nonlinear terms. Nothing about this is an approximation. Under controllability of the standard linearization and detectability of the cost, the equation has exactly one solution with positive diagonal entries, and it can be computed one entry at a time. This solution is the derivative of the value function. We prove that finite-sections of the solution are exact and converge exponentially to the derivative of the value function near the origin. The scalar case has a closed-form solution. This closed form is our benchmark: every result on the lifted equation is checked against it. Nothing in this structure depends on the state dimension. Our methodology points to a general theory of optimal control in which the HJB equation of analytic nonlinear systems becomes an infinite-dimensional Riccati equation.

发表机构

• Lehigh University（利哈伊大学）
• University of Central Florida（中佛罗里达大学）

机构由 AI 辅助整理，请以论文原文为准。