发表机构
School of Mathematics, Sichuan University; School of Mathematical Sciences, Sichuan Normal University（四川大学数学学院; 四川师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非线性最优控制问题，通过将二阶强制性与矩阵Riccati方程可解性等价，建立了牛顿法在L∞和L2空间中的局部二次收敛性，并给出收敛邻域估计和数值验证。
AI 中文摘要
本文的主要目的是建立非线性最优控制问题中牛顿法的局部二次收敛性。在适当的光滑性假设下，首先证明了代价泛函关于控制是两次Gâteaux可微的。推导了其二阶导数的显式算子表示，并考察了其在不同控制空间中的连续性。然后证明了在$L^2(0,T;\mathbb{R}^m)$中的二阶强制性与相关联的矩阵Riccati微分方程强正则解的存在性等价，从而将无穷维二次型条件转化为矩阵微分方程的可解性。在此基础上，利用线性二次最优控制理论，建立了在$L^\infty(0,T;\mathbb{R}^m)$中牛顿法的局部二次收敛性。在适当的结构性假设下，还建立了在$L^2(0,T;\mathbb{R}^m)$中的局部二次收敛性。推导了相应收敛邻域的显式估计，并通过数值算例说明了理论结果。
英文摘要
The main purpose of this paper is to establish the local quadratic convergence of Newton's method for nonlinear optimal control problems. Under suitable smoothness assumptions, the cost functional is first shown to be twice G{â}teaux differentiable with respect to the control. An explicit operator representation of its second derivative is derived, and its continuity in different control spaces is examined. The second-order coercivity condition in $L^2(0,T;\maathbb{R}^m)$ is then shown to be equivalent to the existence of a strongly regular solution to an associated matrix Riccati differential equation, thereby expressing an infinite-dimensional quadratic-form condition in terms of the solvability of a matrix differential equation. On this basis, local quadratic convergence of Newton's method in $L^\infty(0,T;\maathbb{R}^m)$ is established using linear-quadratic optimal control theory. Under suitable structural assumptions, local quadratic convergence in $L^2(0,T;\maathbb{R}^m)$ is also established. Explicit estimates of the corresponding convergence neighborhoods are derived, and the theoretical results are illustrated by numerical examples.
Comments34pages, 8 figures
