多项式控制仿射系统的$\u003cmath\u003e\b{mca{H}}_\u003c/math\u003e\infty$能量函数的可扩展计算
Scalable Computation of $\mathcal{H}_\infty$ Energy Functions for Polynomial Control-Affine Systems
AI总结：
本研究针对含多项式非线性的控制仿射系统，提出基于克罗内克积的Al'brekht幂级数法以可扩展计算其非线性平衡能量函数，推导了方程结构与复杂度理论，实现了开源软件，可支持千维以上系统。
摘要：
我们提出了一种可扩展方法，用于计算含多项式非线性项的控制仿射系统的非线性平衡能量函数。该方法采用Al'brekht幂级数法求解哈密顿-雅可比-贝尔曼方程，以得到能量函数的多项式近似。本文的贡献在于基于克罗内克积实现了该方法的数值计算，使其可扩展至1000维以上的状态维度。克罗内克积表示所产生的张量结构与对称性是开发高效可扩展算法的关键。我们推导了方程的显式代数结构，给出了方程可解性与算法复杂度的严格理论证明，并为所提算法提供了通用开源软件实现。我们在两个简单学术模型上验证了该方法，随后在维度高达$n=1080$的高维半离散化偏微分方程模型上进行了测试。
英文摘要：
We present a scalable approach to computing nonlinear balancing energy functions for control-affine systems with polynomial nonlinearities. Al'brekht's power-series method is used to solve the Hamilton-Jacobi-Bellman equations for polynomial approximations to the energy functions. The contribution of this article lies in the numerical implementation of the method based on the Kronecker product, enabling scalability to over 1000 state dimensions. The tensor structure and symmetries arising from the Kronecker product representation are key to the development of efficient and scalable algorithms. We derive the explicit algebraic structure for the equations, present rigorous theory for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms. The method is illustrated on two simple academic models, followed by a high-dimensional semidiscretized PDE model of dimension as large as $n=1080$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
