非线性控制仿射系统的逼近与可控性：基于半自主神经常微分方程
Approximation and Controllability of Nonlinear Control-Affine Systems via Semiautonomous Neural Ordinary Differential Equations
AI总结：
提出受控半自主神经常微分方程（SA-NODE）逼近非线性控制仿射系统，证明通用逼近定理和可控性保持，数值实验显示参数更少。
摘要：
本文引入受控半自主神经常微分方程（controlled SA-NODEs）用于非线性受控动力系统的逼近与学习。所提框架将半自主神经ODE扩展到控制仿射系统，同时通过时间无关的可训练系数保持参数复杂度降低。我们建立了一个通用逼近定理，表明controlled SA-NODEs在初始条件和容许控制的紧集上一致逼近非线性受控系统的轨迹。在额外的Sobolev和Barron正则性假设下，我们推导出阶为$\mathcal{O}(P^{-1/2}+Q^{-1/2})$的定量逼近估计。我们进一步证明原始非线性系统的近似可控性在controlled SA-NODE逼近下得以保持。在受控摆和Duffing振荡器系统上的数值实验表明，所提框架以比经典神经ODE架构显著更少的可训练参数实现了精确的轨迹重构和可控性性能。
英文摘要：
In this paper, we introduce controlled semiautonomous neural ordinary differential equations (controlled SA-NODEs) for the approximation and learning of nonlinear controlled dynamical systems. The proposed framework extends semiautonomous neural ODEs to control-affine systems while preserving reduced parameter complexity through time-independent trainable coefficients. We establish a universal approximation theorem showing that controlled SA-NODEs approximate trajectories of nonlinear controlled systems uniformly on compact sets of initial conditions and admissible controls. Under additional Sobolev and Barron regularity assumptions, we derive quantitative approximation estimates of order $\mathcal{O}(P^{-1/2}+Q^{-1/2})$. We further prove that approximate controllability properties of the original nonlinear system are preserved under the controlled SA-NODE approximation. Numerical experiments on controlled pendulum and Duffing oscillator systems demonstrate that the proposed framework achieves accurate trajectory reconstruction and controllability performance with significantly fewer trainable parameters than classical neural ODE architectures.
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