发表机构
Louisiana State University（路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立自适应控制与信息几何、变分原理的统一框架，将激励、约束、能量和随机学习关联，连接传统控制与现代信息论语言。
AI 中文摘要
自适应控制通常通过李雅普诺夫稳定性、激励和参数收敛来发展，而相邻领域则使用凸性、信息几何、变分原理和随机热力学来描述学习。本文为这些观点发展了一种受限的共同语言。对于线性参数化模型，激励格兰姆矩阵同时是累积预测损失的Hessian矩阵，并且在高斯观测下，是Fisher信息（至多相差一个缩放因子）。持续、有限和部分激励分别成为均匀、有限时域和受限的时间曲率；记忆方法保留先前获得的曲率。在闭凸参数集上，确定性自适应是投影梯度/Onsager流，切锥和法锥恢复投影，而复合学习提供数据依赖的对称耗散。反射Langevin动力学、无通量Fokker--Planck演化以及受限自由能流提供了随机对应部分。该框架区分了数据提供的信息与正则化或硬约束提供的限制，并将熟悉的自适应控制结构与现代变分和信息论语言联系起来。
英文摘要
Adaptive control is usually developed through Lyapunov stability, excitation, and parameter convergence, whereas neighboring fields describe learning using convexity, information geometry, variational principles, and stochastic thermodynamics. This paper develops a constrained common language for these viewpoints. For linearly parameterized models, the excitation Gramian is simultaneously the Hessian of accumulated prediction loss and, under Gaussian observations, Fisher information up to scaling. Persistent, finite, and partial excitation become uniform, finite-horizon, and restricted temporal curvature; memory methods retain previously acquired curvature. On a closed convex parameter set, deterministic adaptation is a projected gradient/Onsager flow, with tangent and normal cones recovering projection, while composite learning supplies data-dependent symmetric dissipation. Reflected Langevin dynamics, no-flux Fokker--Planck evolution, and constrained free-energy flow provide the stochastic counterpart. The framework distinguishes information supplied by data from confinement supplied by regularization or hard constraints and connects familiar adaptive-control structures to modern variational and information-theoretic language.
