Optimal Control in Geometric Dynamics: Applications To AI Algorithm Optimization

Abstract

This paper extends the fundamental concepts of geometric dynamics and optimal control theory, inspired by the pioneering work of Professor Constantin Udriste, to develop novel optimization algorithms for artificial intelligence systems. We establish connections between nonholonomic macroeconomic systems and reinforcement learning by formulating a multi-time maximum principle framework that integrates sub-Riemannian geometry. Our proposed methodology demonstrates how constrained variational problems can optimize neural network training trajectories through a bang-bang control approach. An empirical case study implements this theoretical framework to optimize a deep reinforcement learning algorithm, showing significant improvements in convergence speed and stability compared to standard approaches. The results demonstrate the practical value of geometric dynamics principles in modern Artificial Intelligence (AI) optimization, establishing a bridge between classical mathematical control theory and contemporary machine learning challenges.