# Detecting the Most Probable High Dimensional Transition Pathway Based on Optimal Control Theory

## TL;DR

This paper develops a high-dimensional optimal control approach to identify the most probable transition pathways in stochastic systems undergoing phase changes, utilizing variational principles and Pontryagin's Maximum Principle.

## Contribution

It introduces a novel application of optimal control theory to high-dimensional stochastic systems for transition pathway detection, surpassing traditional variational methods.

## Key findings

- Successfully applied to double well, Maire-Stein, and Nutrient-Phytoplankton systems.
- Demonstrated effectiveness of successive approximation method for pathway computation.
- Enhanced capability to handle high-dimensional stochastic transition problems.

## Abstract

Many natural systems exhibit phase transition where external environmental conditions spark a shift to a new and sometimes quite different state. Therefore, detecting the behavior of a stochastic dynamic system such as the most probable transition pathway, has made sense.We consider stochastic dynamic systems driven by Brownian motion. Based on variational principle and Onsager-Machlup action functional theory, the variational problem is transformed into deterministic optimal control problem. Different from traditional variational method, optimal control theory can handle high dimensional problems well. This paper describes the following three systems, double well system driven by additive noise, Maire-Stein system driven by multiplicative noise, and Nutrient-Phytoplankton-Nooplanktonr system.For numerical computation, based on Pontryagin's Maximum Principle, we adopt method of successive approximations to compute the optimal solution pathway, which is the most probable transition pathway in the sense of Onsager - Machlup.