Optimal control for stochastic nonlinear Schrodinger equation on graph

TL;DR

This paper develops an optimal control framework for the stochastic nonlinear Schrödinger equation on finite graphs, establishing existence, uniqueness, and optimality conditions using stochastic Wasserstein Hamiltonian flow.

Contribution

It introduces a novel control approach for SNLSE on graphs, including existence proofs and gradient formulas for optimal controls.

Findings

Global existence of unique strong solutions for controlled SNLSE on graphs

Derivation of gradient formulas for optimal control

Characterization of optimal conditions via stochastic differential equations

Abstract

We study the optimal control formulation for stochastic nonlinear Schrodinger equation (SNLSE) on a finite graph. By viewing the SNLSE as a stochastic Wasserstein Hamiltonian flow on density manifold, we show the global existence of a unique strong solution for SNLSE with a linear drift control or a linear diffusion control on graph. Furthermore, we provide the gradient formula, the existence of the optimal control and a description on the optimal condition via the forward and backward stochastic differential equations.