The Koopmanization of controlled nonlinear It\^o stochastic differential systems and its comparison with the Carleman embedding: new results

TL;DR

This paper introduces a Koopman operator framework for controlled nonlinear Itô stochastic differential systems, focusing on eigenfunction construction, operator action, and filtering, and compares it with the Carleman embedding approach.

Contribution

It develops a novel Koopmanization method for stochastic systems and demonstrates its application in filtering, providing a new perspective compared to existing Carleman embedding techniques.

Findings

Koopmanization effectively embeds bilinearization of stochastic systems.

The framework enables filtering of controlled nonlinear Itô systems.

Comparison shows advantages of Koopman approach over Carleman embedding.

Abstract

The Koopmanization embeds the bilinearization via the action of the infinitesimal stochastic Koopman operator on the observables associated with the controlled nonlinear It\^o stochastic differential system without explicit linearizations. The stochastic evolutions of controlled Markov processes assume the structure of controlled nonlinear It\^o stochastic differential equations. This paper sketches a Koopman operator framework for the filtering of the controlled nonlinear It\^o stochastic differential system. The major ingredients of this paper are the construction of the eigenfunctions, action of the infinitesimal stochastic Koopman operator, multi-dimensional It\^o differential rule and filtering concerning the controlled nonlinear It\^o stochastic differential system. In this paper, we illustrate the filtering in the Koopman setting for a polynomial system and compare with the filtering in the Carleman setting.