Global Controllability for Quasilinear Non-negative Definite System of ODEs and SDEs

TL;DR

This paper establishes global controllability for quasi-linear ODEs and SDEs with non-negative definite matrices using a novel approach involving linearization, fixed point theorem, and continuous induction.

Contribution

It introduces a new method combining linearization and continuous induction to prove global controllability for nonlinear stochastic and deterministic systems.

Findings

Proves controllability for large initial data.

Develops a novel control extension method using continuous induction.

Applies fixed point theorem in a new context for nonlinear systems.

Abstract

We consider exact and averaged control problem for a system of quasi-linear ODEs and SDEs with a non-negative definite symmetric matrix of the system. The strategy of the proof is the standard linearization of the system by fixing the function appearing in the nonlinear part of the system, and then applying the Leray-Schauder fixed point theorem. We shall also need the continuous induction arguments to prolong the control to the final state which is a novel approach in the field. This enables us to obtain controllability for arbitrarily large initial data (so called global controllability).