Optimal control of stochastic homogenous systems

TL;DR

This paper extends homogeneous stochastic control theory to nonlinear systems with cone constraints, linking optimal controls to nonlinear BSDEs and providing explicit feedback forms under broad conditions.

Contribution

It introduces a generalized framework connecting nonlinear homogeneous stochastic control problems with nonlinear BSDEs, extending classical LQ results to more complex dynamics and cost functionals.

Findings

Established existence and uniqueness of solutions to the nonlinear BSDEs.

Derived explicit feedback representations for optimal controls.

Unified various known homogeneous LQ results as special cases.

Abstract

This paper investigates a new class of homogeneous stochastic control problems with cone control constraints, extending the classical homogeneous stochastic linear-quadratic (LQ) framework to encompass nonlinear system dynamics and non-quadratic cost functionals. We demonstrate that, analogous to the LQ case, the optimal controls and value functions for these generalized problems are intimately connected to a novel class of highly nonlinear backward stochastic differential equations (BSDEs). We establish the existence and uniqueness of solutions to these BSDEs under three distinct sets of conditions, employing techniques such as truncation functions and logarithmic transformations. Furthermore, we derive explicit feedback representations for the optimal controls and value functions in terms of the solutions to these BSDEs, supported by rigorous verification arguments. Our general solvability conditions allow us to recover many known results for homogeneous LQ problems, including both standard and singular cases, as special instances of our framework.