# Taylor Expansions of the Value Function Associated with a Bilinear Optimal Control Problem

## TL;DR

This paper develops polynomial Taylor expansions of the value function for a bilinear infinite-dimensional optimal control problem, enabling the design of suboptimal feedback laws with proven approximation properties and an application to Fokker-Planck equations.

## Contribution

It introduces a novel method for approximating the value function using multilinear forms derived from generalized Lyapunov equations, advancing control strategies for complex systems.

## Key findings

- Polynomial approximations effectively describe the value function.
- The derived feedback law provides near-optimal control performance.
- Application demonstrates the method's effectiveness on Fokker-Planck equations.

## Abstract

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton-Jacobi-Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker-Planck equation is also provided.