Optimal steering of inertial particles diffusing anisotropically with losses

TL;DR

This paper extends the Schrödinger bridge theory to control inertial particles with anisotropic diffusion and losses, providing new mathematical formulations and computational methods for optimal steering.

Contribution

It introduces a novel extension of Schrödinger bridge theory to particles with losses and anisotropic diffusion, including PDE and semidefinite programming approaches.

Findings

Optimal control law derived from coupled PDEs and Riccati equations.

Semidefinite programming provides a way to compute suboptimal solutions.

Application example demonstrates control of inertial particles with killing rate.

Abstract

Exploiting a fluid dynamic formulation for which a probabilistic counterpart might not be available, we extend the theory of Schroedinger bridges to the case of inertial particles with losses and general, possibly singular diffusion coefficient. We find that, as for the case of constant diffusion coefficient matrix, the optimal control law is obtained by solving a system of two p.d.e.'s involving adjoint operators and coupled through their boundary values. In the linear case with quadratic loss function, the system turns into two matrix Riccati equations with coupled split boundary conditions. An alternative formulation of the control problem as a semidefinite programming problem allows computation of suboptimal solutions. This is illustrated in one example of inertial particles subject to a constant rate killing.