Optimal transport over a linear dynamical system

Abstract

We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator ($\dot{x}(t)=u(t)$) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schrödinger bridge. As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases.

25 pages, 13 figures

References in corpus (5)

• On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint
• Fast cooling for a system of stochastic oscillators
• Optimal steering of a linear stochastic system to a final probability distribution
• Optimal steering of inertial particles diffusing anisotropically with losses
• Optimal steering of a linear stochastic system to a final probability distribution, part II

Cited by in corpus (4)

• Entropic and displacement interpolation: a computational approach using the Hilbert metric
• Optimal perturbations for nonlinear systems using graph-based optimal transport
• Stochastic control, entropic interpolation and gradient flows on Wasserstein product spaces
• Optimal mass transport over bridges