We take a new look at the relation between the optimal transport problem\ud and the Schrödinger bridge problem from a stochastic control perspective. Our aim is\ud to highlight new connections between the two that are richer and deeper than those\ud previously described in the literature. We begin with an elementary derivation of\ud the Benamou–Brenier fluid dynamic version of the optimal transport problem and\ud provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem.\ud We observe that the latter establishes an important connection with optimal transport\ud without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed,\ud the two variational problems differ by a Fisher information functional. We motivate\ud and consider a generalization of optimal mass transport in the form of a (fluid dynamic)\ud problem of optimal transport with prior. This can be seen as the zero-noise limit of\ud Schrödinger bridges when the prior is any Markovian evolution.We finally specialize\ud to the Gaussian case and derive an explicit computational theory based on matrix\ud Riccati differential equations. A numerical example involving Brownian particles is\ud also provided

We consider the problem of steering a linear dynamical system with complete state observation from an initial Gaussian distribution in state-space to a final one with minimum energy control. The system is stochastically driven through the control channels; an example for such a system is that of an inertial particle experiencing random "white noise'' forcing. We show that a target probability distribution can always be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitly by solving a pair of differential Lyapunov equations that are coupled through their boundary values. This result, given its attractive algorithmic nature, appears to have several potential applications such as to quality control, industrial processes as well as to active control of nanomechanical systems and molecular cooling. The problem to steer a diffusion process between end-point marginals has a long history (Schroedinger bridges) and therefore, the present case of steering a linear stochastic system constitutes a Schroedinger bridge for possibly degenerate diffusions. Our results, however, provide the first {\em implementable} form of the optimal control for a general Gauss-Markov process. Illustrative examples of the optimal evolution and control for inertial particles and a stochastic oscillator are provided. A final result establishes directly the property of Schroedinger bridges as the most likely random evolution between given marginals to the present context of linear stochastic systems

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We consider the problem of minimum energy steering of a linear stochastic system to a final prescribed distribution over a finite horizon and the problem to maintain a stationary distribution over an infinite horizon. For both problems the control and noise channels are allowed to be distinct, thereby, placing the results of this paper outside of the scope of previous work both in probability and in control. We present sufficient conditions for optimality in terms of a system of dynamically coupled Riccati equations in the finite horizon case and in terms of algebraic conditions for the stationary case. We then address the question of feasibility for both problems. For the finite-horizon case, provided the system is controllable, we prove that without any restriction on the directionality of the stochastic disturbance it is always possible to steer the state to any arbitrary Gaussian distribution over any specified finite time-interval. For the stationary infinite horizon case, it is not always possible to maintain the state at an arbitrary Gaussian distribution through constant state-feedback. It is shown that covariances of admissible stationary Gaussian distributions are characterized by a certain Lyapunov-like equation and, in fact, they coincide with the class of stationary state covariances that can be attained by a suitable stationary colored noise as input. We finally address the question of how to compute suitable controls numerically. We present an alternative to solving the system of coupled Riccati equations, by expressing the optimal controls in the form of solutions to (convex) semi-definite programs for both cases. We conclude with an example to steer the state covariance of the distribution of inertial particles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.

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 (ẋ(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.

Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Nonstationary reciprocal processes have been extensively studied in the past especially by Jamison et al. The specialization of the nonstationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance extension problem for block-circulant covariance matrices. This generalizes the famous covariance band extension problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem. We show that the maximum entropy principle leads to a complete solution of the problem

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for interpolation of distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of recent developments in physics, probability theory, image processing, time-series analysis, and several other fields. In spite of extensive work and theoretical developments, the computation of OMT for large scale problems has remained a challenging task. An alternative framework for interpolating distributions, rooted in statistical mechanics and large deviations, is that of Schrödinger bridges (entropic interpolation). This may be seen as a stochastic regularization of OMT and can be cast as the stochastic control problem of steering the probability density of the state-vector of a dynamical system between two marginals. In this approach, however, the actual computation of flows had hardly received any attention. In recent work on Schrödinger bridges for Markov chains and quantum evolutions, we noted that the solution can be efficiently obtained from the fixed-point of a map which is contractive in the Hilbert metric. Thus, the purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which i) leads to a new proof of a classical result on Schrödinger bridges and ii) provides an efficient computational scheme for both, Schrödinger bridges and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.

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