We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein-Uhlenbeck process conditioned at the two end-points - a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.

We introduce an optimal mass transport framework on the space of Gaussian mixture models. These models are widely used in statistical inference. Specifically, we treat Gaussian mixture models as a submanifold of probability densities equipped with the Wasserstein metric. The topology induced by optimal transport is highly desirable and natural because, in contrast to total variation and other metrics, the Wasserstein metric is weakly continuous (i.e., convergence is equivalent to convergence of moments). Thus, our approach provides natural ways to compare, interpolate and average Gaussian mixture models. Moreover, the approach has low computational complexity. Different aspects of the framework are discussed and examples are presented for illustration purposes.

We introduce the problem of transporting vector-valued distributions. In this, a salient feature is that mass may ow between vectorial entries as well as across space (discrete or continuous). The theory relies on a first step taken to define an appropriate notion of optimal transport on a graph. The corresponding distance between distributions is readily computable via convex optimization and provides a suitable generalization of Wasserstein-type metrics. Building on this, we define Wasserstein-type metrics on vector-valued distributions supported on continuous spaces as well as graphs. Motivation for developing vector-valued mass transport is provided by applications such as color image processing, multimodality imaging, polarimetric radar, as well as network problems where resources may be vectorial.

We address the problem of steering the state of a linear stochastic system to a prescribed distribution over a finite horizon with minimum energy, and the problem to maintain the state at a stationary distribution over an infinite horizon with minimum power. For both problems the control and Gaussian 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. The special case where the disturbance and control enter through the same channels has been addressed in the first part of this work that was presented as Part I. Herein, 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.

The subject of this work has its roots in the so-called Schrödginer bridge problem (SBP) which asks for the most likely distribution of Brownian particles in their passage between observed empirical marginal distributions at two distinct points in time. Renewed interest in this problem was sparked by a reformulation in the language of stochastic control. In earlier works, presented as Part I and Part II, we explored a generalization of the original SBP that amounts to optimal steering of linear stochastic dynamical systems between state-distributions, at two points in time, under full state feedback. In these works, the cost was quadratic in the control input, i.e., control energy. The purpose of the present work is to detail the technical steps in extending the framework to the case where a quadratic cost in the state is also present. Thus, the main contribution is to derive the optimal control in this case which in fact is given in closed-form (Theorem 1). In the zero-noise limit, we also obtain the solution of a (deterministic) mass transport problem with general quadratic cost.

Consider a linear stochastic system whose initial state is a random vector with a specified Gaussian distribution. Such a distribution may represent a collection of particles abiding by the specified system dynamics. In recent publications, we have shown that, provided the system is controllable, it is always possible to steer the state covariance to any specified terminal Gaussian distribution using state feedback. The purpose of the present work is to show that, in the case where only partial state observation is available, a necessary and sufficient condition for being able to steer the system to a specified terminal Gaussian distribution for the state vector is that the terminal state covariance be greater (in the positive-definite sense) than the error covariance of a corresponding Kalman filter.

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 the Schrödinger bridge problem (SBP), which leads to entropic interpolation. SBP 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. The actual computation of entropic flows, however, has received hardly any attention. In our recent work on Schrödinger bridges for Markov chains and quantum channels, we showed 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 SBP and (ii) provides an efficient computational scheme for both SBP and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images. 2016 Society for Industrial and Applied Mathematics.

We present an overview of our recent work on implementable solutions to the Schrödinger bridge problem and their potential application to optimal transport and various generalizations.

Exploiting a fluid dynamic formulation for which a probabilistic counterpart might not be available, we extend the theory of Schrödinger 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.

We study feedback control of coupled nonlinear stochastic oscillators in a force field. We first consider the problem of asymptotically driving the system to a desired steady state corresponding to reduced thermal noise. Among the feedback controls achieving the desired asymptotic transfer, we find that the most efficient one from an energy point of view is characterized by time-reversibility. We also extend the theory of Schrödinger bridges to this model, thereby steering the system in finite time and with minimum effort to a target steady-state distribution. The system can then be maintained in this state through the optimal steady-state feedback control. The solution, in the finite-horizon case, involves a space-time harmonic functionϕ, and - log ϕ plays the role of an artificial, time-varying potential in which the desired evolution occurs. This framework appears extremely general and flexible and can be viewed as a considerable generalization of existing active control strategies such as macromolecular cooling. In the case of a quadratic potential, the results assume a form particularly attractive from the algorithmic viewpoint as the optimal control can be computed via deterministic matricial differential equations. An example involving inertial particles illustrates both transient and steady state optimal feedback control.