Modeling and control of collective dynamics: From Schroedinger bridges to optimal mass transport

Abstract

We study modeling and control of collective dynamics. More specifically, we consider the problem of steering a particle system from an initial distribution to a final one with minimum energy control during some finite time window. It turns out that this problem is closely related to Optimal Mass Transport (OMT) and the Schroedinger bridge problem (SBP). OMT is concerned with reallocating mass from a specified starting distribution to a final one while incurring minimum cost. The SBP, on the other hand, seeks a most likely density ow to reconcile two marginal distributions with a prior probabilistic model for the flow. Both of these problems can be reformulated as those of controlling a density flow that may represent either a model for the distribution of a collection of dynamical systems or, a model for the uncertainty of the state of single dynamical system. This thesis is concerned with extensions of and point of contact between these two subjects, OMT and SBP. The aim of the work is to provide theory and tools for modeling and control of collections of dynamical systems. The SBP can be seen as a stochastic counterpart of OMT and, as a result, OMT can be recovered as the limit of the SBP as the stochastic excitation vanishes. The link between these two problems gives rise to a novel and fast algorithm to compute solutions of OMT as a suitable limit of SBP. For the special case where the marginal distributions are Gaussian and the underlying dynamics linear, the solution to either problem can be expressed as linear state feedback and computed explicitly in closed form. A natural extension of the work in the thesis concerns OMT and the SBP on discrete spaces and graphs in particular. Along this line we develop a framework to schedule transportation of mass over networks. Control in this context amounts to selecting a transition mechanism that is consistent with initial and final marginal distributions. The SBP on graphs on the other hand can be viewed as an atypical stochastic control problem where, once again, the control consists in suitably modifying the prior transition mechanism. By taking the Ruelle-Bowen random walk as a prior, we obtain scheduling that tends to utilize all paths as uniformly as the topology allows. Effectively, a consequence of such a choice is reduced congestion and increased robustness. The paradigm of Schroedinger bridges as a mechanism for scheduling transport on networks can be adapted to weighted graphs. Thus, our approach may be used to design transportation plans that represent a suitable compromise between robustness and cost of transport.