The purpose of this note is to clarify the importance of the relation $\boldsymbol{gg}^{\top}\propto \boldsymbol{\sigma\sigma}^{\top}$ in solving control-affine Schr\"{o}dinger bridge problems via the Hopf-Cole transform, where $\boldsymbol{g},\boldsymbol{\sigma}$ are the control and noise coefficients, respectively. We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schr\"{o}dinger bridge problems, i.e., without the assumption $\boldsymbol{gg}^{\top}\propto\boldsymbol{\sigma\sigma}^{\top}$, gives a pair of forward-backward PDEs that are neither linear nor equation-level decoupled. We explain how the resulting PDEs can be interpreted as nonlinear forward-backward advection-diffusion-reaction equations, where the nonlinearity stem from additional drift and reaction terms involving the gradient of the log-likelihood a.k.a. the score. These additional drift and reaction vanish when $\boldsymbol{gg}^{\top}\propto\boldsymbol{\sigma\sigma}^{\top}$, and the resulting boundary-coupled system of linear PDEs can then be solved by dynamic Sinkhorn recursions. A key takeaway of our work is that the numerical solution of the generic control-affine Schr\"{o}dinger bridge requires further algorithmic development, possibly generalizing the dynamic Sinkhorn recursion or otherwise.

Schr\"{o}dinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schr\"{o}dinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schr\"{o}dinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schr\"{o}dinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials.

Lambert's problem concerns with transferring a spacecraft from a given initial to a given terminal position within prescribed flight time via velocity control subject to a gravitational force field. We consider a probabilistic variant of the Lambert problem where the knowledge of the endpoint constraints in position vectors are replaced by the knowledge of their respective joint probability density functions. We show that the Lambert problem with endpoint joint probability density constraints is a generalized optimal mass transport (OMT) problem, thereby connecting this classical astrodynamics problem with a burgeoning area of research in modern stochastic control and stochastic machine learning. This newfound connection allows us to rigorously establish the existence and uniqueness of solution for the probabilistic Lambert problem. The same connection also helps to numerically solve the probabilistic Lambert problem via diffusion regularization, i.e., by leveraging further connection of the OMT with the Schr\"odinger bridge problem (SBP). This also shows that the probabilistic Lambert problem with additive dynamic process noise is in fact a generalized SBP, and can be solved numerically using the so-called Schr\"odinger factors, as we do in this work. We explain how the resulting analysis leads to solving a boundary-coupled system of reaction-diffusion PDEs where the nonlinear gravitational potential appears as the reaction rate. We propose novel algorithms for the same, and present illustrative numerical results. Our analysis and the algorithmic framework are nonparametric, i.e., we make neither statistical (e.g., Gaussian, first few moments, mixture or exponential family, finite dimensionality of the sufficient statistic) nor dynamical (e.g., Taylor series) approximations.

We consider the finite horizon optimal steering of the joint state probability distribution subject to the angular velocity dynamics governed by the Euler equation. The problem and its solution amounts to controlling the spin of a rigid body via feedback, and is of practical importance, for example, in angular stabilization of a spacecraft with stochastic initial and terminal states. We clarify how this problem is an instance of the optimal mass transport (OMT) problem with bilinear prior drift. We deduce both static and dynamic versions of the Eulerian OMT, and provide analytical and numerical results for the synthesis of the optimal controller.

Things are not only bigger in Texas, they're also hotter, more sprawling and increasingly traffic-clogged thanks to a population boom that has lasted nearly a decade. In many of the state's fast-growing, car-dependent cities, these realities make for lousy walking conditions and long commutes. For the self-driving car company Drive.ai, Nearly four months after the Mountain View, Calif.-based start-up launched a six-month pilot program in nearby Frisco, Tex., the company deployed its second self-driving service on public roads in Arlington, Tex., on Friday. The service -- which is free to use -- will operate multiple routes in geo-fenced areas in downtown Arlington, according to Drive.ai

Things are not only bigger in Texas, they're also hotter, more sprawling and increasingly traffic-clogged thanks to a population boom that has lasted nearly a decade. In many of the state's fast-growing, car-dependent cities, these realities make for lousy walking conditions and long commutes. For the self-driving car company Drive.ai, Nearly four months after the Mountain View, Calif.-based start-up launched a six-month pilot program in nearby Frisco, Tex., the company deployed its second self-driving service on public roads in Arlington, Tex., on Friday. The service -- which is free to use -- will operate multiple routes in geo-fenced areas in downtown Arlington, according to Drive.ai

Nearly halfway into the NFL season, the Dallas Cowboys are 3–3 and sit 20th out of 32 on ESPN's power ranking index, which gives them a less than 50–50 shot at making the playoffs. So fans of America's Team don't have a whole lot to get excited about. Unless, that is, they like riding in robot cars. Today, startup Drive.ai is launching a self-driving car service in Arlington, Texas, which sits halfway between Dallas and Fort Worth and is home to the Cowboys' AT&T Stadium. The service will run several routes in multiple parts of the city, bustling to and from big venues including that stadium, Globe Life Park (where baseball's Texas Rangers play), and the Arlington Convention Center.

Fingers fly and eyes meet. This orchestra may seem a mess to anyone stuck in the pit at rush hour, but for the most part, it works. Humans may not excel as drivers when it comes to paying attention or keeping calm, but we're masters of communication, even when stuck in our metal boxes. Robots offer this resume in reverse: all-stars when it comes to defeating distraction, noobs when it comes to negotiating the human-filled environment. And for the folks aiming to deploy fleets of self-driving cars into that chaos, this is a problem.

Bibhrajit Halder left the Midwest and a job developing autonomous trucks for Caterpillar Inc. about a year and a half ago to join Ford Motor Co. in the San Francisco Bay Area, where the auto maker is working on self-driving vehicles. The Dearborn, Mich., auto maker, however, soon lost the software engineer to Faraday Future Inc., an electric-car startup luring auto industry veterans with Silicon Valley-like perks including stock options, free health care, catered lunches and foosball tables. "The work is exciting," Mr. Halder said in an interview about six months after joining Faraday, where he says he has more responsibility than at the blue chip companies he left. "The company is dependent on you to deliver." Ford is at the center of a ferocious hiring battle now pitting traditional car makers against startups out to force a shift to electric and autonomous-driving vehicles.

Bibhrajit Halder left the Midwest and a job developing autonomous trucks for Caterpillar Inc. CAT -1.46 % about a year and a half ago to join Ford Motor Co. F -1.21 % in the San Francisco Bay Area, where the auto maker is working on self-driving vehicles. The Dearborn, Mich., auto maker, however, soon lost the software engineer to Faraday Future Inc., an electric-car startup luring auto industry veterans with Silicon Valley-like perks including stock options, free health care, catered lunches and foosball tables. "The work is exciting," Mr. Halder said in an interview about six months after joining Faraday, where he says he has more responsibility than at the blue chip companies he left. "The company is dependent on you to deliver."