  This paper addresses the limitations of standard uncertainty models, e.g.,
robust (norm-bounded) and stochastic (one fixed distribution, e.g., Gaussian),
and proposes to model uncertainty via Optimal Transport (OT) ambiguity sets.
These constitute a very rich uncertainty model, which enjoys many desirable
geometrical, statistical, and computational properties, and which: (1)
naturally generalizes both robust and stochastic models, and (2) captures many
additional real-world uncertainty phenomena (e.g., black swan events). Our
contributions show that OT ambiguity sets are also analytically tractable: they
propagate easily and intuitively through linear and nonlinear (possibly
corrupted by noise) transformations, and the result of the propagation is again
an OT ambiguity set or can be tightly upper bounded by an OT ambiguity set. In
the context of dynamical systems, our results allow us to consider multiple
sources of uncertainty (e.g., initial condition, additive noise, multiplicative
noise) and to capture in closed-form, via an OT ambiguity set, the resulting
uncertainty in the state at any future time. Our results are actionable,
interpretable, and readily employable in a great variety of computationally
tractable control and estimation formulations. To highlight this, we study
three applications in trajectory planning, consensus algorithms, and least
squares estimation. We conclude the paper with a list of exciting open problems
enabled by our results.
