Sub-Riemannian geometry and nonlinear control theory

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In many practical control applications, the systems of interest are non-linear and under-actuated. In most cases of interest, these systems are furthermore non-holonomic. Such systems appear in widely distinct contexts, ranging from classical mechanical systems, such as cars, to the quantum mechanical systems used in Nuclear Magnetic Resonance Imaging. One of the key problems of non-holonomic control is the computation of explicit optimal controls for various applications. This is a challenging problem which has not been entirely settled except in some very special cases. Fundamental research into understanding the properties of these optimal controls, as well as ways to compute or approximate them, is then of considerable importance. The core idea of this research proposal is to study non-holonomic systems by jointly studying the corresponding sub-Riemannian geometry and sub-elliptic diffusion they define. Specifically, we want to investigate the relations between the local sub-Riemannian invariants and the optimal controls. As a starting point, we will study the local geometric invariants of sub-Riemannian contact structures. We will also investigate the properties of the corresponding sub-elliptic diffusions, and we will begin by computing heat kernels or parametrices for these diffusions, since we know they are intimately related to the value function of the optimal control problem. We expect this work will contribute to a deeper understanding of non-holonomic systems and sub-Riemannian geometry, and, in particular, of their relation.