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Geometry of optimal control for control-affine systems. (English) Zbl 1267.93036

Summary: Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with two states and one control and systems with three states and one control, and use Pontryagin’s maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied.

MSC:

93B27 Geometric methods
49K15 Optimality conditions for problems involving ordinary differential equations
53C17 Sub-Riemannian geometry
58A15 Exterior differential systems (Cartan theory)
53C10 \(G\)-structures
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