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Periodic controls in step 2 strictly convex sub-Finsler problems. (English) Zbl 07196170
Summary: We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with a strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all Casimirs linear in momenta on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic. Some related results for other Carnot groups are presented.

MSC:
49K15 Optimality conditions for problems involving ordinary differential equations
49N20 Periodic optimal control problems
53C17 Sub-Riemannian geometry
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