Nilpotent control systems

Combinatorics of Realizations of Nilpotent Control Systems

Optimal control of nilpotent systems: a sub-Riemannian approach

We present a general framework for the optimal control of driftless nonlinear systems defined by means of distributions of smooth vector fields that generate nilpotent Lie algebras. A smooth varying inner product on the planes of the distribution, yields the energy functional that allows to approach the optimal control problem as a sub-Riemannian geodesic problem. This class of systems is relev...

Equivalence operators in nilpotent systems

Optimal Controls for Nilpotent Systems

This note develops explicit formulas for normal forms of the Hamiltonian lifts of trajectories of free nilpotent control systems. These normal forms utilize notions from chronological algebras and rely on combinatorial results about unique factorizations into nonincreasing sequences of generalized Hall words. The normal forms of the Hamilton Jacobi equations exhibit a natural bi-graded structur...

Nilpotent Hopf Bifurcations in Coupled Cell Systems

Network architecture can lead to robust synchrony in coupled systems and, surprisingly, to codimension one bifurcations from synchronous equilibria at which the associated Jacobian is nilpotent. We prove three theorems concerning nilpotent Hopf bifurcations from synchronous equilibria to periodic solutions, where the critical eigenvalues have algebraic multiplicity two and geometric multiplicit...