1,721,021 research outputs found
On the space of symmetric operators with multiple ground states
No full textWe study homological structure of the filtration of the space of selfadjoint operators by the multiplicity of the ground state. We consider only operators acting in a finite dimensional complex or real Hilbert space but infinite dimensional generalizations are easily guessed
Quadratic cohomology
No full textWe study homological invariants of smooth families of real quadratic forms as a step towards a “Lagrange multipliers rule in the large” that intends to describe topology of smooth maps in terms of scalar Lagrange functions
The curvature and hyperbolicity of Hamiltonian systems
No full textCurvature-type invariants of Hamiltonian systems generalize sectional curvatures of Riemannian manifolds: the negativity of the curvature is an indicator of the hyperbolic behavior of the Hamiltonian flow. In this paper, we give a self-contained description of the related constructions and facts; they lead to a natural extension of the classical results about Riemannian geodesic flows and indicate some new phenomena
Feedback--invariant optimal control theory and differential geometry, II. Jacobi curves for singular extremals
No full textThis is the second article in the series that began in [4]. Jacobi curves were defined, computed, and studied in that paper for regular extremals of smooth control systems. Here we do the same for singular extremals. The last section contains a feedback classification and normal forms of generic single-input affine in control systems on a 3-dimensional manifold
Hamiltonian systems and optimal control
No full textSolutions of any optimal control problem are described by trajectories of a Hamiltonian system. The system is intrinsically associated to the problem by a procedure that is a geometric elaboration of the Lagrange multipliers rule. The intimate relation of the optimal control and Hamiltonian dynamics is fruitful for both domains; among other things, it leads to a clarification and a far going generalization of important classical results about Riemannian geodesic flows
Switching in Time-Optimal Problem: the 3D Case with 2D Control
No full textWe study local structure of time-optimal controls and trajectories for a 3-dimensional control-affine system with a 2-dimensional control parameter with values in the disk. In particular, we give sufficient conditions, in terms of Lie bracket relations, for optimal controls to be smooth or to have only isolated jump discontinuities
Topics in sub-Riemannian geometry
Full text linkThis thesis is concerned with three different problems in sub-Riemannian geometry faced during my PhD. The first one is a problem in differential geometry and is about the local conformal classification of a certain class of sub-Riemannian structures. In the second one we deal with topology, and our main result establish some path-fibration properties for the Endpoint map. In the third and last problem, we begin the development of some variational calculus around critical points of the endpoint map, called abnormal controls, and we estabilish a counterpart of the classical Morse deformation techniques and of the Min-Max variational principle
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