Any sub-Riemannian metric has points of smoothness (2009)
We prove the result stated in the title that is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. In the case of a complete real-analytic sub-Riemannian...
Controllability on the group of diffeomorphisms (2008)
Agrachev, Andrei A., Caponigro, Marco
Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of...
The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups (2008)
Agrachev, Andrei A., Boscain, Ugo, Gauthier, Jean-Paul, Rossi, Francesco
We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery. This definition...
Agrachev, Andrei A., Stefani, Gianna
Incluye bibliografía
SIAM J. CONTROL OPTIM. c (2007)
Andrei A. Agrachev, Daniel Liberzon
Abstract. It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that...
The curvature and hyperbolicity of Hamiltonian systems (2007)
Curvature-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...
Vector fields on n-foliated 2n-dimensional manifolds (2007)
Agrachev, Andrei A., Gamkrelidze, R.V.
In this paper, we study basic differential invariants of the pair (vector field, foliation). As a result, we establish a dynamic interpretation and a generalization of the Levi-Civita connection and...
Agrachev, Andrei A., Chtcherbakova, Natalia N., Zelenko, Igor
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic...
Hamiltonian systems of negative curvature are hyperbolic (2007)
Agrachev, Andrei A., Chtcherbakova, Natalia N.
The curvature and the reduced curvature are basic differential invariants of the pair: (Hamiltonian system, Lagrange distribution) on the symplectic manifold. We show that negativity of the curvature...
On feedback classification of control-affine systems with one and two-dimensional inputs (2007)
Agrachev, Andrei A., Zelenko, Igor
The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback...
Control theory and optimal mass transport (2007)
Agrachev, Andrei A., Lee, Paul
We study the Monge’s optimal transportation problem where the cost is given by optimal control cost. We prove the existence and uniqueness of optimal map under certain regularity conditions on the...
An estimation of the controllability time for single-input systems on compact Lie Groups (2007)
Agrachev, Andrei A., Chambrion, Thomas
Geometric control theory and Riemannian techniques are used to describe the reachable set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to estimate...
On finite-dimensional projections of distributions for solutions of randomly forced PDE's (2007)
Agrachev, Andrei A., Kuksin, Sergei, Sarychev, Andrey, Shirikyan, Armen
The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has...
Smooth optimal synthesis for infinite horizon variational problems (2007)
Chittaro, Francesca C., Agrachev, Andrei A.
We study Hamiltonian systems which generate extremal flows of regular variational problems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a system implies...
Rolling balls and Octonions (2006)
In this semi-expository paper we disclose hidden symmetries of a classical nonholonomic kinematic model and try to explain geometric meaning of basic invariants of vector distributions.
Gauss-Bonnet-like Formula on Two-Dimensional Almost-Riemannian (2006)
Agrachev, Andrei A., Boscain, Ugo, Sigalotti, Mario
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and...
A Gauss-Bonnet-like Formula on Two-Dimensional almost-Riemannian Manifolds (2006)
Agrachev, Andrei A., Boscain, Ugo, Sigalotti, Mario
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and...
A Gauss-Bonnet-like Formula on Two-Dimensional almost-Riemannian Manifolds (2006)
Agrachev, Andrei A., Boscain, Ugo, Sigalotti, Mario
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and...
A Gauss-Bonnet-like Formula on Two-Dimensional Almost-Riemannian Manifolds (2006)
Agrachev, Andrei A., Boscain, Ugo, Sigalotti, Mario
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and...
Geometry of optimal control problems and Hamiltonian systems (2006)
These notes are based on the mini-course given in June 2004 in Cetraro, Italy, in the frame of a C.I.M.E. school. Of course, they contain much more material that I could present in the 6 hours...
Controllability of 2D Euler and Navier-Stokes equations by forcing 4 modes (2006)
Agrachev, Andrei A., Sarychev, Andrey V.
We study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid...
A Gauss-Bonnet-like Formula on Two-Dimensional almost-Riemannian Manifolds (2006)
Agrachev, Andrei A., Boscain, Ugo, Sigalotti, Mario
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and...
A Gauss-Bonnet-like Formula on Two-Dimensional almost-Riemannian Manifolds (2006)
Agrachev, Andrei A., Boscain, Ugo, Sigalotti, Mario
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and...
Navier–Stokes Equations: Controllability by Means of Low Modes Forcing (2005)
Agrachev, Andrei A., Sarychev, Andrey V.
We study controllability issues for 2D and 3D Navier–Stokes (NS) systems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of...
Hamiltonian systems of negative curvature are hyperbolic (2004)
Agrachev, Andrei A., Chtcherbakova, Natalia N.
The {\it curvature} and the {\it reduced curvature} are basic differential invariants of the pair: (Hamiltonian system, Lagrange distribution) on the symplectic manifold. We show that negativity of...
Nonholonomic tangent spaces: intrinsic construction and rigid dimensions (2003)
Agrachev, Andrei A., Marigo, Alessia
A nonholonomic space is a smooth manifold equipped with a bracket generating family of vector fields. Its infinitesimal version is a homogeneous space of a nilpotent Lie group endowed with a dilation...
On the local structure of optimal trajectories in R3 (2002)
Agrachev, Andrei A., Sigalotti, Mario
We analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system $\dot q =f(q)+u\, g(q)$ in a three-dimensional manifold, near a point where some...
Geometry of Jacobi Curves I (2002)
Agrachev, Andrei A., Zelenko, Igor
Jacobi curves are deep generalizations of the spaces of "Jacobi fields" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop...
Geometry of Jacobi Curves I (2002)
Agrachev, Andrei A., Zelenko, Igor
Jacobi curves are deep generalizations of the spaces of "Jacobi fields" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop...
A “Gauss–Bonnet formula” for contact sub-Riemannian manifolds (2001)
We study 3-dimensional manifolds endowed with oriented contact sub-Riemannian structures. The Euler characteristic class of the contact structure is presented as the rotation class of a volume...
Lie-Algebraic Stability Criteria for Switched Systems (2001)
Andrei A. Agrachev, Daniel Liberzon
It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the...
An intrinsic approach to the control of rolling bodies (1999)
Agrachev, Andrei A., Sachkov, Yuri L.
We apply the principal tools of geometric control theory to an intrinsic geometric model of a pair of rolling rigid bodies. The controllability problem is solved completely: in particular, the system...
Lie-algebraic stability criteria for switched systems (1999)
Agrachev, Andrei A., Liberzon, Daniel
It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the...
Lie-Algebraic Conditions for Exponential Stability of Switched Systems (1999)
Andrei A. Agrachev, Daniel Liberzon
It has recently been shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which means that the...