We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure.

}, doi = {10.2140/gt.2015.19.1569}, author = {Andrei A. Agrachev and Alessandro Gentile and Antonio Lerario} } @article {2013, title = {On conjugate times of LQ optimal control problems}, number = {Journal of Dynamical and Control Systems}, year = {2014}, note = {14 pages, 1 figure}, publisher = {Springer}, abstract = {Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $\vec{H}$. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if $\vec{H}$ has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of $\vec{H}$.}, keywords = {Optimal control, Lagrange Grassmannian, Conjugate point}, doi = {10.1007/s10883-014-9251-6}, url = {http://hdl.handle.net/1963/7227}, author = {Andrei A. Agrachev and Luca Rizzi and Pavel Silveira} } @article {2013, title = {The curvature: a variational approach}, number = {arXiv:1306.5318;}, year = {2013}, note = {88 pages, 10 figures, (v2) minor typos corrected, (v3) added sections on Finsler manifolds, slow growth distributions, Heisenberg group}, institution = {SISSA}, abstract = {The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.}, keywords = {Crurvature, subriemannian metric, optimal control problem}, url = {http://hdl.handle.net/1963/7226}, author = {Andrei A. Agrachev and Davide Barilari and Luca Rizzi} } @article {2013, title = {Quadratic cohomology}, year = {2013}, note = {24 pages}, publisher = {SISSA}, abstract = {We study homological invariants of smooth families of real quadratic forms as\r\na step towards a \"Lagrange multipliers rule in the large\" that intends to\r\ndescribe topology of smooth vector functions in terms of scalar Lagrange\r\nfunctions.}, author = {Andrei A. Agrachev} } @article {2013, title = {Some open problems}, number = {arXiv:1304.2590;}, year = {2013}, publisher = {SISSA}, abstract = {We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.}, keywords = {Geometry}, url = {http://hdl.handle.net/1963/7070}, author = {Andrei A. Agrachev} } @article {2012, title = {On the Hausdorff volume in sub-Riemannian geometry}, journal = {Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388}, number = {arXiv:1005.0540;}, year = {2012}, publisher = {SISSA}, abstract = {For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative\r\nof the spherical Hausdorff measure with respect to a smooth volume. We prove\r\nthat this is the volume of the unit ball in the nilpotent approximation and it\r\nis always a continuous function. We then prove that up to dimension 4 it is\r\nsmooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4\r\non every smooth curve) but in general not C^5. These results answer to a\r\nquestion addressed by Montgomery about the relation between two intrinsic\r\nvolumes that can be defined in a sub-Riemannian manifold, namely the Popp and\r\nthe Hausdorff volume. If the nilpotent approximation depends on the point (that\r\nmay happen starting from dimension 5), then they are not proportional, in\r\ngeneral.}, doi = {10.1007/s00526-011-0414-y}, url = {http://hdl.handle.net/1963/6454}, author = {Andrei A. Agrachev and Davide Barilari and Ugo Boscain} } @article {2012, title = {Introduction to Riemannian and sub-Riemannian geometry}, number = {SISSA;09/2012/M}, year = {2012}, institution = {SISSA}, url = {http://hdl.handle.net/1963/5877}, author = {Andrei A. Agrachev and Davide Barilari and Ugo Boscain} } @article {2012, title = {On robust Lie-algebraic stability conditions for switched linear systems}, journal = {Systems and Control Letters. Volume 61, Issue 2, February 2012, Pages 347-353}, year = {2012}, abstract = {This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novelty feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters.}, doi = {10.1016/j.sysconle.2011.11.016}, url = {http://hdl.handle.net/1963/6455}, author = {Andrei A. Agrachev and Yurij Baryshnikov and Daniel Liberzon} } @article {2012, title = {Sub-Riemannian structures on 3D Lie groups}, journal = {Journal of Dynamical and Control Systems. Volume 18, Issue 1, January 2012, Pages 21-44}, number = {arXiv:1007.4970;}, year = {2012}, publisher = {SISSA}, abstract = {We give a complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^+(\mathbb{R})\times S^1$, where $A^+(\mathbb{R})$ denotes the group of orientation preserving affine maps on the real line.

}, doi = {10.1007/s10883-012-9133-8}, url = {http://hdl.handle.net/1963/6453}, author = {Andrei A. Agrachev and Davide Barilari} } @article {2012, title = {Systems of Quadratic Inequalities}, journal = {Proceedings of the London Mathematical Society, Volume 105, Issue 3, September 2012, Pages 622-660}, number = {arXiv:1012.5731;}, year = {2012}, publisher = {SISSA}, abstract = {We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit formula for the differential d_2.}, doi = {DOI: 10.1112/plms/pds010}, url = {http://hdl.handle.net/1963/7072}, author = {Andrei A. Agrachev and Antonio Lerario} } @article {2011, title = {Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry}, year = {2011}, note = {25 pages}, publisher = {SISSA}, abstract = {We prove a Bishop volume comparison theorem and a Laplacian comparison\r\ntheorem for three dimensional contact subriemannian manifolds with symmetry.}, url = {http://hdl.handle.net/1963/6508}, author = {Andrei A. Agrachev and Paul Lee} } @article {2011, title = {Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds}, number = {arXiv:0903.2550;}, year = {2011}, note = {This is a revised extended version that contains new results.}, publisher = {SISSA}, url = {http://hdl.handle.net/1963/6507}, author = {Andrei A. Agrachev and Paul Lee} } @article {2011, title = {The geometry of Maximum Principle}, journal = {Proceedings of the Steklov Institute of mathematics. vol. 273 (2011), page: 5-27 ; ISSN: 0081-5438}, number = {Proceedings of the Steklov Institute of mathematics;v.273}, year = {2011}, abstract = {An invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.}, url = {http://hdl.handle.net/1963/6456}, author = {Andrei A. Agrachev and Revaz Gamkrelidze} } @article {2011, title = {On the Space of Symmetric Operators with Multiple Ground States}, journal = {Functional Analysis and its Applications, Volume 45, Issue 4, December 2011, Pages 241-251}, number = {arXiv:1107.3010;}, year = {2011}, publisher = {SISSA}, abstract = {We study homological structure of the filtrations of the spaces of self-adjoint 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.}, keywords = {Multiple eigenvalue}, doi = {DOI: 10.1007/s10688-011-0026-5}, url = {http://hdl.handle.net/1963/7069}, author = {Andrei A. Agrachev} } @article {2010, title = {Continuity of optimal control costs and its application to weak KAM theory}, journal = {Calculus of Variations and Partial Differential Equations. Volume 39, Issue 1, 2010, Pages 213-232}, number = {arXiv:0909.3826;}, year = {2010}, note = {23 pages, 1 figures}, publisher = {SISSA}, abstract = {We prove continuity of certain cost functions arising from optimal control of\\r\\naffine control systems. We give sharp sufficient conditions for this\\r\\ncontinuity. As an application, we prove a version of weak KAM theorem and\\r\\nconsider the Aubry-Mather problems corresponding to these systems.}, doi = {10.1007/s00526-010-0308-4}, url = {http://hdl.handle.net/1963/6459}, author = {Andrei A. Agrachev and Paul Lee} } @article {2010, title = {Dynamics control by a time-varying feedback}, journal = {Journal of Dynamical and Control Systems. Volume 16, Issue 2, April 2010, Pages :149-162}, year = {2010}, publisher = {SISSA}, abstract = {We consider a smooth bracket generating control-affine system in R^d and show that any orientation preserving diffeomorphism of R^d can be approximated, in the very strong sense, by a diffeomorphism included in the flow generated by a time-varying feedback control which is polynomial with respect to the state variables and trigonometric-polynomial with respect to the time variable.}, keywords = {Discrete-time dynamics}, doi = {10.1007/s10883-010-9087-7}, url = {http://hdl.handle.net/1963/6461}, author = {Andrei A. Agrachev and Marco Caponigro} } @article {2010, title = {Invariant Lagrange submanifolds of dissipative systems}, journal = {Russian Mathematical Surveys. Volume 65, Issue 5, 2010, Pages: 977-978}, number = {arXiv:0912.2248;}, year = {2010}, publisher = {SISSA}, abstract = {We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) =\\r\\n0, q \\\\in M, on a compact manifold M .}, doi = {10.1070/RM2010v065n05ABEH004707}, url = {http://hdl.handle.net/1963/6457}, author = {Andrei A. Agrachev} } @article {2010, title = {Two-dimensional almost-Riemannian structures with tangency points}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire }, volume = {27}, number = {arXiv.org;0908.2564v1}, year = {2010}, pages = {793-807}, publisher = {Elsevier}, abstract = {Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

}, doi = {10.1016/j.anihpc.2009.11.011}, url = {http://hdl.handle.net/1963/3870}, author = {Andrei A. Agrachev and Ugo Boscain and Gr{\'e}goire Charlot and Roberta Ghezzi and Mario Sigalotti} } @article {2010, title = {Well-posed infinite horizon variational problems on a compact manifold}, journal = {Proceedings of the Steklov Institute of Mathematics. Volume 268, Issue 1, 2010, Pages 17-31}, number = {arXiv:0906.4433;}, year = {2010}, publisher = {SISSA}, abstract = {We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i. e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian submanifold (well-projected to M) of the flow of extremals in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics}, doi = {10.1134/S0081543810010037}, url = {http://hdl.handle.net/1963/6458}, author = {Andrei A. Agrachev} } @article {2009, title = {Controllability on the group of diffeomorphisms}, journal = {Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire 26 (2009) 2503-2509}, number = {SISSA;79/2008/M}, year = {2009}, abstract = {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 identity of the group of diffeomorphisms of M.}, doi = {10.1016/j.anihpc.2009.07.003}, url = {http://hdl.handle.net/1963/3396}, author = {Andrei A. Agrachev and Marco Caponigro} } @article {2009, title = {The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups}, journal = {J. Funct. Anal. 256 (2009) 2621-2655}, number = {SISSA;33/2008/M}, year = {2009}, abstract = {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 generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.}, doi = {10.1016/j.jfa.2009.01.006}, url = {http://hdl.handle.net/1963/2669}, author = {Andrei A. Agrachev and Ugo Boscain and Jean-Paul Gauthier and Francesco Rossi} } @article {2009, title = {Optimal transportation under nonholonomic constraints}, journal = {Trans. Amer. Math. Soc. 361 (2009) 6019-6047}, number = {SISSA;68/2007/M}, year = {2009}, abstract = {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 Lagrangian, absolute continuity of the measures and most importantly the absent of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2-generating distribution and cost given by d2, where d is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane.}, doi = {10.1090/S0002-9947-09-04813-2}, url = {http://hdl.handle.net/1963/2176}, author = {Andrei A. Agrachev and Paul Lee} } @article {2008, title = {A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds}, journal = {Discrete Contin. Dyn. Syst. 20 (2008) 801-822}, number = {SISSA;55/2006/M}, year = {2008}, abstract = {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 $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of rank-varying sub-Riemannian structures, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular for what concerns the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.}, doi = {10.3934/dcds.2008.20.801}, url = {http://hdl.handle.net/1963/1869}, author = {Andrei A. Agrachev and Ugo Boscain and Mario Sigalotti} } @article {2007, title = {On feedback classification of control-affine systems with one and two-dimensional inputs}, journal = {SIAM J. Control Optim. 46 (2007) 1431-1460}, number = {arXiv.org;math/050231v2}, year = {2007}, abstract = {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 transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intrinsic numbers of functional moduli of each prescribed number of variables on which a classification problem depends. In order to classify affine systems with scalar input we associate with such a system the canonical frame by normalizing some structural functions in a commutative relation of the vector fields, which define our control system. Then, using this canonical frame, we introduce the canonical coordinates and find a complete system of state-feedback invariants of the system. It also gives automatically the micro-local (i.e. local in state-input space) classification of the generic non-affine n-dimensional control system with scalar input for n>2. Further we show how the problem of feedback-equivalence of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine systems with scalar input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension 4 and all abnormal extremals in dimension 5 of the time optimal problem, defined by the original control system. In each classification problem under consideration we find the intrinsic numbers of functional moduli of each prescribed number of variables according to its Poincare series.}, doi = {10.1137/050623711}, url = {http://hdl.handle.net/1963/2186}, author = {Andrei A. Agrachev and Igor Zelenko} } @article {2007, title = {On finite-dimensional projections of distributions for solutions of randomly forced PDE\\\'s}, journal = {Ann. Inst. Henri Poincare-Prob. Stat. 43 (2007) 399-415}, number = {arXiv.org;math/0603295}, year = {2007}, abstract = {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 a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier-Stokes equations perturbed by various random forces of low dimension.}, doi = {10.1016/j.anihpb.2006.06.001}, url = {http://hdl.handle.net/1963/2012}, author = {Andrei A. Agrachev and Sergei Kuksin and Andrey Sarychev and Armen Shirikyan} } @article {2006, title = {An estimation of the controllability time for single-input systems on compact Lie Groups}, journal = {ESAIM Control Optim. Calc. Var. 12 (2006) 409-441}, year = {2006}, abstract = {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 the minimal time needed to reach any point from identity. This method provides an effective way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum system with a drift from a given initial position to a given final position. The bounds include diameters of the flag manifolds; the latter are also explicitly computed in the paper.}, doi = {10.1051/cocv:2006007}, url = {http://hdl.handle.net/1963/2135}, author = {Andrei A. Agrachev and Thomas Chambrion} } @article {2006, title = {Experimental and modeling studies of desensitization of P2X3 receptors.}, journal = {Molecular pharmacology. 2006 Jul; 70(1):373-82}, number = {PMID:16627751;}, year = {2006}, publisher = {the American Society for Pharmacology and Experimental Therapeutics}, abstract = {The function of ATP-activated P2X3 receptors involved in pain sensation is modulated by desensitization, a phenomenon poorly understood. The present study used patch-clamp recording from cultured rat or mouse sensory neurons and kinetic modeling to clarify the properties of P2X3 receptor desensitization. Two types of desensitization were observed, a fast process (t1/2 = 50 ms; 10 microM ATP) following the inward current evoked by micromolar agonist concentrations, and a slow process (t1/2 = 35 s; 10 nM ATP) that inhibited receptors without activating them. We termed the latter high-affinity desensitization (HAD). Recovery from fast desensitization or HAD was slow and agonist-dependent. When comparing several agonists, there was analogous ranking order for agonist potency, rate of desensitization and HAD effectiveness, with 2-methylthioadenosine triphosphate the strongest and beta,gamma-methylene-ATP the weakest. HAD was less developed with recombinant (ATP IC50 = 390 nM) than native P2X3 receptors (IC50 = 2.3 nM). HAD could also be induced by nanomolar ATP when receptors seemed to be nondesensitized, indicating that resting receptors could express high-affinity binding sites. Desensitization properties were well accounted for by a cyclic model in which receptors could be desensitized from either open or closed states. Recovery was assumed to be a multistate process with distinct kinetics dependent on the agonist-dependent dissociation rate from desensitized receptors. Thus, the combination of agonist-specific mechanisms such as desensitization onset, HAD, and resensitization could shape responsiveness of sensory neurons to P2X3 receptor agonists. By using subthreshold concentrations of an HAD-potent agonist, it might be possible to generate sustained inhibition of P2X3 receptors for controlling chronic pain.}, doi = {10.1124/mol.106.023564}, url = {http://hdl.handle.net/1963/4974}, author = {Elena Sokolova and Andrei Skorinkin and Igor Moiseev and Andrei A. Agrachev and Andrea Nistri and Rashid Giniatullin} } @article {2005, title = {On curvatures and focal points of distributions of dynamical Lagrangian distributions and their reductions by first integrals}, journal = {J. Dyn. Control Syst. 11 (2005) 297-327}, number = {SISSA;58/2004/M}, year = {2005}, abstract = {Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and the curvature form. These invariants can be seen as generalizations of the classical curvature tensor in Riemannian Geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. The interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point w.r.t. the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points w.r.t. its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical $N$-body problem.}, doi = {10.1007/s10883-005-6581-4}, url = {http://hdl.handle.net/1963/2254}, author = {Andrei A. Agrachev and Natalia N. Chtcherbakova and Igor Zelenko} } @article {2003, title = {On the local structure of optimal trajectories in R3}, journal = {SIAM J. Control Optim. 42 (2003) 513-531}, number = {SISSA;41/2002/M}, year = {2003}, publisher = {SISSA Library}, abstract = {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 nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four.}, doi = {10.1137/S0363012902409246}, url = {http://hdl.handle.net/1963/1612}, author = {Andrei A. Agrachev and Mario Sigalotti} } @article {2002, title = {Geometry of Jacobi Curves I}, journal = {J. Dynam. Control Systems 8 (2002) 93-140}, number = {SISSA;75/2001/M}, year = {2002}, publisher = {Springer}, abstract = {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 differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.\\nIn the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of \\\"flat curves\\\".}, doi = {10.1023/A:1013904801414}, url = {http://hdl.handle.net/1963/3110}, author = {Andrei A. Agrachev and Igor Zelenko} } @article {2002, title = {Geometry of Jacobi curves II}, journal = {J. Dynam. Control Systems 8 (2002), no. 2, 167--215}, number = {SISSA;18/2002/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1023/A:1015317426164}, url = {http://hdl.handle.net/1963/1589}, author = {Andrei A. Agrachev and Igor Zelenko} } @article {2001, title = {On the subanalyticity of Carnot-Caratheodory distances}, journal = {Ann. I. H. Poincare - An., 2001, 18, 359}, number = {SISSA;25/00/M}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0294-1449(00)00064-0}, url = {http://hdl.handle.net/1963/1483}, author = {Andrei A. Agrachev and Jean-Paul Gauthier} } @inbook {2000, title = {Principal invariants of Jacobi curves}, booktitle = {Nonlinear control in the Year 2000 / Alberto Isidori, Francoise Lamnabhi-Lagarrigue, Witold Respondek (eds.) - Springer : Berlin, 2001. - (Lecture notes in control and information sciences ; 258). - ISBN 1-85233-363-4 (v. 1). - p. 9-22.}, year = {2000}, publisher = {Springer}, organization = {Springer}, abstract = {Jacobi curves are far going generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present paper we mainly discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmanian providing the curve with a natural projective structure, and a fundamental form, which is a 4-oder differential on the curve.}, doi = {10.1007/BFb0110204}, url = {http://hdl.handle.net/1963/3825}, author = {Andrei A. Agrachev and Igor Zelenko} }