Publications of the research group on Geometric Control Theory



Lecture notes

A.A. Agrachev, Yu.L. Sachkov, Control Theory from the Geometric Viewpoint, Preprint SISSA, November 2002, SISSA, Trieste, Italy.
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Controllability of nonlinear systems, in particular, global controllability of right-invariant systems on Lie groups and high-order conditions for local controllability.

A.A. Agrachev, A. Sarychev, On reductions of smooth systems, linear in the control. Matem. sbornik, 1986, v.130.

A.A. Agrachev, Newton diagrams and tangent cones to attainable sets. In the book: Analysis of Controlled Dynamical Systems. B. Bonnard, B. Bride, J.P. Gauthier, I. Kupka, Eds. Proc. col. Int. Lyon, France, 3-6 juillet, 1990. Birkhauser, 1991.

A.A. Agrachev, R. Gamkrelidze, Local controllability for families of diffeomorphisms. Systems and Control Letters, 1993, v.20.

A.A. Agrachev, R. Gamkrelidze, Local controllability and semigroups of diffeomorphisms. Acta Appl. Math., 1993, v.32.

A.A. Agrachev, Is it possible to recognize local controllability in a finite number of differentiations? In the book: Open Problems in Mathematical Systems Theory and Control. Springer Verlag, London, 1999, 15--18.

Yu.L. Sachkov, Controllability of Hypersurface and Solvable Invariant Systems, Journal of Dynamical and Control Systems, 2 (1996), 1: 55--67.

Yu.L. Sachkov, On Positive Orthant Controllability of Bilinear Systems in Small Codimensions, SIAM Journ. Control and Optimization, 35 (1997), 1: 29--35.

Yu.L. Sachkov, Controllability of Right-Invariant Systems on Solvable Lie Groups, Journal of Dynamical and Control Systems, 3 (1997), 4: 531--564.

Yu.L. Sachkov, On Invariant Orthants of Bilinear Systems, Journal of Dynamical and Control Systems, 4 (1998), 1: 137--147.

Yu.L. Sachkov, Controllability of Invariant Systems on Lie Groups and Homogeneous Spaces, J. Math. Sci., in: Progress in Science and Technology, Series on Contemporary Mathematics and Applications, Thematical Surveys, Vol. 59, Dynamical Systems-8, All-Russian Institute for Scientific and Technical Information (VINITI), Ross. Akad. Nauk, Moscow, 1998 (to appear).

Yu.L. Sachkov, Classification of controllable systems on low-dimensional solvable Lie groups, Journal of Dynamical and Control Systems, 6 (2000), 2: 159--217.



Approximation with nilpotent systems, series expansions of the input-output map.


A.A. Agrachev, R. Gamkrelidze, The exponential representation of flows and the chronological calculus. Matem. sbornik,1978, v.107.

A.A. Agrachev, R. Gamkrelidze, The chronological algebras and nonstationary vector fields. Itogi nauki. VINITI. Problemy geometrii, 1980, v.11.

A.A. Agrachev, S. Vakhrameev, Chronological series and Cauchy -- Kovalevska theorem. Itogi nauki. VINITI. Problemy geometrii, 1981, v.12.

A.A. Agrachev, A. Sarychev, Filtrations of a Lie algebra of vector fields and nilpotent approximation of controlled systems. Dokl. Acad. Nauk SSSR, 1987, v.295.

A.A. Agrachev, R. Gamkrelidze, and A. Sarychev, Local invariants of smooth control systems. Acta Appl. Math., 1989, v.14.

A.A. Agrachev, R. Gamkrelidze, and S. Vakhrameev, Ordinary differential equations on vector bundles and the chronological calculus. Itogi nauki. VINITI. Sovremennye problemy matematiki. Novejshie dostigeniya, 1989, v.35.

A.A. Agrachev, R. Gamkrelidze, Volterra series and groups of permutations. Itogi nauki. VINITI. Sovremennye problemy matematiki. Novejshie dostideniya, 1991, v.39.

A.A. Agrachev, R. Gamkrelidze, Shuffle products and symmetric groups. In the book: Differential Equations, Dynamical Systems, and Control Science. A Festchrift in Honor of Laurence Markus. Marcel Dekker, 1994.

A. Bressan, Local asymptotic approximation of nonlinear control systems, International J. Control, 41 (1985), 1331-1336.

A. Bressan, Nilpotent approximations and optimal trajectories, in ``Analysis of Controlled Dynamical Systems, B. Bonnard, B. Bride, J.P. Gauthier and I. Kupka eds., Birkh\"auser, Boston (1991), 103-117.



Feedback stabilization by discontinuous feedback with controlled singularities.

F. Ancona, A. Bressan, Patchy vector fields and asymptotic stabilization, ESAIM; Control, Optimization and Calculus of Variations, to appear.

A. Bressan, Singularities of stabilizing feedbacks, Rend. Mat. Univ. Torino, to appear.



Invariants and normal forms of nonlinear systems with respect to state and feedback transformations.

A.A. Agrachev, Feedback--invariant optimal control theory and differential geometry, II. Jacobi curves for singular extremals. J. Dynamical and Control Systems, 1998, v.4, 583--604.



Necessary and sufficient conditions for optimality, regularity properties of optimal trajectories and minimum time function.

A. Bressan, Sulla funzione tempo minimo nei sistemi non lineari, Atti Accad. Naz. Lincei, Serie VIII, vol. LXVI (1979), 383-388.

A. Bressan, On two conjectures by Hajek, Funkc. Ekvac. 23, (1980), 221-227.

A.A. Agrachev, R. Gamkrelidze , A second order optimality principle for time-optimal problems. Matem. sbornik, 1976, v.100.

A.A. Agrachev, A second order necessary condition for optimality in the general nonlinear case. Matem. sbornik, 1977, V.102.

A.A. Agrachev, R. Gamkrelidze, The extremality index and quasi-extremal controls. Dokl. Acad. Nauk SSSR, 1985, v.284, with

A.A. Agrachev, R. Gamkrelidze, The Morse index and the Maslov index for smooth control systems. Dokl. Acad. Nauk SSSR, 1986, v.287.

A.A. Agrachev, R. Gamkrelidze, The quasi-extremality for controlled systems. Itogi nauki. VINITI. Sovremennye problemy matematiki. Novejshie dostigeniya, 1989, v.35.

A.A. Agrachev, R. Gamkrelidze, Symplectic geometry for optimal control. In the book: Nonlinear Controllability and Optimal Control. H.~Sussmann, Ed. Marcel Dekker, 1990.

A.A. Agrachev, R. Gamkrelidze, Symplectic geometry and necessary conditions for optimality. Matem. sbornik, 1991, v.144.

A.A. Agrachev, S. Vakhrameev, Morse theory and optimal control problems. In the book: Nonlinear Synthesis. C.~I.~Byrnes, A.~Kurzhansky, Ed. Proc. Int. Conf. Sopron, Hungary, 5-9 June, 1989. Birkhauser, 1991.

A.A. Agrachev, A.V. Sarychev, On abnormal extremals for Lagrange variational problems. Mathematical Systems. Estimation and Control, 1998, v.8, 87--118

A.A. Agrachev, R.V. Gamkrelidze, Symplectic methods in optimization and control. In the book: Geometry of Feedback and Optimal Control. B.~Jakubczyk, W.~Respondek, Eds. Marcel Dekker, 1998, 1--58.

A.A. Agrachev, On regularity properties of extremal controls. J. Dynamical and Control Systems, 1995, v.1, 319-324.

A.A. Agrachev, G. Stefani, P. Zezza, Strong minima in optimal control. Proc. Steklov Math. Inst., 1998, v.220, 4--22.

A.A. Agrachev, G. Stefani and P. Zezza, An invariant second variation in optimal control. Int. J. Control, 1998, v.71.

A.A. Agrachev, G. Stefani, P. Zezza, A Hamiltonian approach to strong minima in optimal control. Proc. Symp. Pure Math., AMS, 1999, v.64, 11--22.

A.A. Agrachev, G. Stefani, P. Zezza, Symplectic methods for strong local optimality in the bang-bang case. Proceed. Int. Conf. on Geometric Control Theory, Mexico, Sept. 2000. Birchauser, to appear. A. Bressan, A high-order test for optimality of bang-bang controls, SIAM J. Control & Optim. 23 (1985), 38-48.

A. Bressan, Directional convexity and finite optimality conditions, J. Math. Anal. & Appl. 125 (1987), 234-246.

A. Bressan, A. Marson, A maximum principle for optimally controlled systems of conservation laws, Rend. Sem. Mat. Univ. Padova, 94 (1995), 79-94.

A. Bressan, Dual variational methods in optimal control theory, in Nonlinear Controllability and Optimal Control, H. Sussmann ed., M. Dekker (1990), 219-235.



Bang-bang property.

A. Bressan, On a bang-bang principle for nonlinear systems, Suppl. Boll. Un. Mat. Ital. 1, (1980), 53-59.

A. Bressan, B. Piccoli, A Baire category approach to the bang-bang property, J. Differential Equations, 116 (1995), 318-337.

A.A. Agrachev, S. Vakhrameev, Nonlinear control systems of constant rank and bang-bang conditions for extremal controls. Dokl. Acad. Nauk SSSR, 1984, v.279.



Optimal synthesis, its singularities and structural stability. Generic classifications in low dimensions.

A. Bressan, The generic local time-optimal stabilizing controls in dimension 3, SIAM J. Control & Optim., 24 (1986), 177-190.

A. Bressan, B. Piccoli, Structural stability for time optimal planar feedbacks, Discrete, Cont. Impuls. Dynamical Systems, 3 (1997), 335-371.

A. Bressan, B. Piccoli, A generic classification of optimal planar stabilizing feedbacks, S.I.A.M. J. Control Optim.36 (1998), 12-32.

U. Boscain, B. Piccoli, Geometric Control Approach To Synthesis Theory, Rendiconti del Seminario Matematico dell'Universit\`a e del Politecnico di Torino, ``Control Theory and related Topics'', accepted.

U. Boscain, B. Piccoli, Projection Singularities of Extremals for Planar Systems, Proceeding of the 38th IEEE Conference on Decision and Control, Phoenix, December 1999, accepted.



Symplectic and topological invariants of optimal control problems and Hamiltonian systems.

A.A. Agrachev, R. Gamkrelidze, Computation of the Euler characteristic of intersections of real quadrics. Dokl. Acad. Nauk SSSR, 1988, v.299.

A.A. Agrachev, Homology of intersections of real quadrics. Dokl. Acad. Nauk SSSR, 1988, v.299.

A.A. Agrachev, Quadratic mappings in the geometric control theory. Itogi nauki. VINITI. Problemy geometrii, 1988, v.20.

A.A. Agrachev, Topology of quadratic mappings and Hessians of smooth mappings. Itogi nauki. VINITI. Algebra. Topologiya. Geometriya, 1988, v.26.

A.A. Agrachev, R. Gamkrelidze, and S. Vakhrameev, Ordinary differential equations on vector bundles and the chronological calculus. Itogi nauki. VINITI. Sovremennye problemy matematiki. Novejshie dostigeniya, 1989, v.35.

A.A. Agrachev, A. Arutyunov, Singularities of level surfaces for smooth mappings and intersections of quadrics. Matem. sbornik, 1991, v.145.

A.A. Agrachev, S. Vakhrameev, Morse theory and optimal control problems. In the book: Nonlinear Synthesis. C. I. Byrnes, A. Kurzhansky, Ed. Proc. Int. Conf. Sopron, Hungary, 5-9 June, 1989. Birkhauser, 1991.

A.A. Agrachev, R.V. Gamkrelidze, Symplectic methods in optimization and control. In the book: Geometry of Feedback and Optimal Control. B.~Jakubczyk, W.~Respondek, Eds. Marcel Dekker, 1998, 1--58.

A.A. Agrachev, Methods of Control Theory in Nonholonomic Geometry. Proc. ICM-94, Z\"urich. Birkh\"auser, 1995, 1473--1483.

A.A. Agrachev, R.V. Gamkrelidze, Feedback--invariant optimal control theory and differential geometry, I. Regular extremals. J. Dynamical and Control Systems, 1997, v.3, 343--389.

A.A. Agrachev, Feedback--invariant optimal control theory and differential geometry, II. Jacobi curves for singular extremals. J. Dynamical and Control Systems, 1998, v.4, 583--604.

A.A. Agrachev, D. Pallaschke, S. Scholtes, On Morse theory for piecewise smooth functions. J. Dynamical and control systems, 1997, v.3, 449--469.

A.A. Agrachev, I. Zelenko, Principal invariants of Jacobi curves. In the book: Nonlinear Control in the Year 2000. Springer Verlag, 2000.

A. Agrachev, I. Zelenko, Geometry of Jacobi curves I. J. Dynamical and Control Systems, to appear.



Sub-Riemannian geometry and geometry of distributions.


A.A. Agrachev, A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Annales de l'Institut Henri Poincar\'e---Analyse non lin\'eaire, 1996, v.13, 635--690.

A.A. Agrachev, A.V. Sarychev, Strong minimality of abnormal geodesics for 2-distributions. J. Dynamical and Control Systems, 1995, v.1, 139--176.

A.A. Agrachev, Methods of Control Theory in Nonholonomic Geometry. Proc. ICM-94, Z\"urich. Birkh\"auser, 1995, 1473--1483.

A.A. Agrachev, Ch.El Alaoui, J.-P. Gauthier, I. Kupka, Generic singularities of sub-Riemannian metrics on $R^3.$ Comptes Rendus Acad. Sci., 1996, v.322, Serie I, 377--384.

A.A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dynamical and Control Systems, 1996, v.2, 321--358.

A.A. Agrachev, Ch.El Alaoui, J.-P. Gauthier, Sub-Riemannian metrics in $R^3$. Proc. Canadian Math. Soc., 1998, v.25, 29--78.

A.A. Agrachev, B. Bonnard, M. Chyba, I. Kupka, Sub-Riemannian sphere in Martinet flat case. J. ESAIM: Control, Optimisation and Calculus of Variations, 1997, v.2, 377--448.

A.A. Agrachev, J.-P. Gauthier, On the Dido problem and plane isoperimetric problems. Acta Appl. Math., 1999, v.57, 287--338.

A.A. Agrachev, On the equivalence of different types of local minima in sub-Riemannian problems. Proc. 37th IEEE Confer. on Decision and Control, 1998, 2240---2243.

A.A. Agrachev, A.V. Sarychev, Sub-Riemannian metrics: minimality of abnormal geodesics versus subanalyticity. J. ESAIM: Control, Optimisation and Calculus of Variations, 1999, v.4, 377--403.

A.A. Agrachev, J.P. Gauthier, Sub-Riemannian metrics and isoperimetric problems in the contact case. Proc. Int. Confer. Pontryagin-90, 1999, v. 3, 5--48.

A.A. Agrachev, Compactness for sub-Riemannian length-minimizers and subanalyticity. Rend. Semin. Mat. Torino, 1998, v.56

A.A. Agrachev, G. Charlot, J.-P. Gauthier, V.M. Zakalyukin, On stability of generic sub-Riemannian caustics in the three-space. Comptes Rendus Acad. Sci., 2000.

A.A. Agrachev, G. Charlot, J.-P. Gauthier, V.M. Zakalyukin, On sub-Riemannian caustics and wave fronts for contact distributions in the three-space. J. Dynamical and Control Systems, 2000, v.6.

A.A. Agrachev, J.-P. Gauthier, Subanalyticty of distance and spheres in sub-Riemannian geometry. In the book: Nonlinear Control in the Year 2000. Springer Verlag, 2000.

A.A. Agrachev, J.-P. Gauthier, On subanalyticity of Carnot-Caratheodory distances. Annales de l'Institut Henri Poincare - Analyse non lineaire, v.18, 359--382, 2001.

A.A. Agrachev, A "Gauss--Bonnet formula" for contact sub-Riemannian manifolds. Russian Math. Dokl., to appear

Yu.L. Sachkov, Symmetries of Flat Rank Two Distributions and Sub-Riemannian Structures, Report No. 98-151, March 1998, Laboratoire de Topologie, Universite` de Bourgogne, Dijon, France.

S. Jacquet, Subanalyticity of the Sub-Riemannian Distance, Journal of Dynamical and Control Systems, 5 (1999), 3: 303--328.

S. Jacquet, Regularity of sub-Riemannian distance and cut locus, Universita degli Studi di Firenze, Dip. di Matematica Appl. "G. Sansone", Preprint No. 35, May 1999.

I. Zelenko, M. Zhitomirskii, Rigid paths of generic 2-distribution on 3-manifolds, Duke Math. Journal, 79 (1995), No. 2, 281-307.

I. Zelenko, Nonregular abnormal extremals of 2-distribution: existence, second variation, and rigidity, Journal of Dynamical and Control Systems, Vol.5, No. 3, 1999, pp. 347-383.



Control of finite dimensional Shrodinger equations

C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(N), Journal of Mathematical Physics,43(5):2051-2062 2002

C. Altafini. On the generation of sequential unitary gates from continuous time Schrodinger equations driven by external fields, Quantum Information Processing, 1(3):207-224, 2002

C. Altafini, Parameter differentiation and quantum state decomposition for time varying Schrodinger equations, Preprint, 2002

C. Altafini. Controllability properties for finite dimensional quantum Markovian master equations, Journal of Mathematical Physics 44(6):2357-2372, 2003

C. Altafini. Tensor of coherences parameterization of multiqubit density operators for entanglement characterization, Physical Review A, 2003.

Ugo Boscain, Gregoire Charlot, Jean-Paul Gauthier, Stephane Guerin, Hans--Rudolf Jauslin, Optimal Control in laser-induced population transfer for two- and three-level quantum systems. Journal of Mathematical Physics, 2002.



Engineering applications of control theory; control of mechanical systems; motion planning.


A. Marigo, A. Bicchi, Rolling bodies with regular surface: Controllability Theory and Applications, Preprint.

A. Marigo, A. Bicchi, Rolling bodies with regular surface: the holonomic case , Differential Geometry and Control, Guillermo Ferreyra, Robert Gardner, Henry Hermes and Hector Sussmann (eds.), Proceedings of Symposia in Pure Mathematics, American Mathematical Society Publ., 1998.

M. Ceccarelli, A. Marigo , S. Piccinocchi, and A. Bicchi, Planning Motions of Polyhedral Parts by Rolling, submitted.

A. Bicchi, A. Marigo, and D. Prattichizzo, Robotic Dexterity via nonholonomy , Workshop on Control Problems in Robotics and Automation, at CDC'97, published by Springer Verlag .

A. Marigo, Y. Chitour, A. Bicchi, Manipulation of Polyhedral parts by rollingc, IEEE Int. Conf. on Robotics and Automation, 1997 (subm.).

A. Bicchi, Y. Chitour, A. Marigo, and D. Prattichizzo, Dexterity through Rolling: Towards Manipulation of Unknown Objects, Third Int. Symp. on Methods and Models for Automation and Robotics, MMAR'96, Miedzyzdroje, Poland, 1996.

Y. Chitour, A. Marigo, D. Prattichizzo, and A. Bicchi, Rolling Polyhedra on a Plane: Analysis of the Reachable Set, 2nd Work. Algorithmic Foundations of Robotics, WAFR'96, 1996.

Y. Chitour, A. Marigo, D. Prattichizzo, and A. Bicchi, Reachability of Rolling Parts, in Advances in Robotics: The ERNET Perspective - C. Bonivento, C. Melchiorri, and H. Tolle, eds., World Scientific Publisher Corporation, 1996.

A.A. Agrachev, A. Sarychev, The control of rotation for asymmetric rigid body. Problems of Control and Information Theory, 1983, v.12.

A.A. Agrachev, A. Sarychev, On reductions of smooth systems, linear in the control. Matem. sbornik, 1986, v.130.

A. Agrachev, Yu. Sachkov, An Intrinsic Approach to the Control of Rolling Bodies , Proceedings of the 38-th IEEE Conference on Decision and Control, Phoenix, Arizona, USA, December 7--10, 1999, vol. 1, 431--435.

C. Altafini. Geometric motion control for a kinematically redundant robotic chain: application to a holonomic mobile manipulator, Journal of Robotic Systems, 20(5):211-227, 2003.

C. Altafini. Redundant robotic chains on Riemannian submersions , IEEE Transactions on Robotics and Automation, to appear 2003.



Impulsive systems.


A. Bressan, F. Rampazzo, On differential systems with vector-valued impulsive controls, Bull. Un. Mat. Ital. (7) 2-B, (1988), 641-656 .

A. Bressan, F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory & Appl. 71 (1991), 67-84.

A. Bressan, F. Rampazzo, On systems with quadratic impulses and their application to Lagrangian mechanics, SIAM J. Control, 31 (1993), 1205-1220.

A. Bressan, F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory & Appl. 81 (1994), 435-457.

A. Bressan, Impulsive control systems, in ``Nonsmooth analysis and Geometric methods in Deterministic Optimal Control", B.S.Mordukhovich and H.J.Sussmann Eds., Springer-Verlag, New York (1995), 1-22.


Discrete-time, hybrid, and switching systems.


A.A. Agrachev, D. Liberzon, Lie-algebraic conditions for exponential stability of switched systems. Proc. 38th IEEE Confer. on Decision and Control, 1999.

C. Altafini. The reachable set of a linear endogenous switching system. Systems and Control Letters , 47(4):343-353, 2002.


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