Chronological Calculus in Systems and Control Theory

[1]  A. S. Dzhumadil’daev,et al.  Zinbiel algebras under q-commutators , 2007 .

[2]  Nikolay Kirov,et al.  Volterra Series and Numerical Approximations of ODEs , 2004, NAA.

[3]  Roger W. Brockett.  Differential geometric methods in system theory , 1971, CDC 1971.

[4]  A. I. Tretyak,et al.  Chronological Calculus, High-Order Necessary Conditions for Optimality, and Perturbation Methods , 1998 .

[5]  Gianna Stefani.  Polynomial approximations to control systems and local controllability , 1985, 1985 24th IEEE Conference on Decision and Control.

[6]  S. A. Vakhrameev.  A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems , 1997 .

[7]  S. Sastry,et al.  Nonholonomic motion planning: steering using sinusoids , 1993, IEEE Trans. Autom. Control..

[8]  Joel W. Burdick,et al.  Control of biomimetic locomotion via averaging theory , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[9]  A. N. Kolmogorov,et al.  IZRAIL' MOISEEVICH GELFAND (on his fiftieth birthday) , 1964 .

[10]  Peter A. Cook,et al.  Nonlinear dynamical systems , 1986 .

[11]  A. Krener,et al.  Nonlinear controllability and observability , 1977 .

[12]  Yuan Wang,et al.  Non-causal Fliess operators and their shuffle algebra , 2008, Int. J. Control.

[13]  N. Bourbaki.  Lie groups and Lie algebras , 1998 .

[14]  Robert L. Grossman,et al.  Hopf-algebraic structure of combinatorial objects and differential operators , 1990 .

[15]  J. Canny,et al.  Nonholonomic Motion Planning , 1992 .

[16]  Francesco Bullo,et al.  Series Expansions for the Evolution of Mechanical Control Systems , 2001, SIAM J. Control. Optim..

[17]  A. Krener,et al.  The existence and uniqueness of volterra series for nonlinear systems , 1977 .

[18]  C. Byrnes,et al.  Theory and Applications of Nonlinear Control Systems , 1986 .

[19]  Gerardo Lafferriere,et al.  A Differential Geometric Approach to Motion Planning , 1993 .

[20]  Sonia Martínez,et al.  Analysis and design of oscillatory control systems , 2003, IEEE Trans. Autom. Control..

[21]  Kurusch Ebrahimi-Fard,et al.  NEW IDENTITIES IN DENDRIFORM ALGEBRAS , 2007, 0705.2636.

[22]  Marcel P. Schützenberger,et al.  Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées , 1959 .

[23]  H. Sussmann,et al.  Control systems on Lie groups , 1972 .

[24]  M. Sigalotti.  Local Regularity of Optimal Trajectories for Control Problems with General Boundary Conditions , 2005 .

[25]  Bronisław Jakubczyk,et al.  Local realizations of nonlinear causal operators , 1986 .

[26]  H. Sussmann.  Lie Brackets and Local Controllability: A Sufficient Condition for Scalar-Input Systems , 1983 .

[27]  Hans Z. Munthe-Kaas,et al.  Foundations of Computational Mathematics on the Hopf Algebraic Structure of Lie Group Integrators , 2022 .

[28]  Andrey Sarychev,et al.  Stability criteria for time-periodic systems via high-order averaging techniques , 2001 .

[29]  C. Reutenauer.  Free Lie Algebras , 1993 .

[30]  Roger W. Brockett.  Volterra series and geometric control theory , 1976, Autom..

[31]  Richard G. Larson,et al.  Hopf-algebraic structure of families of trees , 1989 .

[32]  H. Sussmann.  An extension of a theorem of Nagano on transitive Lie algebras , 1974 .

[33]  E. Rocha.  On computation of the logarithm of the Chen-Fliess series for nonlinear systems , 2003 .

[34]  H. Sussmann.  New Differential Geometric Methods in Nonholonomic Path Finding , 1992 .

[35]  Francesco Bullo,et al.  Averaging and Vibrational Control of Mechanical Systems , 2002, SIAM J. Control. Optim..

[36]  R. Ree,et al.  Lie Elements and an Algebra Associated With Shuffles , 1958 .

[37]  Andrei A. Agrachev,et al.  Local invariants of smooth control systems , 1989 .

[38]  M. Fliess,et al.  Fonctionnelles causales non linaires et indtermines non commutatives , 1981 .

[39]  A. D. Lewis,et al.  Geometric control of mechanical systems : modeling, analysis, and design for simple mechanical control systems , 2005 .

[40]  Matthias Kawski,et al.  Chronological Algebras: Combinatorics and Control , 2001 .

[41]  On variational approach to differential invariants of rank two distributions , 2004, math/0402171.

[42]  H. Hermes,et al.  Nonlinear Controllability via Lie Theory , 1970 .

[43]  Joel W. Burdick,et al.  Trajectory stabilization for a planar carangiform robot fish , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[44]  Li Guo,et al.  Rota–Baxter algebras and dendriform algebras , 2008 .

[45]  P. Crouch.  Dynamical Realizations of Finite Volterra Series , 1981 .

[46]  P. Crouch,et al.  Numerical integration of ordinary differential equations on manifolds , 1993 .

[47]  M. Kawski.  Control variations with an increasing number of switchings , 1988 .

[48]  I. M. Gelfand,et al.  Commutative Normed Rings. , 1967 .

[49]  Matthias Kawski,et al.  The Combinatorics of Nonlinear Controllability and Noncommuting Flows , 2008 .

[50]  Gerardo Lafferriere,et al.  Motion planning for controllable systems without drift , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[51]  J.-L. Loday.  Une version non commutative des algèbres de Lie : les algèbres de Leibniz , 1993 .

[52]  A. Iserles,et al.  Explicit Magnus expansions for nonlinear equations , 2006 .

[53]  Jorge Cortes,et al.  Motion Control Algorithms for Simple Mechanical Systems with Symmetry , 2003 .

[54]  Antonella Zanna,et al.  Iterated commutators, Lie's reduction method and ordinary differential equations on matrix Lie groups , 1997 .

[55]  Kurusch Ebrahimi-Fard,et al.  A Magnus- and Fer-Type Formula in Dendriform Algebras , 2007, Found. Comput. Math..

[56]  H. Sussmann,et al.  Controllability of nonlinear systems , 1972 .

[57]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[58]  Héctor J. Sussmann,et al.  Noncommutative Power Series and Formal Lie-algebraic Techniques in Nonlinear Control Theory , 1997 .

[59]  Andrei A. Agrachev,et al.  Chronological algebras and nonstationary vector fields , 1981 .

[60]  Chronological series and the Cauchy-Kowalevski theorem , 1983 .

[61]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[62]  Marshall Hall,et al.  A basis for free Lie rings and higher commutators in free groups , 1950 .

[63]  Eduardo Sontag,et al.  Generating series and nonlinear systems: Analytic aspects, local realizability, and i/o representations , 1992 .

[64]  Andrei A. Agrachev,et al.  Local controllability and semigroups of diffeomorphisms , 1993 .

[65]  G. Viennot.  Algèbres de Lie Libres et Monoïdes Libres , 1978 .

[66]  M. Krastanov,et al.  Higher Order Approximations of Affinely Controlled Nonlinear Systems , 2003, LSSC.

[67]  J. Neumann.  Mathematische grundlagen der Quantenmechanik , 1935 .

[68]  H. Sussmann.  A general theorem on local controllability , 1987 .

[69]  Andrey V. Sarychev.  Lie- and chronologico-algebraic tools for studying stability of time-varying systems , 2001, Syst. Control. Lett..

[70]  M. Kapranov,et al.  Koszul duality for Operads , 1994, 0709.1228.

[71]  A. Agrachev,et al.  Ordinary differential equations on vector bundles and chronological calculus , 1991 .

[72]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[73]  C. Lobry.  Contr^olabilite des systemes non lineaires , 1970 .

[74]  Salvatore Monaco,et al.  From Chronological Calculus to Exponential Representations of Continuous and Discrete-Time Dynamics: A Lie-Algebraic Approach , 2007, IEEE Transactions on Automatic Control.

[75]  P. E. Crouch,et al.  Algebraic and multiple integral identities , 1989 .

[76]  Sonia Martínez,et al.  On nonlinear controllability and series expansions for Lagrangian systems with dissipative forces , 2002, IEEE Trans. Autom. Control..

[77]  R. Hermann.  On the Accessibility Problem in Control Theory , 1963 .

[78]  R. Gamkrelidze,et al.  THE EXPONENTIAL REPRESENTATION OF FLOWS AND THE CHRONOLOGICAL CALCULUS , 1979 .

[79]  G. Jacob.  MOTION PLANNING BY PIECEWISE CONSTANT OR POLYNOMIAL INPUTS , 1992 .

[80]  A. I. Tretiyak.  Sufficient conditions for local controllability and high-order necessary conditions for optimality. A differential-geometric approach , 1997 .

[81]  Ulysse Serres.  On the curvature of two-dimensional optimal control systems and Zermelo’s navigation problem , 2006 .

[82]  A. Dzhumadil'daev,et al.  Nilpotency of Zinbiel Algebras , 2005 .

[83]  Robert L. Grossman,et al.  The realization of input-output maps using bialgebras , 2020, 2007.09526.

[84]  Kuo-Tsai Chen,et al.  Integration of Paths, Geometric Invariants and a Generalized Baker- Hausdorff Formula , 1957 .

[85]  A. Tretiak.  Higher-order local approximations of smooth control systems and pointwise higher-order optimality conditions , 1998 .

[86]  Guy Melançon,et al.  Lyndon Words, Free Algebras and Shuffles , 1989, Canadian Journal of Mathematics.

[87]  M. Grayson,et al.  Models for Free Nilpotent Lie Algebras , 1990 .

[88]  P. Michor,et al.  The Convenient Setting of Global Analysis , 1997 .

[89]  Ander Murua,et al.  The Hopf Algebra of Rooted Trees, Free Lie Algebras, and Lie Series , 2006, Found. Comput. Math..

[90]  On stability and stabilization of elastic systems by time-variant feedback , 2005, math/0507123.

[91]  Andrey Sarychev,et al.  Navier–Stokes Equations: Controllability by Means of Low Modes Forcing , 2005 .

[92]  H. Munthe-Kaas,et al.  Computations in a free Lie algebra , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[93]  I. Zelenko.  Variational Approach to Differential Invariants of Rank 2 Vector Distributions , 2004 .

[94]  J. Loday,et al.  Leibniz Representations of Lie Algebras , 1996 .