Control-Affine Systems in Low Dimensions: From Small-Time Reachable Sets to Time-Optimal Syntheses

[1]  Philip L. Freeman,et al.  Minimum Jerk Trajectory Planning for Trajectory Constrained Redundant Robots , 2012 .

[2]  Andrei V. Dmitruk,et al.  Quadratic order conditions for bang-singular extremals , 2011, 1107.0161.

[3]  Laura Poggiolini,et al.  Strong Local Optimality for a Bang-Bang Trajectory in a Mayer Problem , 2011, SIAM J. Control. Optim..

[4]  Heinz Schättler,et al.  On classical envelopes in optimal control theory , 2010, 49th IEEE Conference on Decision and Control (CDC).

[5]  Hans Josef Pesch,et al.  The Maximum Principle of optimal control: A history of ingenious ideas and missed opportunities , 2009 .

[6]  Urszula Ledzewicz,et al.  On the Optimality of Singular Controls for a Class of Mathematical Models for Tumor Anti-Angiogenesis , 2009 .

[7]  R. Gamkrelidze Hamiltonian form of the maximum principle , 2009 .

[8]  Maria do Rosário de Pinho,et al.  The nonsmooth maximum principle , 2009 .

[9]  Urszula Ledzewicz,et al.  No . 4 B Singular controls and chattering arcs in optimal control problems arising in biomedicine , 2010 .

[10]  Gianna Stefani,et al.  Control and Cybernetics Optimality and Stability Result for Bang–bang Optimal Controls with Simple and Double Switch Behaviour * † , 2022 .

[11]  Tyrone E. Duncan,et al.  Parameter continuity of the ergodic cost for a growth optimal portfolio with proportional transaction costs , 2008, 2008 47th IEEE Conference on Decision and Control.

[12]  Nicolas Bourbaki,et al.  Lie Groups and Lie Algebras: Chapters 7-9 (Elements of Mathematics) , 2008 .

[13]  Urszula Ledzewicz,et al.  Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. , 2008, Journal of theoretical biology.

[14]  Emmanuel Trélat,et al.  Singular Trajectories of Control-Affine Systems , 2006, SIAM J. Control. Optim..

[15]  Gianna Stefani,et al.  Sufficient optimality conditions for a bang--singular extremal in the minimum time problem , 2008 .

[16]  Andrei Dmitruk,et al.  Jacobi Type Conditions for Singular Extremals , 2008 .

[17]  Héctor J. Sussmann,et al.  Set separation, approximating multicones, and the Lipschitz maximum principle , 2007 .

[18]  Urszula Ledzewicz,et al.  AntiAngiogenic Therapy in Cancer Treatment as an Optimal Control Problem , 2007, SIAM J. Control. Optim..

[19]  John T. Workman,et al.  Optimal Control Applied to Biological Models , 2007 .

[20]  Urszula Ledzewicz,et al.  Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy. , 2007, Mathematical biosciences.

[21]  G. Stefani,et al.  Sufficient optimality conditions for a bang-bang trajectory , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[22]  H. Schättler Local Fields of Extremals for Optimal Control Problems with State Constraints of Relative Degree 1 , 2006 .

[23]  E. Trélat,et al.  Genericity results for singular curves , 2006 .

[24]  B. Mordukhovich Variational Analysis and Generalized Differentiation II: Applications , 2006 .

[25]  M. Chyba,et al.  Autonomous Underwater Vehicles: Singular Extremals and Chattering , 2005, Systems, Control, Modeling and Optimization.

[26]  Urszula Ledzewicz,et al.  Analysis of a Mathematical Model for Tumor Anti-Angiogenesis , 2006 .

[27]  H. Schättler,et al.  A Synthesis of Optimal Controls for a Model of Tumor Growth under Angiogenic Inhibitors , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[28]  U. Ledzewicz,et al.  A model for cancer chemotherapy with state-space constraints , 2005 .

[29]  Urszula Ledzewicz,et al.  Drug resistance in cancer chemotherapy as an optimal control problem , 2005 .

[30]  H. Maurer,et al.  Optimization methods for the verification of second order sufficient conditions for bang–bang controls , 2005 .

[31]  Mario Sigalotti,et al.  Regularity properties of optimal trajectories of single-input control systems in dimension three , 2005 .

[32]  M. Chyba,et al.  Singular Trajectories and Their Role in Control Theory , 2003, IEEE Transactions on Automatic Control.

[33]  H. Sussmann,et al.  Underwater vehicles: the minimum time problem , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[34]  Alberto Gandolfi,et al.  Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999). , 2004, Mathematical biosciences.

[35]  N. Osmolovskii,et al.  Quadratic Extremality Conditions for Broken Extremals in the General Problem of the Calculus of Variations , 2004 .

[36]  Urszula Ledzewicz,et al.  Stratifiable Families of Extremals and Sufficient Conditions for Optimality in Optimal Control Problems , 2004 .

[37]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[38]  B. Piccoli,et al.  Optimal Syntheses for Control Systems on 2-D Manifolds , 2004 .

[39]  Helmut Maurer,et al.  Second Order Sufficient Conditions for Time-Optimal Bang-Bang Control , 2003, SIAM J. Control. Optim..

[40]  Aurelian Cernea,et al.  The connection between the maximum principle and the value function for optimal control problems under state constraints , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[41]  H. Sussmann,et al.  Uniqueness results for the value function via direct trajectory-construction methods , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[42]  Frédéric Jean,et al.  Propriétés génériques des trajectoires singulières , 2003 .

[43]  Emmanuel Trélat,et al.  Optimal Control with State Constraints and the Space Shuttle Re-entry Problem , 2003 .

[44]  Andrei A. Agrachev,et al.  On the Local Structure of Optimal Trajectories in R[sup 3] , 2003, SIAM J. Control. Optim..

[45]  Ursula Felgenhauer,et al.  On Stability of Bang-Bang Type Controls , 2002, SIAM J. Control. Optim..

[46]  Alexey Tret'yakov,et al.  Optimality Conditions for Degenerate Extremum Problems with Equality Constraints , 2003, SIAM J. Control. Optim..

[47]  Rosa-Maria Bianchini,et al.  Needle Variations that Cannot be Summed , 2003, SIAM J. Control. Optim..

[48]  Helmut Maurer,et al.  Second order optimality conditions for bang-bang control problems , 2003 .

[49]  Urszula Ledzewicz,et al.  ANALYSIS OF A CELL-CYCLE SPECIFIC MODEL FOR CANCER CHEMOTHERAPY , 2002 .

[50]  Urszula Ledzewicz,et al.  Optimal Bang-Bang Controls for a Two-Compartment Model in Cancer Chemotherapy , 2002 .

[51]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[52]  H. Schättler,et al.  Sufficient conditions for relative minima of broken extremals in optimal control theory , 2002 .

[53]  Gianna Stefani,et al.  Strong Optimality for a Bang-Bang Trajectory , 2002, SIAM J. Control. Optim..

[54]  Helmut Maurer,et al.  Second Order Sufficient Conditions for Optimal Control Problems with Free Final Time: The Riccati Approach , 2002, SIAM J. Control. Optim..

[55]  R. Vinter,et al.  Existence of Neighboring Feasible Trajectories: Applications to Dynamic Programming for State-Constrained Optimal Control Problems , 2000, CDC 2000.

[56]  Héctor J. Sussmann,et al.  Regular Synthesis and Sufficiency Conditions for Optimality , 2000, SIAM J. Control. Optim..

[57]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[58]  Urszula Ledzewicz,et al.  A High-Order Generalized Local Maximum Principle , 2000, SIAM J. Control. Optim..

[59]  Wolfgang Kliemann,et al.  The dynamics of control , 2000 .

[60]  P. Hahnfeldt,et al.  Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. , 1999, Cancer research.

[61]  Heinz Schättler,et al.  Parametrized Families of Extremals and Singularities in Solutions to the Hamilton--Jacobi--Bellman Equation , 1999 .

[62]  Urszula Ledzewicz,et al.  High-Order Approximations and Generalized Necessary Conditions for Optimality , 1999 .

[63]  John N. McDonald,et al.  A course in real analysis , 1999 .

[64]  H. Schättler,et al.  A high-order generalization of the Lyusternik theorem , 1998 .

[65]  M. Zelikin,et al.  The structure of optimal synthesis in a neighbourhood of singular manifolds for problems that are affine in control , 1998 .

[66]  A. A. Mili︠u︡tin,et al.  Calculus of variations and optimal control , 1998 .

[67]  B. Piccoli,et al.  A Generic Classification of Time-Optimal Planar Stabilizing Feedbacks , 1998 .

[68]  B. Jakubczyk,et al.  Geometry of feedback and optimal control , 1998 .

[69]  H. Schättler,et al.  High-Order Extended Maximum Principles for Optimal Control Problems with Non-Regular Constraints , 1998 .

[70]  Eduardo D. Sontag,et al.  Mathematical Control Theory Second Edition , 1998 .

[71]  Aram V. Arutyunov,et al.  Second-order conditions in extremal problems. The abnormal points , 1998 .

[72]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[73]  R. Vinter,et al.  Necessary Conditions for Optimal Control Problems Involving Nonlinear Differential Algebraic Equations , 1997 .

[74]  H. Schättler,et al.  An extended maximum principle , 1997 .

[75]  Andrei V. Sarychev,et al.  First- and Second-Order Sufficient Optimality Conditions for Bang-Bang Controls , 1997 .

[76]  Bernard Bonnard,et al.  Generic properties of singular trajectories , 1997 .

[77]  C. Shin TIME-OPTIMAL BANG-BANG TRAJECTORIES USING BIFURCATION RESULT , 1997 .

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

[79]  V. Jurdjevic Geometric control theory , 1996 .

[80]  B. Piccoli Classification of Generic Singularities for the Planar Time-Optimal Synthesis , 1996 .

[81]  Andrei A. Agrachev,et al.  Abnormal sub-riemannian geodesics : Morse index and rigidity , 1996 .

[82]  Gianna Stefani,et al.  Optimality Conditions for a Constrained Control Problem , 1996 .

[83]  A Swierniak,et al.  Optimal control problems arising in cell‐cycle‐specific cancer chemotherapy , 1996, Cell proliferation.

[84]  Heinz Schättler,et al.  Small-Time Reachable Sets and Time-Optimal Feedback Control , 1996 .

[85]  H. Sussmann,et al.  Nonsmooth analysis and geometric methods in deterministic optimal control , 1996 .

[86]  H. Frankowska,et al.  CONJUGATE POINTS AND SHOCKS IN NONLINEAR OPTIMAL CONTROL , 1996 .

[87]  B. Bonnard,et al.  Toward a Geometric Theory in the Time-Minimal Control of Chemical Batch Reactors , 1995 .

[88]  U. Ledzewicz,et al.  Second-order conditions for extremum problems with nonregular equality constraints , 1995 .

[89]  W. Grantham,et al.  Neighbouring extremals for nonlinear systems with control constraints , 1995 .

[90]  Kevin A. Grasse,et al.  Reachability of Interior States by Piecewise Constant Controls , 1995 .

[91]  M. I. Zelikin,et al.  Theory of Chattering Control: with applications to Astronautics, Robotics, Economics, and Engineering , 1994 .

[92]  Vladimir Borisov,et al.  Theory of Chattering Control , 1994 .

[93]  K. Teo,et al.  Optimal Control of Drug Administration in Cancer Chemotherapy , 1993 .

[94]  Bernard Bonnard,et al.  Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal , 1993 .

[95]  Heinz Schättler,et al.  A synthesis of time-optimal controls in the presence of saturated singular arcs , 1993 .

[96]  F. Jones Lebesgue Integration on Euclidean Space , 1993 .

[97]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[98]  H. Frankowska,et al.  Unicité des solutions optimales et absence de chocs pour les équations d'Hamilton-Jacobi-Bellman et de Riccati , 1992 .

[99]  Andrej V. Sarychev,et al.  Morse index and sufficient optimality conditions for bang-bang pontryagin extremals , 1992 .

[100]  Heinz Schättler,et al.  A local feedback synthesis of time-optimal stabilizing controls in dimension three , 1991, Math. Control. Signals Syst..

[101]  Jean-Paul Gauthier,et al.  Analysis of Controlled Dynamical Systems , 1991 .

[102]  M. Zelikin,et al.  Optimal Synthesis Containing Chattering Arcs and Singular Arcs of the Second Order , 1991 .

[103]  Heinz Schättler,et al.  Extremal Trajectories, Small-time Reachable Sets and Local Feedback Synthesis: a Synopsis of the Three-dimensional Case , 1991 .

[104]  G. W. Swan Role of optimal control theory in cancer chemotherapy. , 1990, Mathematical biosciences.

[105]  Sussmann Nonlinear Controllability and Optimal Control , 1990 .

[106]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[107]  Héctor J. Sussmann,et al.  Envelopes, high-order optimality conditions and Lie brackets , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[108]  Heinz Schättler,et al.  Conjugate points and intersections of bang-bang trajectories , 1989 .

[109]  E. P. Avakov Necessary extremum conditions for smooth anormal problems with equality- and inequality-type constraints , 1989 .

[110]  Joel W. Burdick,et al.  On the inverse kinematics of redundant manipulators: characterization of the self-motion manifolds , 1989, Proceedings, 1989 International Conference on Robotics and Automation.

[111]  Heinz Schättler,et al.  The structure of small-time reachable sets in low dimensions , 1989 .

[112]  Halina Frankowska,et al.  Contingent cones to reachable sets of control systems , 1989 .

[113]  Heinz Schättler,et al.  The local structure of time-optimal trajectories in dimension three under generic conditions , 1988 .

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

[115]  A. Nowakowski Field theories in the modern calculus of variations , 1988 .

[116]  H. Schättler On the local structure of time-optimal bang-bang trajectories in R 3 , 1988 .

[117]  H. Frankowska An open mapping principle for set-valued maps , 1987 .

[118]  Héctor J. Sussmann,et al.  Regular synthesis for time-optimal control of single-input real analytic systems in the plane , 1987 .

[119]  A. Fordy APPLICATIONS OF LIE GROUPS TO DIFFERENTIAL EQUATIONS (Graduate Texts in Mathematics) , 1987 .

[120]  Héctor J. Sussmann,et al.  The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case , 1987 .

[121]  A. Dmitruk,et al.  QUADRATIC CONDITIONS FOR A PONTRYAGIN MINIMUM IN AN OPTIMUM CONTROL PROBLEM LINEAR IN THE CONTROL. I: A DECODING THEOREM , 1987 .

[122]  H. Sussmann The Structure of Time-Optimal Trajectories for Single-Input Systems in the Plane: the C , 1987 .

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

[124]  I. Kupka,et al.  GEOMETRIC THEORY OF EXTREMALS IN OPTIMAL CONTROL PROBLEMS: I THE FOLD AND MAXWELL CASE , 1987 .

[125]  E. R. Avakov,et al.  Extremum conditions for smooth problems with equality-type constraints , 1986 .

[126]  Alberto Bressan,et al.  The generic local time-optimal stabilizing controls in dimension 3 , 1986 .

[127]  Héctor J. Sussmann,et al.  Envelopes, Conjugate Points, and Optimal Bang-Bang Extremals , 1986 .

[128]  P. Olver Applications of Lie Groups to Differential Equations , 1986 .

[129]  Bernard Bonnard,et al.  On Singular Extremals in the Time Minimal Control Problem in $\mathbb{R}^3 $ , 1985 .

[130]  Héctor J. Sussmann,et al.  Lie brackets and real analyticity in control theory , 1985 .

[131]  M. Golubitsky,et al.  The Recognition Problem , 1985 .

[132]  Stanislaw Walczak,et al.  Some properties of cones in normed spaces and their application to investigating extremal problems , 1984 .

[133]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[134]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[135]  F. Clarke Optimization And Nonsmooth Analysis , 1983 .

[136]  H. Nijmeijer,et al.  Controlled invariance for nonlinear systems: two worked examples : (preprint) , 1984 .

[137]  A V Saryčev,et al.  THE INDEX OF THE SECOND VARIATION OF A CONTROL SYSTEM , 1982 .

[138]  Greg Knowles,et al.  An Introduction to Applied Optimal Control , 1982 .

[139]  W. Greub Linear Algebra , 1981 .

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

[141]  Pavol Brunovský,et al.  Regular synthesis for the linear-quadratic optimal control problem with linear control constraints , 1980 .

[142]  R. Bishop,et al.  Tensor Analysis on Manifolds , 1980 .

[143]  R. Lewis,et al.  Definitions of Order and Junction Conditions in Singular Optimal Control Problems , 1980 .

[144]  Thomas Kailath,et al.  Linear Systems , 1980 .

[145]  Hector Sussmann,et al.  A bang-bang theorem with bounds on the number of switchings , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

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

[147]  H. Sussmann Subanalytic sets and feedback control , 1979 .

[148]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[149]  D. Rebhuhn On the Stability of the Existence of Singular Controls under Perturbation of the Control System , 1978 .

[150]  B.D.O. Anderson,et al.  Singular optimal control problems , 1975, Proceedings of the IEEE.

[151]  A. Zygmund,et al.  Measure and integral : an introduction to real analysis , 1977 .

[152]  A. Krener The High Order Maximal Principle and Its Application to Singular Extremals , 1977 .

[153]  F. Clarke The Maximum Principle under Minimal Hypotheses , 1976 .

[154]  W. Fleming,et al.  Deterministic and Stochastic Optimal Control , 1975 .

[155]  H. Maurer On Optimal Control Problems with Bounded State Variables and Control Appearing Linearly , 1975, Optimization Techniques.

[156]  Helmut Maurer,et al.  An Example of a Continuous Junction for a Singular Control Problem of Even Order , 1975 .

[157]  W. Boothby An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[158]  L. Berkovitz Optimal Control Theory , 1974 .

[159]  J. Marsden,et al.  Elementary classical analysis , 1974 .

[160]  H. Gardner Moyer,et al.  Sufficient Conditions for a Strong Minimum in Singular Control Problems , 1973 .

[161]  C. Marchal,et al.  Chattering arcs and chattering controls , 1973 .

[162]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[163]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[164]  R. Brockett System Theory on Group Manifolds and Coset Spaces , 1972 .

[165]  F. M. Kirillova,et al.  High Order Necessary Conditions for Optimality , 1972 .

[166]  B. T. Poljak,et al.  Lectures on mathematical theory of extremum problems , 1972 .

[167]  G. Leitmann,et al.  Mathematical Methods of Optimal Control , 1971 .

[168]  F. W. Warner Foundations of Differentiable Manifolds and Lie Groups , 1971 .

[169]  L. Young,et al.  Lectures on the Calculus of Variations and Optimal Control Theory. , 1971 .

[170]  D. Jacobson,et al.  Necessary and sufficient conditions for optimality for singular control problems - A limit approach , 1971 .

[171]  J. P. Mcdanell,et al.  Necessary Conditions Joining Optimal Singular and Nonsingular Subarcs , 1971 .

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

[173]  D. H. Jacobson,et al.  A New Necessary Condition of Optimality for Singular Control Problems , 1969 .

[174]  J. P. Lasalle,et al.  functional analysis and time Optimal Control , 1969 .

[175]  Harry Pollard,et al.  A non-classical variational problem arising from an optimal filter problem , 1967 .

[176]  Richard Bellman,et al.  Introduction to the mathematical theory of control processes , 1967 .

[177]  H. Gardner Moyer,et al.  3 Singular Extremals , 1967 .

[178]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[179]  M. Hestenes Calculus of variations and optimal control theory , 1966 .

[180]  B. Goh Necessary Conditions for Singular Extremals Involving Multiple Control Variables , 1966 .

[181]  V. Boltyanskii Sufficient Conditions for Optimality and the Justification of the Dynamic Programming Method , 1966 .

[182]  R. Bellman Dynamic Programming , 1957, Science.

[183]  Michael Athans,et al.  Optimal Control , 1966 .

[184]  J. Milnor Topology from the differentiable viewpoint , 1965 .

[185]  H. Kelley A second variation test for singular extremals , 1964 .

[186]  W. M. Wonham,et al.  NOTE ON A PROBLEM IN OPTIMAL NONLINEAR CONTROL , 1963 .

[187]  A. Fuller Study of an Optimum Non-linear Control System† , 1963 .

[188]  A. Bryson,et al.  Optimization and Control of Nonlinear Systems Using the Second Variation , 1963 .

[189]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[190]  H. Whitney Elementary Structure of Real Algebraic Varieties , 1957 .

[191]  T. Broadbent Measure and Integral , 1957, Nature.

[192]  R. Thom Les singularites des applications differentiables , 1956 .

[193]  F. John Partial differential equations , 1967 .

[194]  G. Bliss Lectures on the calculus of variations , 1946 .

[195]  A. Kneser,et al.  Lehrbuch der Variationsrechnung , 1925 .