The Calculus of Variations: A Historical Perspective

  • Heinz Schättler
  • Urszula Ledzewicz
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 38)


We begin with an introduction to the historical origin of optimal control theory, the calculus of variations. But it is not our intention to give a comprehensive treatment of this topic. Rather, we introduce the fundamental necessary and sufficient conditions for optimality by fully analyzing two of the cornerstone problems of the theory, the brachistochrone problem and the problem of determining surfaces of revolution with minimum surface area, so-called minimal surfaces. Our emphasis is on illustrating the methods and techniques required for getting complete solutions for these problems. More generally, we use the so-called fixed-endpoint problem, the problem of minimizing a functional over all differentiable curves that satisfy given boundary conditions, as a vehicle to introduce the classical results of the theory: (a) the Euler–Lagrange equation as the fundamental first-order necessary condition for optimality, (b) the Legendre and Jacobi conditions, both in the form of necessary and sufficient second-order conditions for local optimality, (c) the Weierstrass condition as additional necessary condition for optimality for so-called strong minima, and (d) its connection with field theory, the fundamental idea in any sufficiency theory. Throughout our presentation, we emphasize geometric constructions and a geometric interpretation of the conditions. For example, we present the connections between envelopes and conjugate points of a fold type and use these arguments to give a full solution for the minimum surfaces of revolution.


Optimal Control Problem Minimum Surface Lagrange Equation Jacobi Equation Optimal Control Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Differential Geometric Control Theory, R. Brockett, R. Millman, H. Sussmann, eds., Progress in Mathematics, Birkhäuser, Boston, 1983.Google Scholar
  2. 2.
    Differential Geometry and Control, Proc. Symposia in Pure Mathematics, Vol. 64, G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann, eds., American Mathematical Society, 1999.Google Scholar
  3. 3.
    Fifty Years of Optimal Control, A. Ioffe. K. Malanowski, and F. Tröltzsch, eds., Control and Cybernetics, 38 (2009).Google Scholar
  4. 4.
    Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek, eds., Marcel Dekker, New York, 1998.Google Scholar
  5. 5.
    Modern Optimal Control, E.O. Roxin, ed., Marcel Dekker, New York, 1989.Google Scholar
  6. 6.
    Nonlinear Controllability and Optimal Control, H. Sussmann, ed., Marcel Dekker, New York, 1990.Google Scholar
  7. 7.
    Nonlinear Synthesis, C.I. Byrnes and A. Kurzhansky, eds., Birkhäuser, Boston, 1991.Google Scholar
  8. 8.
    Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, B.S. Mordukhovich and H. Sussmann, eds., The IMA Volumes in Mathematics and Its Applications, Springer-Verlag, New York, 1996.Google Scholar
  9. 9.
    Optimal Control of Differential Equations, N. Pavel, ed., Marcel Dekker, New York, 1994.Google Scholar
  10. 10.
    Optimal Control: Theory, Algorithms and Applications, W.W. Hager and P.M. Pardalos, Kluwer Academic Publishers, 1998.Google Scholar
  11. 11.
    A.A. Agrachev and R.V. Gamkrelidze, Exponential representation of flows and chronological calculus, Math. USSR Sbornik, 35 (1979), pp. 727–785.MATHGoogle Scholar
  12. 12.
    A.A. Agrachev and R.V. Gamkrelidze, Chronological algebras and nonstationary vector fields, J. of Soviet Mathematics, 17 (1979), pp. 1650–1672.Google Scholar
  13. 13.
    A.A. Agrachev and R.V. Gamkrelidze, Symplectic geometry for optimal control, in: Nonlinear Controllability and Optimal Control (H. Sussmann, ed.), Marcel Dekker, (1990), pp. 263–277.Google Scholar
  14. 14.
    A.A. Agrachev and R.V. Gamkrelidze, Symplectic methods for optimization and control, in: Geometry of Feedback and Optimal Control, (B. Jakubczyk and W. Respondek, eds.), Marcel Dekker, (1998), pp. 19–77.Google Scholar
  15. 15.
    A.A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, 2004.Google Scholar
  16. 16.
    A.A. Agrachev and A.V. Sarychev, On abnormal extremals for Lagrange variational problems, J. of Mathematical Systems, Estimation and Control, 5 (1995), pp. 127–130.Google Scholar
  17. 17.
    A.A. Agrachev and A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. Henri Poincaré, 13 (1996), pp. 635–690.MathSciNetMATHGoogle Scholar
  18. 18.
    A.A. Agrachev and M. Sigalotti, On the local structure of optimal trajectories in 3, SIAM J. on Control and Optimization, 42 (2003), pp. 513—-531.Google Scholar
  19. 19.
    A.A. Agrachev, G. Stefani, and P.L. Zezza, A Hamiltonian approach to strong minima in optimal control, in: Differential Geometry and Control, (G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann, eds.), American Mathematical Society, 1999, pp. 11–22.Google Scholar
  20. 20.
    A.A. Agrachev, G. Stefani, and P.L. Zezza, Strong optimality for a bang-bang trajectory, SIAM J. Control and Optimization, 41 (2002), pp. 991–1014.MathSciNetMATHGoogle Scholar
  21. 21.
    V.M. Alekseev, V.M. Tikhomirov, and S.V. Fomin, Optimal Control, Contemporary Soviet Mathematics, 1987.Google Scholar
  22. 22.
    M.S. Aronna, J.F. Bonnans, A.V. Dmitruk and P.A. Lotito, Quadratic conditions for bang-singular extremals, Numerical Algebra, Control and Optimization, (2012), to appear.Google Scholar
  23. 23.
    A.V. Arutyunov, Higher-order conditions in anormal extremal problems with constraints of equality type, Soviet Math. Dokl., 42 (1991), pp. 799–804.MathSciNetGoogle Scholar
  24. 24.
    A.V. Arutyunov, Second-order conditions in extremal problems. The abnormal points, Trans. of the American Mathematical Society, 350 (1998), pp. 4341–4365.Google Scholar
  25. 25.
    M. Athans and P. Falb, Optimal Control, McGraw Hill, 1966.Google Scholar
  26. 26.
    J.P. Aubin and H. Frankowska, Set–Valued Analysis, Birkhäuser, Boston, 1990.MATHGoogle Scholar
  27. 27.
    E.R. Avakov, Extremum conditions for smooth problems with equality–type constraints, USSR Comput. Math. and Math. Phys., 25 (1985), pp. 24–32, [translated from Zh. Vychisl. Mat. Fiz., 25 (1985)].Google Scholar
  28. 28.
    E.R. Avakov, Necessary conditions for a minimum for nonregular problems in Banach spaces. Maximum principle for abnormal problems of optimal control, Trudy Mat. Inst. AN. SSSR, 185 (1988), pp. 3–29 [in Russian].Google Scholar
  29. 29.
    E.R. Avakov, Necessary extremum conditions for smooth abnormal problems with equality and inequality-type constraints, Math. Zametki, 45 (1989), pp. 3–11.MathSciNetGoogle Scholar
  30. 30.
    M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Modern Birkhäuser Classics, Birkhäuser, 2008.MATHGoogle Scholar
  31. 31.
    D.J. Bell and D.H. Jacobson, Singular Optimal Control Problems, Academic Press, New York, 1975.MATHGoogle Scholar
  32. 32.
    R. Bellman, Dynamic Programming, Princeton University Press, 1961.Google Scholar
  33. 33.
    L.D. Berkovitz, Optimal Control Theory, Springer-Verlag, 1974.Google Scholar
  34. 34.
    L.D. Berkovitz and H. Pollard, A non-classical variational problem arising from an optimal filter problem, Arch. Rational Mech. Anal., 26 (1967), pp. 281–304.MathSciNetGoogle Scholar
  35. 35.
    L.D. Berkovitz and H. Pollard, A non-classical variational problem arising from an optimal filter problem II, Arch. Rational Mech. Anal., 38 (1971), pp. 161–172.MathSciNetGoogle Scholar
  36. 36.
    R.M. Bianchini, Good needle-like variations, Proceedings of Symposia in Pure Mathematics, Vol. 64, American Mathematical Society, (1999), pp. 91–101.Google Scholar
  37. 37.
    R.M. Bianchini and M. Kawski, Needle variations that cannot be summed, SIAM J. Control and Optimization, 42 (2003), pp. 218–238.MathSciNetMATHGoogle Scholar
  38. 38.
    R.L. Bishop and S.I. Goldberg, Tensor Analysis on Manifolds, Dover, (1980).Google Scholar
  39. 39.
    G. Bliss, Calculus of Variations, The Mathematical Association of America, 1925.Google Scholar
  40. 40.
    G. Bliss, Lectures on the Calculus of Variations, The University of Chicago Press, Chicago and London, 1946.MATHGoogle Scholar
  41. 41.
    V.G. Boltyansky, Sufficient conditions for optimality and the justification of the dynamic programming method, SIAM J. on Control, 4 (1966), pp. 326–361.Google Scholar
  42. 42.
    V.G. Boltyansky, Mathematical Methods of Optimal Control, Holt, Rinehart and Winston, Inc., (1971).Google Scholar
  43. 43.
    B. Bonnard, On singular extremals in the time minimal control problem in 3, SIAM J. Control and Optimization, 23 (1985), pp. 794–802.MathSciNetMATHGoogle Scholar
  44. 44.
    B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathématiques & Applications, Vol. 40, Springer-Verlag, Paris, 2003.Google Scholar
  45. 45.
    B. Bonnard, L. Faubourg, G. Launay, and E. Trélat, Optimal control with state space constraints and the space shuttle re–entry problem, J. of Dynamical and Control Systems, 9 (2003), pp. 155–199.MATHGoogle Scholar
  46. 46.
    B. Bonnard, J.P. Gauthier, and J. de Morant, Geometric time-optimal control for batch reactors, Part I, in: Analysis of Controlled Dynamical Systems, (B. Bonnard, B. Bride, J.P. Gauthier and I. Kupka, eds.), Birkhäuser, (1991); Part II, in: Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, United Kingdom, (1991).Google Scholar
  47. 47.
    B. Bonnard and J. de Morant, Toward a geometric theory in the time-minimal control of chemical batch reactors, SIAM J. Control and Optimization, 33 (1995), pp. 1279–1311.MathSciNetMATHGoogle Scholar
  48. 48.
    B. Bonnard and I.A.K. Kupka, Théorie des singularités de l’application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Forum Matematicum, 5 (1993), pp. 111–159.MathSciNetMATHGoogle Scholar
  49. 49.
    B. Bonnard and I.A.K. Kupka, Generic properties of singular trajectories, A. Inst. H. Poincaré, Anal. Non Linéaire, 14 (1997), pp. 167–186.Google Scholar
  50. 50.
    W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975.MATHGoogle Scholar
  51. 51.
    U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Mathématiques & Applications, Vol. 43, Springer-Verlag, Paris, 2004.Google Scholar
  52. 52.
    N. Bourbaki, Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 1–3, Springer-Verlag, 1989.Google Scholar
  53. 53.
    J.V. Breakwell, J.L. Speyer, and A.E. Bryson, Optimization and control of nonlinear systems using the second variation, SIAM J. on Control, 1 (1963), pp. 193–223MATHGoogle Scholar
  54. 54.
    A. Bressan, The generic local time-optimal stabilizing controls in dimension 3, SIAM J. Control Optim., 24 (1986), pp. 177–190.MathSciNetMATHGoogle Scholar
  55. 55.
    A. Bressan and B. Piccoli, A generic classification of time-optimal planar stabilizing feedbacks, SIAM J. Control Optim., 36 (1998), pp. 12–32.MathSciNetMATHGoogle Scholar
  56. 56.
    A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences (AIMS), 2007.Google Scholar
  57. 57.
    R.W. Brockett, Finite Dimensional Linear Systems, John Wiley and Sons, New York, 1970.MATHGoogle Scholar
  58. 58.
    R.W. Brockett, System theory on group manifolds and coset spaces, SIAM J. Control Optim., 10 (1972), pp. 265–284.MathSciNetMATHGoogle Scholar
  59. 59.
    A.B. Bruckner, J.B. Bruckner and B.S. Thomson, Real Analysis, Prentice Hall, 1997.Google Scholar
  60. 60.
    P. Brunovsky, Every normal linear system has a regular time–optimal synthesis, Math. Slovaca, 28 (1979), pp. 81–100.MathSciNetGoogle Scholar
  61. 61.
    P. Brunovsky, Regular synthesis for the linear-quadratic optimal control problem with linear control constraints, J. of Differential Equations, 38 (1980), pp. 344–360.MathSciNetMATHGoogle Scholar
  62. 62.
    P. Brunovsky, Existence of a regular synthesis for general control problems, J. of Differential Equations, 38 (1980), pp. 81–100.Google Scholar
  63. 63.
    J. Burdick, On the inverse kinematics of redundant manipulators: characterization of the self-motion manifolds, in: Proc. of the 1989 IEEE International Conference on Robotics and Automation, Vol. 1, (1989), pp. 264–270.Google Scholar
  64. 64.
    A.E. Bryson, Jr. and Y.C. Ho, Applied Optimal Control, Revised Printing, Hemisphere Publishing Company, New York, 1975.Google Scholar
  65. 65.
    C.I. Byrnes and H. Frankowska, Unicité des solutions optimales et absence de chocs pour les équations d’Hamilton–Jacobi–Bellman et de Riccati, C.R. Acad. Sci. Paris, 315, Série I (1992), pp. 427–431.Google Scholar
  66. 66.
    C.I. Byrnes and A. Jhemi, Shock waves for Riccati Partial Differential Equations Arising in Nonlinear Optimal Control, in: Systems, Models and Feedback: Theory and Applications, (A. Isidori and T.J. Tarn, eds.), Birkhäuser, (1992), pp. 211–225.Google Scholar
  67. 67.
    C. Carathéodory, Variationsrechnung und Partielle Differential Gleichungen erster Ordnung, Teubner Verlag, Leipzig, 1936; translated as Calculus of Variations and Partial Differential Equations of First Order, American Mathematical Society, 3rd edition, 1999.Google Scholar
  68. 68.
    N. Caroff and H. Frankowska, Conjugate points and shocks in nonlinear optimal control, Trans. of the American Mathematical Society, 348 (1996), pp. 3133–3153.MathSciNetMATHGoogle Scholar
  69. 69.
    A. Cernea and H. Frankowska, The connection between the maximum principle and the value function for optimal control problems under state constraints, Proc. of the 43rd IEEE Conference on Decision and Control, Nassau, The Bahamas, (2004), pp. 893–898.Google Scholar
  70. 70.
    L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983.Google Scholar
  71. 71.
    Y. Chitour, F. Jean, E. Trélat, Propriétés génériques des trajectoires singulères, C.R. Math. Acad. Sci. Paris, 337 (2003), pp. 49–52.Google Scholar
  72. 72.
    Y. Chitour, F. Jean, E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), pp. 45–73.MathSciNetMATHGoogle Scholar
  73. 73.
    Y. Chitour, F. Jean, E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control and Optimization, 47 (2008), pp. 1078–1095.MathSciNetMATHGoogle Scholar
  74. 74.
    M. Chyba and T. Haberkorn, Autonomous underwater vehicles: singular extremals and chattering, in: Systems, Control, Modeling and Optimization, (F. Cergioli et al., eds.), Springer-Verlag, (2003), pp. 103–113.Google Scholar
  75. 75.
    M. Chyba, H. Sussmann, H. Maurer, and G. Vossen, Underwater vehicles: The minimum time problem, Proc. of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, (2004), pp. 1370–1375.Google Scholar
  76. 76.
    F.H. Clarke, The maximum principle under minimal hypothesis, SIAM J. Control Optim., 14 (1976), pp. 1078–1091.MATHGoogle Scholar
  77. 77.
    F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley–Interscience, 1983.Google Scholar
  78. 78.
    F.H. Clarke and M.D.R. de Pinho, The nonsmooth maximum principle, Contol and Cybernetics, 38 (2009), pp. 1151–1168.MATHGoogle Scholar
  79. 79.
    F. Colonius and W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000.Google Scholar
  80. 80.
    M.G. Crandall and P.L. Lions, Viscosity Solutions of Hamilton–Jacobi Equations, Trans. of the American Mathematical Society, 277 (1983), pp. 1–42.MathSciNetMATHGoogle Scholar
  81. 81.
    F.H. Croom, Principles of Topology, Saunders Publishing, 1983.Google Scholar
  82. 82.
    M.d.R. de Pinho and M.M.A. Ferreira, Optimal Control Problems with Constraints, Seria Matematica Aplicata Si Industriala, Editura Universitii Din Pitesti, Romania, 2002.Google Scholar
  83. 83.
    M.d.R. de Pinho and R.B. Vinter, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations, J. Math. Anal. Appl., 212 (1997), pp. 493–516.Google Scholar
  84. 84.
    A. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems, Soviet Math. Doklady, 18 (1977).Google Scholar
  85. 85.
    A. Dmitruk, Quadratic conditions for a Pontryagin minimum in an optimal control problem, linear in the control, Mathematics of the USSR, Izvestija, 28 (1987), pp. 275–303.Google Scholar
  86. 86.
    A. Dmitruk, Jacobi type conditions for singular extremals, Control and Cybernetics, 37 (2008), pp. 285–306.MathSciNetMATHGoogle Scholar
  87. 87.
    A. d’Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999), Math. Biosciences, 191 (2004), pp. 159–184.Google Scholar
  88. 88.
    T. Duncan, B. Pasik–Duncan, and L. Stettner, Parameter continuity of the ergodic cost for a growth optimal portfolio with proportional transaction costs, Proc. of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, (2008), pp. 4275–4279.Google Scholar
  89. 89.
    U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control Optim., 41 (2003), pp. 1843–1867.MathSciNetMATHGoogle Scholar
  90. 90.
    U. Felgenhauer, Lipschitz stability of broken extremals in bang-bang control problems, in: Large-Scale Scientific Computing (Sozopol 2007), (I. Lirkov et al., eds.), Lecture Notes in Computer Science, vol. 4818, Springer-Verlag (2008), pp. 317–325.Google Scholar
  91. 91.
    U. Felgenhauer, L. Poggiolini, and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour. in: 50 Years of Optimal Control, (A. Ioffe, K. Malanowski, F. Tröltzsch, eds.), Control and Cybernetics, vol. 38(4B) (2009), pp. 1305–1325.Google Scholar
  92. 92.
    M.M. Ferreira, U. Ledzewicz, M. do Rosario de Pinho, and H. Schättler, A model for cancer chemotherapy with state space constraints, Nonlinear Analysis, 63 (2005), pp. 2591–2602.Google Scholar
  93. 93.
    M.E. Fisher, W.J. Grantham, and K.L. Teo, Neighbouring extremals for nonlinear systems with control constraints, Dynamics and Control, 5 (1995), pp. 225–240.MathSciNetMATHGoogle Scholar
  94. 94.
    A.T. Fuller, Study of an optimum non–linear system, J. Electronics Control, 15 (1963), pp. 63–71.MathSciNetGoogle Scholar
  95. 95.
    W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, 1975.Google Scholar
  96. 96.
    W.H. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.MATHGoogle Scholar
  97. 97.
    H. Frankowska, An open mapping principle for set-valued maps, J. Math. Analysis and Applications, 127 (1987), pp. 172–180.MathSciNetMATHGoogle Scholar
  98. 98.
    H. Frankowska, Contingent cones to reachable sets of control systems, SIAM J. Control Optim., 27 (1989), pp. 170–198.MathSciNetMATHGoogle Scholar
  99. 99.
    H. Frankowska and R. Vinter, Existence of neighbouring feasible trajectories: applications to dynamic programming for state constrained optimal control problems, J. Optimization Theory and Applications, 104 (2000), pp. 21–40.MathSciNetMATHGoogle Scholar
  100. 100.
    P. Freeman, Minimum jerk trajectory planning for trajectory constrained redundant robots, D. Sc. thesis, Washington University, 2012Google Scholar
  101. 101.
    R. Gabasov and F.M. Kirillova, High order necessary conditions for optimality, SIAM J. Control, 10 (1972), pp. 127–168.MathSciNetMATHGoogle Scholar
  102. 102.
    S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, Springer-Verlag, 2nd edition, 1990.Google Scholar
  103. 103.
    R.V. Gamkrelidze, Hamiltonian form of the Maximum Principle, Control and Cybernetics, 38 (2009), pp. 959–972.MathSciNetMATHGoogle Scholar
  104. 104.
    H. Gardner-Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), pp. 620–636.MathSciNetGoogle Scholar
  105. 105.
    I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall, Englewood Cliffs, 1963.Google Scholar
  106. 106.
    I.V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, Vol. 67, Springer-Verlag, Heidelberg, 1972.Google Scholar
  107. 107.
    B.S. Goh, Necessary conditions for singular extremals involving multiple control variables, SIAM J. Control, 5 (1966), pp. 716–731.MathSciNetGoogle Scholar
  108. 108.
    M. Golubitsky and V. Guillemin, Stable Mappings and their Singularities, Springer-Verlag, New York, 1973.MATHGoogle Scholar
  109. 109.
    M. Golubitsky and D.G. Schaefer, Singularities and Groups in Bifurcation Theory, Vol. I, Applied Mathematical Sciences, vol. 51, Springer-Verlag, New York, 1985.Google Scholar
  110. 110.
    M. Golubitsky, I.N. Stewart, and D.G. Schaefer, Singularities and Groups in Bifurcation Theory, Vol. II, Applied Mathematical Sciences, vol. 69, Springer-Verlag, New York, 1988.Google Scholar
  111. 111.
    K. Grasse, Reachability of interior states by piecewise constant controls, Forum Matematicum, 7 (1995), pp. 607–628.MathSciNetMATHGoogle Scholar
  112. 112.
    K. Grasse and H. Sussmann, Global controllability by nice controls, in: Nonlinear Controllability and Optimal Control, (H. Sussmann, ed.), Marcel Dekker, New York, (1990), pp. 33–79.Google Scholar
  113. 113.
    W. Greub, Linear Algebra, Graduate Texts in Mathematics, Vol. 23, Springer-Verlag, New York, 1975.Google Scholar
  114. 114.
    P. Hahnfeldt, D. Panigrahy, J. Folkman, and L. Hlatky, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), pp. 4770–4775.Google Scholar
  115. 115.
    H. Hermes, Feedback synthesis and positive local solutions to Hamilton–Jacobi–Bellman equations, Proc. of the 1987 Conference on Mathematical Theory of Networks and Systems, (MTNS), Phoenix, AZ, (1987), pp. 155–164Google Scholar
  116. 116.
    H. Hermes, Nilpotent approximations of control systems, in: Modern Optimal Control, (E.O. Roxin, ed.), Marcel Dekker, New York, 1989, pp. 157–172.Google Scholar
  117. 117.
    H. Hermes and J.P. Lasalle, Functional Analysis and Time Optimal Control, Academic Press, New York, 1969.MATHGoogle Scholar
  118. 118.
    M.R. Hestenes, Calculus of Variations and Optimal Control Theory, Krieger Publishing Co., Huntington, New York, 1980.MATHGoogle Scholar
  119. 119.
    A.D. Ioffe and V.M. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.MATHGoogle Scholar
  120. 120.
    A. Isidori, Nonlinear Contol Systems, Springer-Verlag, Berlin, 1989.Google Scholar
  121. 121.
    D.H. Jacobson, A new necessary condition for optimality of singular control problems, SIAM J. Control, 7 (1969), pp. 578–595.MathSciNetMATHGoogle Scholar
  122. 122.
    D.H. Jacobson and J.L. Speyer, Necessary and sufficient conditions for optimality for singular control problems: a limit approach, J. Mathematical Analysis and Applications, 34 (1971), pp. 239–266.MathSciNetGoogle Scholar
  123. 123.
    N. Jacobson, Lie Algebras, Dover, (1979).Google Scholar
  124. 124.
    F. John, Partial Differential Equations, Springer-Verlag, 1982.Google Scholar
  125. 125.
    F. Jones, Lebesgue Integration on Euclidean Space, Jones and Bartlett Publishers, Boston, 1993.MATHGoogle Scholar
  126. 126.
    V. Jurdjevic, Geometric Control Theory, Cambridge Studies in Advanced Mathematics, Vol. 51, Cambridge University Press, 1977.Google Scholar
  127. 127.
    T. Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, NJ, 1980.MATHGoogle Scholar
  128. 128.
    M. Kawski, Control variations with an increasing number of switchings, Bul. of the American Mathematical Society, 18 (1988), pp. 149–152.MathSciNetMATHGoogle Scholar
  129. 129.
    M. Kawski, High-order small-time local controllability, in: Nonlinear Controllability and Optimal Control, (H. Sussmann, ed.), Marcel Dekker, New York, (1990), pp. 431–467.Google Scholar
  130. 130.
    M. Kawski and H. Sussmann, Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, in: Operators, Systems and Linear Algebra: Three Decades of Algebraic Systems Theory, (U. Helmke, D. Praetzel-Wolters, and E. Zerz, eds.), B.G. Teubner, Stuttgart, (1997), pp. 111–129.Google Scholar
  131. 131.
    H.J. Kelley, A second variation test for singular extremals, AIAA (American Institute of Aeronautics and Astronautics) J., 2 (1964), pp. 1380–1382.Google Scholar
  132. 132.
    H.J. Kelley, R. Kopp, and H.G. Moyer, Singular extremals, in: Topics in Optimization (G. Leitmann, ed.), Academic Press, 1967.Google Scholar
  133. 133.
    H.K. Khalil, Nonlinear Systems, 3rd. ed., Prentice Hall, 2002.Google Scholar
  134. 134.
    M. Kiefer, On Singularities in Solutions to the Hamilton–Jacobi–Bellman Equation and Their Implications for the Optimal Control Problem, D.Sc. Thesis, Washington University, 1997.Google Scholar
  135. 135.
    M. Kiefer and H. Schättler, Parametrized families of extremals and singularities in solutions to the Hamilton–Jacobi–Bellman equation, SIAM J. on Control and Optimization, 37 (1999), pp. 1346–1371.MATHGoogle Scholar
  136. 136.
    A. Kneser, Lehrbuch der Variationsrechnung, Braunschweig, 1925.Google Scholar
  137. 137.
    H.W. Knobloch, Higher Order Necessary Conditions in Optimal Control Theory, Lecture Notes in Control and Information Sciences, Vol. 34, Springer-Verlag, Berlin, 1981.Google Scholar
  138. 138.
    H.W. Knobloch and H. Kwakernaak, Lineare Kontrolltheorie, Springer-Verlag, Berlin, 1985.MATHGoogle Scholar
  139. 139.
    G. Knowles, An Introduction to Applied Optimal Control, Academic Press, 1981.Google Scholar
  140. 140.
    A.J. Krener, The high order maximum principle and its application to singular extremals, SIAM J. Control Optim., 15 (1977), pp. 256–293.MathSciNetMATHGoogle Scholar
  141. 141.
    A.J. Krener and H. Schättler, The structure of small time reachable sets in low dimension, SIAM J. Control Optim., 27 (1989), pp. 120–147.MathSciNetMATHGoogle Scholar
  142. 142.
    I.A.K. Kupka, Geometric theory of extremals in optimal control problems I: The fold and Maxwell cases, Trans. of the American Mathematical Society, 299 (1987), pp. 225–243.MathSciNetMATHGoogle Scholar
  143. 143.
    I.A.K. Kupka, The ubiquity of Fuller’s phenomenon, in: Nonlinear Controllability and Optimal Control (H. Sussmann, ed.), Marcel Dekker, (1990), pp. 313–350.Google Scholar
  144. 144.
    H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley–Interscience, (1972).Google Scholar
  145. 145.
    I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Vol. I: Abstract Parabolic Systems, Cambridge University Press, 2000.Google Scholar
  146. 146.
    I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories,Vol. II: Abstract Hyperbolic-Like Systems over a Finite Time-Horizon, Cambridge University Press, 2000.Google Scholar
  147. 147.
    E.B. Lee and L. Marcus, Foundations of Optimal Control Theory, Wiley, New York, 1967.MATHGoogle Scholar
  148. 148.
    U. Ledzewicz, A. Nowakowski, and H. Schättler, Stratifiable families of extremals and sufficient conditions for optimality in optimal control problems, J. of Optimization Theory and Applications (JOTA), 122 (2004), pp. 105–130.Google Scholar
  149. 149.
    U. Ledzewicz and H. Schättler, Second order conditions for extremum problems with nonregular equality constraints, J. of Optimization Theory and Applications (JOTA), 86 (1995), pp. 113–144.MATHGoogle Scholar
  150. 150.
    U. Ledzewicz and H. Schättler, An extended maximum principle, Nonlinear Analysis, 29 (1997), pp. 159–183.MathSciNetMATHGoogle Scholar
  151. 151.
    U. Ledzewicz and H. Schättler, High order extended maximum principles for optimal control problems with non-regular constraints, in: Optimal Control: Theory, Algorithms and Applications, (W.W. Hager and P.M. Pardalos, eds.), Kluwer Academic Publishers, (1998), pp. 298–325.Google Scholar
  152. 152.
    U. Ledzewicz and H. Schättler, A high-order generalization of the Lyusternik theorem, Nonlinear Analysis, 34 (1998), pp. 793–815.MathSciNetMATHGoogle Scholar
  153. 153.
    U. Ledzewicz and H. Schättler, High-order approximations and generalized necessary conditions for optimality, SIAM J. Contr. Optim., 37 (1999), pp. 33–53.MATHGoogle Scholar
  154. 154.
    U. Ledzewicz and H. Schättler, A high-order generalized local Maximum Principle, SIAM J. on Control and Optimization, 38 (2000), pp. 823–854.MATHGoogle Scholar
  155. 155.
    U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications - JOTA, 114 (2002), pp. 609–637.MATHGoogle Scholar
  156. 156.
    U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), pp. 183–206.MATHGoogle Scholar
  157. 157.
    U. Ledzewicz and H. Schättler, A synthesis of optimal controls for a model of tumor growth under angiogenic inhibitors, Proc. of the 44th IEEE Conference on Decision and Control, Sevilla, Spain, (2005), pp. 934–939.Google Scholar
  158. 158.
    U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete and Continuous Dynamical Systems, Series B, 6 (2006), pp. 129–150.Google Scholar
  159. 159.
    U. Ledzewicz and H. Schättler, Application of optimal control to a system describing tumor anti–angiogenesis, Proc. of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Kyoto, Japan, (2006), pp. 478–484.Google Scholar
  160. 160.
    U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), pp. 1052–1079.MATHGoogle Scholar
  161. 161.
    U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Mathematical Biosciences, 206 (2007), pp. 320–342.MathSciNetMATHGoogle Scholar
  162. 162.
    U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti–angiogenesis, J. of Theoretical Biology, 252 (2008), pp. 295–312.Google Scholar
  163. 163.
    U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis, Optimal Control, Applications and Methods, 29 (2008), pp. 41–57.Google Scholar
  164. 164.
    U. Ledzewicz and H. Schättler, On the optimality of singular controls for a class of mathematical models for tumor anti-angiogenesis, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), pp. 691–715.Google Scholar
  165. 165.
    U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), pp. 1501–1523.MathSciNetMATHGoogle Scholar
  166. 166.
    U. Ledzewicz and H. Schaettler, Applications of Geometric Optimal Control to Biomedical Problems, Springer Verlag, to appear.Google Scholar
  167. 167.
    S. Lenhart and J.T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC, Mathematical & Computational Biology, 2007.Google Scholar
  168. 168.
    E.S. Levitin, A.A. Milyutin, and N.P. Osmolovskii, Higher order conditions for local minima in problems with constraints, Uspekhi Mat. Nauk, translated as: Russian Mathematical Surveys, 33 (1978) pp. 97–165.Google Scholar
  169. 169.
    R.M. Lewis, Definitions of order and junction conditions in singular optimal control problems, SIAM J. Control and Optimization, 18 (1980), pp. 21–32.MathSciNetMATHGoogle Scholar
  170. 170.
    C. Lobry, Contrôlabilité des Systèmes nonlinéaires, SIAM J. Control, 8 (1970), pp. 573–605.MathSciNetMATHGoogle Scholar
  171. 171.
    R. Martin and K.L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994.MATHGoogle Scholar
  172. 172.
    C. Marchal, Chattering arcs and chattering controls, J. of Optimization Theory and Applications (JOTA), 11, (1973), pp. 441–468.MathSciNetMATHGoogle Scholar
  173. 173.
    J.P. McDanell and W.J. Powers, Necessary conditions for joining optimal singular and nonsingular subarcs, SIAM J. Control, 9 (1971), pp. 161–173.MathSciNetMATHGoogle Scholar
  174. 174.
    J. McDonald and N. Weiss, A Course in Real Analysis, Academic Press, 1999.Google Scholar
  175. 175.
    E.J. McShane, Integration, Princeton University Press, 1944.Google Scholar
  176. 176.
    H. Nijmeier and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer Verlag, 1990.Google Scholar
  177. 177.
    J.E. Marsden and M.J. Hoffman, Elementary Classical Analysis, W.H. Freeman, New York, second edition, 1993.MATHGoogle Scholar
  178. 178.
    H. Maurer, An example of a continuous junction for a singular control problem of even order, SIAM J. Control, 13 (1975), pp. 899–903.MathSciNetMATHGoogle Scholar
  179. 179.
    H. Maurer, On optimal control problems with bounded state variables and control appearing linearly, SIAM J. on Control and Optimization, 15 (1977), pp. 345–362.MATHGoogle Scholar
  180. 180.
    H. Maurer, C. Büskens, J.H. Kim, and Y. Kaja, Optimization techniques for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control, Applications and Methods, 26 (2005), pp. 129–156.Google Scholar
  181. 181.
    H. Maurer and H.J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach SIAM J. Control and Optimization, 41 (2002), pp. 380–403.Google Scholar
  182. 182.
    H. Maurer and N. Osmolovskii, Second order optimality conditions for bang-bang control problems, Control and Cybernetics, 32 (2003), pp. 555–584.MATHGoogle Scholar
  183. 183.
    H. Maurer and N. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control problems, SIAM J. Control and Optimization, 42 (2003), pp. 2239–2263.MathSciNetGoogle Scholar
  184. 184.
    J.W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, VA, 1965.MATHGoogle Scholar
  185. 185.
    A.A. Milyutin and N.P. Osmolovskii, Calculus of Variations and Optimal Control, American Mathematical Society, 1998.Google Scholar
  186. 186.
    B. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330, Springer-Verlag, 2006.Google Scholar
  187. 187.
    B. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 331, Springer-Verlag, 2006.Google Scholar
  188. 188.
    H. Nijmeijer and A. van der Schaft, Controlled Invariance for nonlinear systems: two worked examples, IEEE Transactions on Automatic Control, 29 (1984), pp. 361–364.MATHGoogle Scholar
  189. 189.
    J. Noble and H. Schättler, Sufficient conditions for relative minima of broken extremals, J. of Mathematical Analysis and Applications, 269 (2002), pp. 98–128.MATHGoogle Scholar
  190. 190.
    A. Nowakowski, Field theories in the modern calculus of variations, Transactions of the Americam Mathematical Society, 309 (1988), pp. 725–752.MathSciNetMATHGoogle Scholar
  191. 191.
    P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.Google Scholar
  192. 192.
    N.P. Osmolovskii, Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations, J. of Mathematical Sciences, 123 (2004), pp. 3987–4122.MathSciNetGoogle Scholar
  193. 193.
    L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964.MATHGoogle Scholar
  194. 194.
    H.J. Pesch and M. Plail, The maximum principle of optimal control: a history of ingenious ideas and missed opportunities, Control and Cybernetics, 38 (2009), pp. 973–996.MathSciNetMATHGoogle Scholar
  195. 195.
    B. Piccoli, Classification of generic singularities for the planar time-optimal synthesis, SIAM J. Control and Optimization, 34 (1996), pp. 1914–1946.MathSciNetMATHGoogle Scholar
  196. 196.
    B. Piccoli and H. Sussmann, Regular synthesis and sufficient conditions for optimality, SIAM J. on Control and Optimization, 39 (2000), pp. 359–410.MathSciNetMATHGoogle Scholar
  197. 197.
    L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem, SIAM J. Control and Optimization, 49 (2011), pp. 140–161.MathSciNetMATHGoogle Scholar
  198. 198.
    L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bang-bang trajectory, in: Proc. 45th IEEE Conference on Decision and Control, (2006).Google Scholar
  199. 199.
    L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bangsingular extremal in the minimum time problem, Control and Cybernetics, 37 (2008), pp. 469–490.MathSciNetMATHGoogle Scholar
  200. 200.
    V. Ramakrishna and H. Schättler, Controlled invariant distributions and group invariance, J. of Mathematical Systems and Control, 1 (1991), pp. 209–240.Google Scholar
  201. 201.
    D. Rebhuhn, On the stability of the existence of singular controls under perturbation of the control system, SIAM J. Control and Optimization, 18 (1978), pp. 463–472.MathSciNetGoogle Scholar
  202. 202.
    R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970.Google Scholar
  203. 203.
    E.O. Roxin, Reachable zones in autonomous differential systems, Bol. Soc. Mat. Mexicana, (1960), pp. 125–135.Google Scholar
  204. 204.
    E.O. Roxin, Reachable sets, limit sets and holding sets in control systems, in: Nonlinear Analysis and Applications, (V. Lakshmikantham, ed.,) Marcel Dekker, (1987), pp. 401–407.Google Scholar
  205. 205.
    E.O. Roxin, Control Theory and Its Applications, Gordon and Breach Scientific Publishers, 1997.Google Scholar
  206. 206.
    A. Sarychev, The index of second variation of a control system, Math. USSR Sbornik, 41 (1982), pp. 383–401.Google Scholar
  207. 207.
    A. Sarychev, Morse index and sufficient optimality conditions for bang-bang Pontryagin extremals, in: System Modeling and Optimization, Lecture Notes in Control and Information Sciences, Vol. 180, (1992), pp. 440–448.MathSciNetGoogle Scholar
  208. 208.
    A. Sarychev, First and second order sufficient optimality conditions for bang-bang controls, SIAM J. on Control and Optimization, 35, (1997), pp. 315–340.MathSciNetMATHGoogle Scholar
  209. 209.
    H. Schättler, On the local structure of time-optimal trajectories for a single-input control-linear system in dimension 3, Ph.D. thesis, Rutgers, The State University of New Jersey, 1986.Google Scholar
  210. 210.
    H. Schättler, On the local structure of time-optimal bang-bang trajectories in 3, SIAM J. on Control and Optimization, 26 1988, pp. 186–204.MATHGoogle Scholar
  211. 211.
    H. Schättler, The local structure of time-optimal trajectories in dimension 3 under generic conditions, SIAM J. Control Optim., 26 (1988), pp. 899–918.MathSciNetMATHGoogle Scholar
  212. 212.
    H. Schättler, Conjugate points and intersections of bang-bang trajectories, Proc. of the 28th IEEE Conference on Decision and Control, Tampa, Florida, (1989), pp. 1121–1126.Google Scholar
  213. 213.
    H. Schättler, Regularity properties of optimal trajectories: recently developed techniques, in: Nonlinear Controllability and Optimal Control (H. Sussmann, ed.), Marcel Dekker, (1990), pp. 351–381.Google Scholar
  214. 214.
    H. Schättler, Extremal trajectories, small-time reachable sets and local feedback synthesis: a synopsis of the three-dimensional case, in: Nonlinear Synthesis, (C.I. Byrnes and A. Kurzhansky, eds.), Birkhäuser, Boston, September 1991, pp. 258–269.Google Scholar
  215. 215.
    H. Schättler, A local feedback-synthesis of time-optimal stabilizing controls in dimension three, Mathematics of Control, Signals and Systems, 4 (1991), pp. 293–313.Google Scholar
  216. 216.
    H. Schättler, A geometric approach to optimal control, in: WCNA-92, Proceedings of the First World Congress of Nonlinear Analysts, Vol. II, (V. Lakhshmikantham, ed.), Walter de Gruyter Publishers, (1996), pp. 1579–1590.Google Scholar
  217. 217.
    H. Schättler, Small-time reachable sets and time-optimal feedback control for nonlinear systems, in: Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, (B. Mordukhovich and H.J. Sussmann, eds.), IMA Volumes in Mathematics and its Applications, Vol. 78, Springer-Verlag, 1996, pp. 203–225.Google Scholar
  218. 218.
    H. Schättler, Time-optimal feedback control for nonlinear systems, in: Geometry of Feedback and Optimal Control, (B. Jakubczyk and W. Respondek, eds.), Marcel Dekker, New York, (1998), pp. 383–421.Google Scholar
  219. 219.
    H. Schättler, On classical envelopes in optimal control theory, Proc. of the 49th IEEE Conference on Decision and Control, Atlanta, USA, (2010), pp. 1879–1884.Google Scholar
  220. 220.
    H. Schättler, A local field of extremals for optimal control problems with state constraints of relative degree 1, J. of Dynamical and Control Systems, 12 (2006), pp. 563–599.MATHGoogle Scholar
  221. 221.
    H. Schättler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturated singular arcs, Forum Matematicum, 5 (1993), pp. 203–241.MATHGoogle Scholar
  222. 222.
    Ch.E. Shin, On the structure of time-optimal stabilizing controls in 4, Bollettino U.M.I., 7 (1995), pp. 299–320.Google Scholar
  223. 223.
    Ch.E. Shin, Time-optimal bang-bang trajectories using bifurcation results, J. Korean Math.Soc., 34 (1997), pp. 553–567.MathSciNetMATHGoogle Scholar
  224. 224.
    M. Sigalotti, Regularity properties of optimal trajectories of single-input control systems in dimension three, J. of Math. Sci., 126 (2005), pp. 1561–1573.MathSciNetMATHGoogle Scholar
  225. 225.
    E.D. Sontag, Mathematical Control Theory, second edition, Springer-Verlag, (1998).Google Scholar
  226. 226.
    G. Stefani, Higher order variations: how can they be defined in order to have good properties? in: Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, IMA Volumes in Mathematics and Its Applications, Vol. 78, Springer-Verlag, New York, (1996), pp. 227–237.Google Scholar
  227. 227.
    G. Stefani and P.L. Zezza, Optimality conditions for a constrained control problem, SIAM J. Control and Optimization, 34 (1996), pp. 635–659.MathSciNetMATHGoogle Scholar
  228. 228.
    J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York, 1990.Google Scholar
  229. 229.
    H.J. Sussmann, Lie brackets, real analyticity and geometric control, in: Differential Geometric Control Theory, (R. Brockett, R. Millman, H. Sussmann, eds.), Progress in Mathematics, Birkhäuser, Boston, (1983), pp. 1–116.Google Scholar
  230. 230.
    H.J. Sussmann, Time-optimal control in the plane, in: Feedback Control of Linear and Nonlinear Systems, Lecture Notes in Control and Information Sciences, Vol. 39, Springer-Verlag, Berlin, (1982), pp. 244–260.Google Scholar
  231. 231.
    H.J. Sussmann, Lie brackets and real analyticity in control theory, in: Mathematical Control Theory, Banach Center Publications, Vol. 14, Polish Scientific Publishers, Warsaw, Poland, (1985), pp. 515–542.Google Scholar
  232. 232.
    H.J. Sussmann, A bang-bang theorem with bounds on the number of switchings, SIAM J. Control Optim., 17 (1979), pp. 629–651.MathSciNetMATHGoogle Scholar
  233. 233.
    H.J. Sussmann, Subanalytic sets and feedback control, J. of Differential Equations, 31 (1979), pp. 31–52.MathSciNetMATHGoogle Scholar
  234. 234.
    H.J. Sussmann, A product expansion for the Chen series, in: Theory and Applications of Nonlinear Control Systems, (C. Byrnes and A. Lindquist, eds.), North-Holland, Amsterdam, (1986), pp. 323–335.Google Scholar
  235. 235.
    H.J. Sussmann, A general theorem on local controllability, SIAM J. Control Optim., 25 (1987), pp. 158–194.MathSciNetMATHGoogle Scholar
  236. 236.
    H.J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the C nonsingular case, SIAM J. Control Optim., 25 (1987), pp. 433–465.MathSciNetGoogle Scholar
  237. 237.
    H.J. Sussmann, The structure of time–optimal trajectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optim., 25 (1987), pp. 868–904.MathSciNetMATHGoogle Scholar
  238. 238.
    H.J. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optim., 25 (1987), pp. 1145–1162.MathSciNetMATHGoogle Scholar
  239. 239.
    H.J. Sussmann, Recent developments in the regularity theory of optimal trajectories, Rend. Sem. Mat. Univ. Politec. Torino, Fasc. Spec. Control Theory, (1987), pp. 149–182.Google Scholar
  240. 240.
    H.J. Sussmann, Envelopes, conjugate points, and optimal bang-bang extremals, Proc. of the 1985 Paris Conference on Nonlinear Systems (M. Fliess, M. Hazewinkel, eds.), Reidel Publ., The Netherlands, (1987).Google Scholar
  241. 241.
    H.J. Sussmann, Thirty years of optimal control: was the path unique? Modern Optimal Control, (E.O. Roxin, ed.), Marcel Dekker, New York, 1989, pp. 359–375.Google Scholar
  242. 242.
    H.J. Sussmann, Envelopes, high-order optimality conditions and Lie brackets, Proc. of the 28th IEEE Conference on Decision and Control, Tampa, Florida, December 1989, pp. 1107–1112.Google Scholar
  243. 243.
    H.J. Sussmann, Synthesis, presynthesis, sufficient conditions for optimality and subanalytic sets, in: Nonlinear Controllability and Optimal Control, (H. Sussmann, ed.), Marcel Dekker, (1990), pp. 1–19.Google Scholar
  244. 244.
    H.J. Sussmann, A strong version of the Lojasiewicz Maximum Principle, in: Optimal Control of Differential Equations, (N. Pavel, ed.), Marcel Dekker, New York, 1994, pp. 293–310.Google Scholar
  245. 245.
    H.J. Sussmann and J.C. Willems, 300 years of optimal control: from the Brachistochrone to the maximum principle, IEEE Control Systems, (1997), pp. 32–44.Google Scholar
  246. 246.
    H.J. Sussmann, Uniqueness results for the value function via direct trajectory-construction methods, in: Proc. of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, (2003), pp. 3293–3298.Google Scholar
  247. 247.
    H.J. Sussmann, Multidifferential calculus: chain rule, open mapping and transversal intersection theorems, in: Optimal Control: Theory, Algorithms and Applications, (W.W. Hager and P.M. Pardalos, eds.), Kluwer Academic Publishers, (1998), pp. 436–487.Google Scholar
  248. 248.
    H.J. Sussmann, Set separation, transversality and the Lipschitz maximum principle, J. of Differential Equations, 243 (2007), pp. 446–488.MathSciNetGoogle Scholar
  249. 249.
    H.J. Sussmann and G. Tang, Shortest paths for the Reeds-Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control, Report SYCON 91–10, 1991.Google Scholar
  250. 250.
    R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, 6 (1955–56), pp. 43–87.Google Scholar
  251. 251.
    O.A. Brezhneva and A.A. Tret’yakov, Optimality Conditions for Degenerate Extremum Problems with Equality Constraints, SIAM J. Control Optim., 42 (2003), pp. 729–743.Google Scholar
  252. 252.
    G.W. Swan, Role of optimal control in cancer chemotherapy, Math. Biosci., 101 (1990), pp. 237–284.MATHGoogle Scholar
  253. 253.
    A. Swierniak, A. Polanski, and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell prolif., 29 (1996), pp. 117–139.Google Scholar
  254. 254.
    R.B. Vinter, Optimal Control Theory, Birkhäuser, Boston, 2000.Google Scholar
  255. 255.
    S. Walczak, On some properties of cones in normed spaces and their application to investigating extremal problems, J. of Optimization Theory and Applications, 42 (1984), pp. 559–582.MathSciNetGoogle Scholar
  256. 256.
    F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer Verlag, 1983.Google Scholar
  257. 257.
    R.L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977.MATHGoogle Scholar
  258. 258.
    H. Whitney, Elementary structure of real algebraic varieties, Ann. Math., 66 (1957), pp. 545–556.MathSciNetMATHGoogle Scholar
  259. 259.
    W.M. Wonham, Note on a problem in optimal nonlinear control, J. Electronics Control, 15 (1963), pp. 59–62.MathSciNetGoogle Scholar
  260. 260.
    L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W.B. Saunders, Philadelphia, 1969.MATHGoogle Scholar
  261. 261.
    M.I. Zelikin and V.F. Borisov, Optimal synthesis containing chattering arcs and singular arcs of the second order, in: Nonlinear Synthesis, (C.I. Byrnes and A. Kurzhansky, eds.), Birkhäuser, Boston, (1991), pp. 283–296.Google Scholar
  262. 262.
    M.I. Zelikin and V.F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering, Birkhäuser, 1994.Google Scholar
  263. 263.
    M.I. Zelikin and L.F. Zelikina, The structure of optimal synthesis in a neighborhood of singular manifolds for problems that are affine in control, Sbornik: Mathematics, 189 (1998), pp. 1467–1484.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Heinz Schättler
    • 1
  • Urszula Ledzewicz
    • 2
  1. 1.Washington UniversitySt. LouisUSA
  2. 2.Southern Illinois UniversityEdwardsvilleUSA

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