Advanced Search
Article Contents
Article Contents

Necessity for isoperimetric inequality constraints

The second author was supported by the PASPA-DGAPA program from Universidad Nacional Autónoma de México.
Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • In this paper we deal with second order necessary conditions for the problem of Lagrange in the calculus of variations posed over piecewise smooth trajectories and involving inequality and equality isoperimetric constraints. We provide a review of different approaches to derive second order necessary conditions for this problem and prove that, surprisingly, though the solution set to the problem where the conditions hold may vary, all approaches impose the same strong assumption of normality relative to the set defined by equality constraints for active indices. Based on these approaches, we also give some applications to certain optimization problems with mixed constraints.

    Mathematics Subject Classification: Primary:49K15, 49K21;Secondary:34H05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, Translated from the German by Gerhard Metzen, de Gruyter Studies in Mathematics 13, Berlin, 1990. doi: 10.1515/9783110853698.
    [2] A. V. Arutyunov and F. L. Pereira, Second-order necessary optimality conditions for problems without a priori normality assumptions, Mathematics of Operations Research, 31 (2006), 1-12.  doi: 10.1287/moor.1050.0172.
    [3] A. V. Arutyunov and Y. S. Vereshchagina, On necessary second-order conditions in optimal control problems, Differential Equations, 38 (2002), 1531-1540.  doi: 10.1023/A:1023624602611.
    [4] K. L. Cortez del Río and J. F. Rosenblueth, Normality and regularity for inequality control constraints, Journal of Convex Analysis, (submitted).
    [5] M. R. de Pinho and J. F. Rosenblueth, Mixed constraints in optimal control: An implicit function theorem approach, IMA Journal of Mathematical Control and Information, 24 (2007), 197-218.  doi: 10.1093/imamci/dnl008.
    [6] G. Giorgi, A. Guerraggio and J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier, Amsterdam, 2004.
    [7] E. G. Gilbert and D. S. Bernstein, Second order necessary conditions in optimal control: Accessory-problem results without normality conditions, Journal of Optimization Theory & Applications, 41 (1983), 75-106.  doi: 10.1007/BF00934437.
    [8] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.
    [9] M. R. Hestenes, Optimization Theory. The Finite Dimensional Case, John Wiley, New York, 1975.
    [10] E. LevitinA. Milyutin and N. P. Osomolovskiǐ, Conditions of high order for a local minimum for problems with constraints, Russian Math. Surveys, 33 (1978), 97-168.  doi: 10.1070/RM1978v033n06ABEH003885.
    [11] P. D. Loewen and H. Zheng, Generalized conjugate arcs in optimal control, Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista, Florida, 4 (1994), 4004-4008. 
    [12] P. D. Loewen and H. Zheng, Generalized conjugate points for optimal control problems, Nonlinear Analysis, Theory, Methods & Applications, 22 (1994), 771-791.  doi: 10.1016/0362-546X(94)90226-7.
    [13] H. Maurer and S. Pickenhain, Second order sufficient conditions for control problems with mixed control-state constraints, Journal of Optimization Theory and Applications, 86 (1995), 649-667.  doi: 10.1007/BF02192163.
    [14] H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach, SIAM Journal on Control and Optimization, 41 (2002), 380-403.  doi: 10.1137/S0363012900377419.
    [15] E. J. McShane, The Lagrange multiplier rule, The American Mathematical Monthl, 80 (1973), 922-925.  doi: 10.2307/2319406.
    [16] A. A. Milyutin and N. P. Osmolovskiǐ, Calculus of Variations and Optimal Control, Translations of Mathematical Monographs, 180, American Mathematical Society, Providence, Rhode Island, 1998.
    [17] N. P. Osmolovskiǐ, Second order conditions for a weak local minimum in an optimal control problem (necessity sufficiency), Soviet Math. Dokl., 16 (1975), 1480-1484. 
    [18] J. F. Rosenblueth, A new notion of conjugacy for isoperimetric problems, Applied Mathematics and Optimization, 50 (2004), 209-228.  doi: 10.1007/s00245-004-0800-3.
    [19] J. F. Rosenblueth, Modified critical directions for inequality control constraints, WSEAS Transactions on Systems and Control, 10 (2015), 215-227. 
    [20] J. F. Rosenblueth and Licea G. Sánchez, Cones of critical directions in optimal control, International Journal of Applied Mathematics and Informatics, 7 (2013), 55-67. 
    [21] I. B. Russak, Second order necessary conditions for problems with state inequality constraints, SIAM Journal on Control, 13 (1975), 372-388.  doi: 10.1137/0313021.
    [22] I. B. Russak, Second order necessary conditions for general problems with state inequality constraints, Journal of Optimization Theory and Applications, 17 (1975), 43-92.  doi: 10.1007/BF00933916.
    [23] G. Stefani and P. L. Zezza, Optimality conditions for a constrained control problem, SIAM Journal on Control & Optimization, 34 (1996), 635-659.  doi: 10.1137/S0363012994260945.
    [24] F. A. Valentine, The problem of Lagrange with differential inequalities as added side conditions, Contributions to the Calculus of Variations 1933-37, The University of Chicago Press, 1937, 34 (1996), 635-659.  doi: 10.1007/978-3-0348-0439-4_16.
    [25] J. Warga, A second-order Lagrangian condition for restricted control problems, Journal of Optimization Theory & Applications, 24 (1996), 475-483.  doi: 10.1007/BF00932890.
  • 加载中

Article Metrics

HTML views(1512) PDF downloads(182) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint