# American Institute of Mathematical Sciences

March  2017, 37(3): 1129-1158. doi: 10.3934/dcds.2017047

## Necessity for isoperimetric inequality constraints

 IIMAS, Universidad Nacional Autóonoma de México, Apartado Postal 20-126, México DF 01000

Received  May 2016 Revised  September 2016 Published  December 2016

Fund Project: The second author was supported by the PASPA-DGAPA program from Universidad Nacional Autónoma de México.

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.

Citation: Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047
##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] K. L. Cortez del Río and J. F. Rosenblueth, Normality and regularity for inequality control constraints, Journal of Convex Analysis, (submitted). Google Scholar [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.  Google Scholar [6] G. Giorgi, A. Guerraggio and J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier, Amsterdam, 2004.  Google Scholar [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.  Google Scholar [8] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.  Google Scholar [9] M. R. Hestenes, Optimization Theory. The Finite Dimensional Case, John Wiley, New York, 1975.  Google Scholar [10] E. Levitin, A. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] E. J. McShane, The Lagrange multiplier rule, The American Mathematical Monthl, 80 (1973), 922-925.  doi: 10.2307/2319406.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [19] J. F. Rosenblueth, Modified critical directions for inequality control constraints, WSEAS Transactions on Systems and Control, 10 (2015), 215-227.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] K. L. Cortez del Río and J. F. Rosenblueth, Normality and regularity for inequality control constraints, Journal of Convex Analysis, (submitted). Google Scholar [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.  Google Scholar [6] G. Giorgi, A. Guerraggio and J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier, Amsterdam, 2004.  Google Scholar [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.  Google Scholar [8] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.  Google Scholar [9] M. R. Hestenes, Optimization Theory. The Finite Dimensional Case, John Wiley, New York, 1975.  Google Scholar [10] E. Levitin, A. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] E. J. McShane, The Lagrange multiplier rule, The American Mathematical Monthl, 80 (1973), 922-925.  doi: 10.2307/2319406.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [19] J. F. Rosenblueth, Modified critical directions for inequality control constraints, WSEAS Transactions on Systems and Control, 10 (2015), 215-227.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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