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April  2011, 29(2): 559-575. doi: 10.3934/dcds.2011.29.559

On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems

Sofia O. Lopes 1, , Fernando A. C. C. Fontes 2, and  Maria do Rosário de Pinho 2,

1. 

CMAT e Departamento de Matemática e Aplicações, Universidade do Minho, 4800-058 Guimarães, Portugal
2. 

ISR-Porto, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal, Portugal

Received  September 2009 Revised  April 2010 Published  October 2010

We address necessary conditions of optimality (NCO), in the form of a maximum principle, for optimal control problems with state constraints. In particular, we are interested in the NCO that are strengthened to avoid the degeneracy phenomenon that occurs when the trajectory hits the boundary of the state constraint. In the literature on this subject, we can distinguish two types of constraint qualifications (CQ) under which the strengthened NCO can be applied: CQ involving the optimal control and CQ not involving it. Each one of these types of CQ has its own merits. The CQs involving the optimal control are not so easy to verify, but, are typically applicable to problems with less regularity on the data. In this article, we provide conditions under which the type of CQ involving the optimal control can be reduced to the other type. In this way, we also provide nondegenerate NCO that are valid under a different set of hypotheses.
Keywords: Optimal Control, State Constraints, Maximum Principle, Constraint Qualifications., Degeneracy Phenomenon.
Mathematics Subject Classification: 49K15, 49K3.
Citation: Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
References:
[1]

A. V. Arutyunov and S. M. Aseev, State constraints in optimal control. The degeneracy phenomenon, Systems Control Lett., 26 (1995), 267-273. doi: doi:10.1016/0167-6911(95)00021-Z.

[2]

A. V. Arutyunov and S. M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim., 35 (1997), 930-952. doi: doi:10.1137/S036301299426996X.

[3]

J. Abadie, On the Kuhn-Tucker theorem, in "Nonlinear Programming" (J. Abadie, ed.), North Holland, Amsterdam, 1967, 21-56.

[4]

A. V. Arutyunov, On necessary conditions for optimality in a problem with phase constraints, Dokl. Akad. Nauk SSSR, 280 (1985), 1033-1037.

[5]

Aram V. Arutyunov, "Optimality Conditions. Abnormal and Degenerate Problems," Mathematics and its Applications, vol. 526, Kluwer Academic Publishers, Dordrecht, 2000, Translated from the Russian by S. A. Vakhrameev.

[6]

A. V. Arutyunov and N. T. Tynyanskiy, The maximum principle in a problem with phase constraints, Izv. Akad. Nauk SSSR Tekhn. Kibernet, (1984), 60-68, 235.

[7]

P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems, J. Differential Equations, 243 (2007), 256-269. doi: doi:10.1016/j.jde.2007.05.005.

[8]

A. Cernea and H. Frankowska, A connection between the maximum principle and dynamic programming for constrained control problems, SIAM J. Control Optim., 44 (2005), 673-703 (electronic). doi: doi:10.1137/S0363012903430585.

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1983, A Wiley-Interscience Publication.

[10]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998.

[11]

A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems under constraints, Dokl. Akad. Nauk SSSR, 149 (1963), 759-762.

[12]

M. d. R. de Pinho, R. B. Vinter and H. Zheng, A maximum principle for optimal control problems with mixed constraints, IMA J. Math. Control Inform., 18 (2001), 189-205. doi: doi:10.1093/imamci/18.2.189.

[13]

M. M. A. Ferreira, F. A. C. C. Fontes and R. B. Vinter, Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints, J. Math. Anal. Appl., 233 (1999), 116-129. doi: doi:10.1006/jmaa.1999.6270.

[14]

F. A. C. C. Fontes, "Optimisation-based Control of Constrained Nonlinear Systems," Ph.D. thesis, Imperial College of Science Technology and Medicine, University of London, London SW7 2BY, U.K., 1999.

[15]

F. A. C. C. Fontes, "Normality in the Necessary Conditions of Optimality for Control Problems with State Constraints," Proceedings of the IASTED Conference on Control and Applications (Cancun, Mexico), 2000.

[16]

F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers, Systems Control Lett., 42 (2001), 127-143. doi: doi:10.1016/S0167-6911(00)00084-0.

[17]

F. A. C. C. Fontes, Nondegenerate necessary conditions of optimality for control problems with state constraints, in "Nonlinear Control Systems" (A. B. Kurzhanski and A. L. Fradkov, eds.), (2002), 45-50.

[18]

M. M. A. Ferreira and R. B. Vinter, When is the maximum principle for state constrained problems nondegenerate?, J. Math. Anal. Appl., 187 (1994), 438-467. doi: doi:10.1006/jmaa.1994.1366.

[19]

F. John, "Extremum Problems as Inequalities as Subsidiary Conditions," Studies and Essays: Courant Anniversary Volume (K. O. Friedrichs, O. E. Neugebauer and J. J. Stoker, eds.), Interscience Publishers, New York, 1948.

[20]

H. W. Kuhn and A. W. Tucker, Nonlinear programming, in "Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability" (J. Neyman, ed.), University of California Press, Berkeley, 1951, 481-492.

[21]

K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52 (2003), 75-91. doi: doi:10.1080/0233193021000058940.

[22]

O. L. Mangasarian, "Nonlinear Programming," McGraw-Hill, New York, 1969.

[23]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation. I," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, Springer-Verlag, Berlin, 2006, Basic theory.

[24]

L. W. Neustadt, A general theory of extremals, Journal of Computer and System Sciences, 3 (1969), 57-92.

[25]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Wiley Interscience, New York, 1962.

[26]

F. Rampazzo and R. B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA J. Math. Control Inform., 16 (1999), 335-351. doi: doi:10.1093/imamci/16.4.335.

[27]

F. Rampazzo and R. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim., 39 (2000), 989-1007 (electronic). doi: doi:10.1137/S0363012998340223.

[28]

R. Vinter, "Optimal Control," Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2000.

[29]

R. B. Vinter and G. Pappas, A maximum principle for nonsmooth optimal-control problems with state constraints, J. Math. Anal. Appl., 89 (1982), 212-232. doi: doi:10.1016/0022-247X(82)90099-3.

[30]

R. B. Vinter and H. Zheng, Necessary conditions for optimal control problems with state constraints, Trans. Amer. Math. Soc., 350 (1998), 1181-1204. doi: doi:10.1090/S0002-9947-98-02129-1.

[31]

J. Warga, "Optimal Control of Differential and Functional Equations," Academic Press, New York, 1972.

show all references

References:
[1]

A. V. Arutyunov and S. M. Aseev, State constraints in optimal control. The degeneracy phenomenon, Systems Control Lett., 26 (1995), 267-273. doi: doi:10.1016/0167-6911(95)00021-Z.

[2]

A. V. Arutyunov and S. M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim., 35 (1997), 930-952. doi: doi:10.1137/S036301299426996X.

[3]

J. Abadie, On the Kuhn-Tucker theorem, in "Nonlinear Programming" (J. Abadie, ed.), North Holland, Amsterdam, 1967, 21-56.

[4]

A. V. Arutyunov, On necessary conditions for optimality in a problem with phase constraints, Dokl. Akad. Nauk SSSR, 280 (1985), 1033-1037.

[5]

Aram V. Arutyunov, "Optimality Conditions. Abnormal and Degenerate Problems," Mathematics and its Applications, vol. 526, Kluwer Academic Publishers, Dordrecht, 2000, Translated from the Russian by S. A. Vakhrameev.

[6]

A. V. Arutyunov and N. T. Tynyanskiy, The maximum principle in a problem with phase constraints, Izv. Akad. Nauk SSSR Tekhn. Kibernet, (1984), 60-68, 235.

[7]

P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems, J. Differential Equations, 243 (2007), 256-269. doi: doi:10.1016/j.jde.2007.05.005.

[8]

A. Cernea and H. Frankowska, A connection between the maximum principle and dynamic programming for constrained control problems, SIAM J. Control Optim., 44 (2005), 673-703 (electronic). doi: doi:10.1137/S0363012903430585.

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1983, A Wiley-Interscience Publication.

[10]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998.

[11]

A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems under constraints, Dokl. Akad. Nauk SSSR, 149 (1963), 759-762.

[12]

M. d. R. de Pinho, R. B. Vinter and H. Zheng, A maximum principle for optimal control problems with mixed constraints, IMA J. Math. Control Inform., 18 (2001), 189-205. doi: doi:10.1093/imamci/18.2.189.

[13]

M. M. A. Ferreira, F. A. C. C. Fontes and R. B. Vinter, Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints, J. Math. Anal. Appl., 233 (1999), 116-129. doi: doi:10.1006/jmaa.1999.6270.

[14]

F. A. C. C. Fontes, "Optimisation-based Control of Constrained Nonlinear Systems," Ph.D. thesis, Imperial College of Science Technology and Medicine, University of London, London SW7 2BY, U.K., 1999.

[15]

F. A. C. C. Fontes, "Normality in the Necessary Conditions of Optimality for Control Problems with State Constraints," Proceedings of the IASTED Conference on Control and Applications (Cancun, Mexico), 2000.

[16]

F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers, Systems Control Lett., 42 (2001), 127-143. doi: doi:10.1016/S0167-6911(00)00084-0.

[17]

F. A. C. C. Fontes, Nondegenerate necessary conditions of optimality for control problems with state constraints, in "Nonlinear Control Systems" (A. B. Kurzhanski and A. L. Fradkov, eds.), (2002), 45-50.

[18]

M. M. A. Ferreira and R. B. Vinter, When is the maximum principle for state constrained problems nondegenerate?, J. Math. Anal. Appl., 187 (1994), 438-467. doi: doi:10.1006/jmaa.1994.1366.

[19]

F. John, "Extremum Problems as Inequalities as Subsidiary Conditions," Studies and Essays: Courant Anniversary Volume (K. O. Friedrichs, O. E. Neugebauer and J. J. Stoker, eds.), Interscience Publishers, New York, 1948.

[20]

H. W. Kuhn and A. W. Tucker, Nonlinear programming, in "Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability" (J. Neyman, ed.), University of California Press, Berkeley, 1951, 481-492.

[21]

K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems, Optimization, 52 (2003), 75-91. doi: doi:10.1080/0233193021000058940.

[22]

O. L. Mangasarian, "Nonlinear Programming," McGraw-Hill, New York, 1969.

[23]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation. I," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, Springer-Verlag, Berlin, 2006, Basic theory.

[24]

L. W. Neustadt, A general theory of extremals, Journal of Computer and System Sciences, 3 (1969), 57-92.

[25]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Wiley Interscience, New York, 1962.

[26]

F. Rampazzo and R. B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA J. Math. Control Inform., 16 (1999), 335-351. doi: doi:10.1093/imamci/16.4.335.

[27]

F. Rampazzo and R. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim., 39 (2000), 989-1007 (electronic). doi: doi:10.1137/S0363012998340223.

[28]

R. Vinter, "Optimal Control," Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2000.

[29]

R. B. Vinter and G. Pappas, A maximum principle for nonsmooth optimal-control problems with state constraints, J. Math. Anal. Appl., 89 (1982), 212-232. doi: doi:10.1016/0022-247X(82)90099-3.

[30]

R. B. Vinter and H. Zheng, Necessary conditions for optimal control problems with state constraints, Trans. Amer. Math. Soc., 350 (1998), 1181-1204. doi: doi:10.1090/S0002-9947-98-02129-1.

[31]

J. Warga, "Optimal Control of Differential and Functional Equations," Academic Press, New York, 1972.

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