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

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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

Abstract
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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 & Continuous Dynamical Systems - A, 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. 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. doi: doi:10.1137/S036301299426996X.
[3]	J. Abadie, On the Kuhn-Tucker theorem,, in, (1967), 21.
[4]	A. V. Arutyunov, On necessary conditions for optimality in a problem with phase constraints,, Dokl. Akad. Nauk SSSR, 280 (1985), 1033.
[5]	Aram V. Arutyunov, "Optimality Conditions. Abnormal and Degenerate Problems,", Mathematics and its Applications, 526 (2000).
[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.
[7]	P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems,, J. Differential Equations, 243 (2007), 256. 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. doi: doi:10.1137/S0363012903430585.
[9]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).
[10]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).
[11]	A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems under constraints,, Dokl. Akad. Nauk SSSR, 149 (1963), 759.
[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. 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. doi: doi:10.1006/jmaa.1999.6270.
[14]	F. A. C. C. Fontes, "Optimisation-based Control of Constrained Nonlinear Systems,", Ph.D. thesis, (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, (2000).
[16]	F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers,, Systems Control Lett., 42 (2001), 127. 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, (2002), 45.
[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. 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, (1948).
[20]	H. W. Kuhn and A. W. Tucker, Nonlinear programming,, in, (1951), 481.
[21]	K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75. doi: doi:10.1080/0233193021000058940.
[22]	O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill, (1969).
[23]	B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation. I,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330 (2006).
[24]	L. W. Neustadt, A general theory of extremals,, Journal of Computer and System Sciences, 3 (1969), 57.
[25]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Wiley Interscience, (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. 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. doi: doi:10.1137/S0363012998340223.
[28]	R. Vinter, "Optimal Control,", Systems & Control: Foundations & Applications, (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. 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. doi: doi:10.1090/S0002-9947-98-02129-1.
[31]	J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (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. 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. doi: doi:10.1137/S036301299426996X.
[3]	J. Abadie, On the Kuhn-Tucker theorem,, in, (1967), 21.
[4]	A. V. Arutyunov, On necessary conditions for optimality in a problem with phase constraints,, Dokl. Akad. Nauk SSSR, 280 (1985), 1033.
[5]	Aram V. Arutyunov, "Optimality Conditions. Abnormal and Degenerate Problems,", Mathematics and its Applications, 526 (2000).
[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.
[7]	P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems,, J. Differential Equations, 243 (2007), 256. 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. doi: doi:10.1137/S0363012903430585.
[9]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).
[10]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).
[11]	A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems under constraints,, Dokl. Akad. Nauk SSSR, 149 (1963), 759.
[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. 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. doi: doi:10.1006/jmaa.1999.6270.
[14]	F. A. C. C. Fontes, "Optimisation-based Control of Constrained Nonlinear Systems,", Ph.D. thesis, (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, (2000).
[16]	F. A. C. C. Fontes, A general framework to design stabilizing nonlinear model predictive controllers,, Systems Control Lett., 42 (2001), 127. 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, (2002), 45.
[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. 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, (1948).
[20]	H. W. Kuhn and A. W. Tucker, Nonlinear programming,, in, (1951), 481.
[21]	K. Malanowski, On normality of Lagrange multipliers for state constrained optimal control problems,, Optimization, 52 (2003), 75. doi: doi:10.1080/0233193021000058940.
[22]	O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill, (1969).
[23]	B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation. I,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330 (2006).
[24]	L. W. Neustadt, A general theory of extremals,, Journal of Computer and System Sciences, 3 (1969), 57.
[25]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Wiley Interscience, (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. 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. doi: doi:10.1137/S0363012998340223.
[28]	R. Vinter, "Optimal Control,", Systems & Control: Foundations & Applications, (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. 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. doi: doi:10.1090/S0002-9947-98-02129-1.
[31]	J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (1972).

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