Second-order sufficient conditions for strong solutions to optimal control problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, p. 704-724
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

In this article, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem.

DOI : https://doi.org/10.1051/cocv/2013080
Classification:  49K15,  34K35,  90C48
Keywords: optimal control, second-order sufficient conditions, quadratic growth, bounded strong solutions, Pontryagin multipliers, pure state and mixed control-state constraints, decomposition principle
@article{COCV_2014__20_3_704_0,
     author = {Fr\'ed\'eric Bonnans, J. and Dupuis, Xavier and Pfeiffer, Laurent},
     title = {Second-order sufficient conditions for strong solutions to optimal control problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {3},
     year = {2014},
     pages = {704-724},
     doi = {10.1051/cocv/2013080},
     zbl = {1293.49039},
     mrnumber = {3264220},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2014__20_3_704_0}
}
Frédéric Bonnans, J.; Dupuis, Xavier; Pfeiffer, Laurent. Second-order sufficient conditions for strong solutions to optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 704-724. doi : 10.1051/cocv/2013080. http://www.numdam.org/item/COCV_2014__20_3_704_0/

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000). | MR 1857292 | Zbl 0957.49001

[2] J.F. Bonnans, X. Dupuis and L. Pfeiffer, Second-order necessary conditions in Pontryagin form for optimal control problems. Inria Research Report RR-8306. INRIA (2013). | MR 3285893

[3] J.F. Bonnans and A. Hermant, No-gap second-order optimality conditions for optimal control problems with a single state constraint and control. Math. Program. B 117 (2009) 21-50. | MR 2421298 | Zbl 1167.49021

[4] J.F. Bonnans and A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009) 561-598. | Numdam | MR 2504044 | Zbl 1158.49023

[5] J.F. Bonnans and N.P. Osmolovskiĭ, Second-order analysis of optimal control problems with control and initial-final state constraints. J. Convex Anal. 17 (2010) 885-913. | MR 2731283 | Zbl 1207.49020

[6] J.F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer-Verlag, New York (2000). | MR 1756264 | Zbl 0966.49001

[7] R. Cominetti, Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Opt. 21 (1990) 265-287. | MR 1036588 | Zbl 0692.49018

[8] A.V. Dmitruk, Maximum principle for the general optimal control problem with phase and regular mixed constraints. Software and models of systems analysis. Optimal control of dynamical systems. Comput. Math. Model. 4 (1993) 364-377. | MR 1323870

[9] A.Ja. Dubovickiĭ and A.A. Miljutin, Extremal problems with constraints. Ž. Vyčisl. Mat. i Mat. Fiz. 5 (1965) 395-453. | MR 191691

[10] A.Ja. Dubovickiĭ and A.A. Miljutin, Necessary conditions for a weak extremum in optimal control problems with mixed constraints of inequality type. Ž. Vyčisl. Mat. i Mat. Fiz. 8 (1968) 725-779. | MR 236795 | Zbl 0215.21603

[11] M.R. Hestenes, Calculus of variations and optimal control theory. John Wiley & Sons Inc., New York (1966). | MR 203540 | Zbl 0481.49001

[12] R.P. Hettich and H.Th. Jongen, Semi-infinite programming: conditions of optimality and applications, in Optimization techniques, Proc. 8th IFIP Conf., Würzburg 1977, Part 2. Vol. 7 of Lecture Notes in Control and Information Sci. Springer, Berlin (1978) 1-11. | MR 496690 | Zbl 0381.90085

[13] H. Kawasaki, An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems. Math. Program. 41 (Ser. A) (1988) 73-96. | MR 941318 | Zbl 0661.49012

[14] K. Malanowski and H. Maurer, Sensitivity analysis for optimal control problems subject to higher order state constraints. Optimization with data perturbations II. Ann. Oper. Res. 101 (2001) 43-73. | MR 1851988 | Zbl 1005.49021

[15] H. Mäurer, First and second order sufficient optimality conditions in mathematical programming and optimal control. Math. Program. Oberwolfach Proc. Conf. Math. Forschungsinstitut, Oberwolfach (1979). Math. Programming Stud. 14 (1981) 163-177. | MR 600128 | Zbl 0448.90069

[16] A.A. Milyutin and N.P. Osmolovskii, Calculus of variations and optimal control, vol. 180 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1998). | MR 1641590 | Zbl 0911.49001

[17] N.P. Osmolovskii, Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. Optimal control and dynamical systems. J. Math. Sci. 123 (2004) 3987-4122. | MR 2096266 | Zbl 1106.49003

[18] N.P. Osmolovskii, Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J. Math. Sci., 173 (2011) 1-106. | MR 3139265 | Zbl 1234.49032

[19] N.P. Osmolovskii, Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints. ESAIM: COCV 18 (2012) 452-482. | Numdam | MR 2954634 | Zbl 1246.49017

[20] N.P. Osmolovskii and H. Maurer, Applications to regular and bang-bang control, vol. 24 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2012). | MR 3012263 | Zbl 1263.49002

[21] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Translated from the Russian by K.N. Trirogoff, edited by L.W. Neustadt. Interscience Publishers John Wiley & Sons, Inc. New York-London (1962). | MR 166037 | Zbl 0117.31702

[22] G. Stefani and P. Zezza, Optimality conditions for a constrained control problem. SIAM J. Control Optim 34 (1996) 635-659. | MR 1377716 | Zbl 0859.49023