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Second-order sufficient conditions for strong solutions to optimal control problems∗

Published online by Cambridge University Press:  14 March 2014

J. Frédéric Bonnans ,
Xavier Dupuis  and
Laurent Pfeiffer
J. Frédéric Bonnans
Affiliation:
Inria Saclay and CMAP, Ecole Polytechnique. Route de Saclay, 91128 Palaiseau Cedex, France. frederic.bonnans@inria.fr; xavier.dupuis@cmap.polytechnique.fr; laurent.pfeiffer@polytechnique.edu
Xavier Dupuis
Affiliation:
Inria Saclay and CMAP, Ecole Polytechnique. Route de Saclay, 91128 Palaiseau Cedex, France. frederic.bonnans@inria.fr; xavier.dupuis@cmap.polytechnique.fr; laurent.pfeiffer@polytechnique.edu
Laurent Pfeiffer
Affiliation:
Inria Saclay and CMAP, Ecole Polytechnique. Route de Saclay, 91128 Palaiseau Cedex, France. frederic.bonnans@inria.fr; xavier.dupuis@cmap.polytechnique.fr; laurent.pfeiffer@polytechnique.edu
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Abstract

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.

Keywords

Optimal controlsecond-order sufficient conditionsquadratic growthbounded strong solutionsPontryagin multiplierspure state and mixed control-state constraintsdecomposition principle

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 704 - 724
Copyright
© EDP Sciences, SMAI, 2014

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References

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).
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).
Bonnans, J.F. and Hermant, A., No-gap second-order optimality conditions for optimal control problems with a single state constraint and control. Math. Program. B 117 (2009) 21–50. Google Scholar
Bonnans, J.F. and Hermant, A., 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. Google Scholar
Bonnans, J.F. and Osmolovskiĭ, N.P., Second-order analysis of optimal control problems with control and initial-final state constraints. J. Convex Anal. 17 (2010) 885–913. Google Scholar
J.F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer-Verlag, New York (2000).
Cominetti, R., Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Opt. 21 (1990) 265–287. Google Scholar
Dmitruk, A.V., 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. Google Scholar
Dubovickiĭ, A.Ja. and Miljutin, A.A., Extremal problems with constraints. Ž. Vyčisl. Mat. i Mat. Fiz. 5 (1965) 395–453. Google Scholar
Dubovickiĭ, A.Ja. and Miljutin, A.A., 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. Google Scholar
M.R. Hestenes, Calculus of variations and optimal control theory. John Wiley & Sons Inc., New York (1966).
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.
Kawasaki, H., 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. Google Scholar
Malanowski, K. and Maurer, H., Sensitivity analysis for optimal control problems subject to higher order state constraints. Optimization with data perturbations II. Ann. Oper. Res. 101 (2001) 43–73. Google Scholar
Mäurer, H., 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. Google Scholar
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).
Osmolovskii, N.P., 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. Google Scholar
Osmolovskii, N.P., Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J. Math. Sci., 173 (2011) 1–106. Google Scholar
Osmolovskii, N.P., Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints. ESAIM: COCV 18 (2012) 452–482. Google Scholar
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).
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).
Stefani, G. and Zezza, P., Optimality conditions for a constrained control problem. SIAM J. Control Optim 34 (1996) 635–659. Google Scholar