Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

Bayen, Térence; Bonnans, J. Frédéric; Silva, Francisco

doi:10.1090/S0002-9947-2013-05961-2

Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations
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by Térence Bayen, J. Frédéric Bonnans and Francisco J. Silva PDF

Trans. Amer. Math. Soc. 366 (2014), 2063-2087 Request permission

Abstract:

In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $J$ in the sense of strong solutions. This means that the function $J$ grows quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin’s principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of the Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Hans Benker and Michael Handschug, An algorithm for abstract optimal control problems using maximum principles and applications to a class of distributed parameter systems, Optimal control (Freiburg, 1991) Internat. Ser. Numer. Math., vol. 111, Birkhäuser, Basel, 1993, pp. 31–42. MR 1297996
J. F. Bonnans, Second-order analysis for control constrained optimal control problems of semilinear elliptic systems, Appl. Math. Optim. 38 (1998), no. 3, 303–325. MR 1646703, DOI 10.1007/s002459900093
J. F. Bonnans, Pontryagin’s principle for the optimal control of semilinear elliptic systems with state constraints, Proceedings of the 30th Conference on Decision and Control, Brigthon, England 2 (December 1991), 1976–1979.
Joseph Frédéric Bonnans, On an algorithm for optimal control using Pontryagin’s maximum principle, SIAM J. Control Optim. 24 (1986), no. 3, 579–588. MR 838058, DOI 10.1137/0324034
Joseph Frédéric Bonnans and Eduardo Casas, Un principe de Pontryagine pour le contrôle des systémes semilinéaires elliptiques, J. Differential Equations 90 (1991), no. 2, 288–303 (French, with English summary). MR 1101241, DOI 10.1016/0022-0396(91)90149-4
Joseph Frédéric Bonnans and Eduardo Casas, Maximum principles in the optimal control of semilinear elliptic systems, 5th IFAC Conference on Distributed Parameter Systems, A. El Jai ed. (Perpignan), Pergamon Press 33-1 (26-29 June 1989), 274–298.
J. Frédéric Bonnans and N. P. Osmolovskiĭ, Second-order analysis of optimal control problems with control and initial-final state constraints, J. Convex Anal. 17 (2010), no. 3-4, 885–913. MR 2731283
J. Frédéric Bonnans and Nikolai P. Osmolovskii, Characterization of a local quadratic growth of the Hamiltonian for control constrained optimal control problems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19 (2012), no. 1-2, 1–16. MR 2918246
J. Frédéric Bonnans and Alexander Shapiro, Perturbation analysis of optimization problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. MR 1756264, DOI 10.1007/978-1-4612-1394-9
J. Frédéric Bonnans and Francisco J. Silva, Asymptotic expansion for the solutions of control constrained semilinear elliptic problems with interior penalties, SIAM J. Control Optim. 49 (2011), no. 6, 2494–2517. MR 2873193, DOI 10.1137/090778602
J. Frédéric Bonnans and Housnaa Zidani, Optimal control problems with partially polyhedric constraints, SIAM J. Control Optim. 37 (1999), no. 6, 1726–1741. MR 1720134, DOI 10.1137/S0363012998333724
Eduardo Casas, Juan Carlos de los Reyes, and Fredi Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim. 19 (2008), no. 2, 616–643. MR 2425032, DOI 10.1137/07068240X
E. Casas and F. Tröltzsch, Second-order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations, Appl. Math. Optim. 39 (1999), no. 2, 211–227. MR 1665676, DOI 10.1007/s002459900104
Eduardo Casas and Fredi Tröltzsch, Second-order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim. 13 (2002), no. 2, 406–431. MR 1951028, DOI 10.1137/S1052623400367698
Eduardo Casas and Fredi Tröltzsch, Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM Control Optim. Calc. Var. 16 (2010), no. 3, 581–600. MR 2674627, DOI 10.1051/cocv/2009010
Eduardo Casas and Fredi Tröltzsch, Second order analysis for optimal control problems: improving results expected from abstract theory, SIAM J. Optim. 22 (2012), no. 1, 261–279. MR 2902693, DOI 10.1137/110840406
E. Casas, F. Tröltzsch, and A. Unger, Second order sufficient optimality conditions for a nonlinear elliptic boundary control problem, Z. Anal. Anwendungen 15 (1996), no. 3, 687–707. MR 1406083, DOI 10.4171/ZAA/723
Eduardo Casas, Fredi Tröltzsch, and Andreas Unger, Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations, SIAM J. Control Optim. 38 (2000), no. 5, 1369–1391. MR 1766420, DOI 10.1137/S0363012997324910
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
A. V. Fursikov, Optimal control of distributed systems. Theory and applications, Translations of Mathematical Monographs, vol. 187, American Mathematical Society, Providence, RI, 2000. Translated from the 1999 Russian original by Tamara Rozhkovskaya. MR 1726442, DOI 10.1090/mmono/187
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29 (1977), no. 4, 615–631. MR 481060, DOI 10.2969/jmsj/02940615
Alan J. Hoffman, On approximate solutions of systems of linear inequalities, J. Research Nat. Bur. Standards 49 (1952), 263–265. MR 0051275, DOI 10.6028/jres.049.027
Xun Jing Li and Jiong Min Yong, Optimal control theory for infinite-dimensional systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. MR 1312364, DOI 10.1007/978-1-4612-4260-4
F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22 (1976), no. 2, 130–185 (French). MR 0423155, DOI 10.1016/0022-1236(76)90017-3
A. A. Milyutin and N. P. Osmolovskii, Calculus of variations and optimal control, Translations of Mathematical Monographs, vol. 180, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Dimitrii Chibisov. MR 1641590, DOI 10.1090/mmono/180
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962. Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt. MR 0166037
U. E. Raĭtum, The maximum principle in optimal control problems for an elliptic equation, Z. Anal. Anwendungen 5 (1986), no. 4, 291–306 (Russian, with English and German summaries). MR 855633, DOI 10.4171/zaa/200
R. Tyrrell Rockafellar, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Lecture Notes in Math., Vol. 543, Springer, Berlin, 1976, pp. 157–207. MR 0512209
A. Rösch and F. Tröltzsch, Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints, SIAM J. Optim. 17 (2006), no. 3, 776–794. MR 2257208, DOI 10.1137/050625850
Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258 (French). MR 192177, DOI 10.5802/aif.204
Fredi Tröltzsch, Optimal control of partial differential equations, Graduate Studies in Mathematics, vol. 112, American Mathematical Society, Providence, RI, 2010. Theory, methods and applications; Translated from the 2005 German original by Jürgen Sprekels. MR 2583281, DOI 10.1090/gsm/112