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Second Order Optimality Conditions and Their Role in PDE Control

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Abstract

If \(f: \mathbb{R}^{n} \to \mathbb{R}\) is twice continuously differentiable, f′(u)=0 and f″(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order sufficient optimality condition to the case \(f: U \to \mathbb{R}\), where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How have second order sufficient optimality conditions to be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled?

It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where f″(u) exists can be useless to ensure positive definiteness of the quadratic form v↦f″(u)v 2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the theory is developed for problems in function spaces with simple box constraints of the form α≤u≤β. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of f″(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense.

As a first application of second order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.

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References

  1. Alt, W., Griesse, R., Metla, N., Rösch, A.: Lipschitz stability for elliptic optimal control problems with mixed control-state constraints. Optimization 59(5–6), 833–849 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alt, W., Malanowski, K.: The Lagrange-Newton method for nonlinear optimal control problems. Comput. Optim. Appl. 2, 77–100 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23, 201–229 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bayen, T., Bonnans, J.F., Silva, F.J.: Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations. Trans. Am. Math. Soc. 366(4), 2063–2087 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bonnans, F., Casas, E.: Une principe de Pontryagine pour le contrôle des systèmes semilinéaires elliptiques. J. Differ. Equ. 90, 288–303 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  7. Bonnans, J.F.: Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38(3), 303–325 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bonnans, J.F., Jaisson, P.: Optimal control of a parabolic equation with time-dependent state constraints. SIAM J. Control Optim. 48(7), 4550–4571 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bonnans, J.F., Zidani, H.: Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37(6), 1726–1741 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cartan, H.: Calcul Différentiel. Formes Différentielles. Hermann, Paris (1967)

    Google Scholar 

  11. Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 4, 1309–1322 (1986)

    Article  MathSciNet  Google Scholar 

  12. Casas, E.: Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints. ESAIM Control Optim. Calc. Var. 14, 575–589 (2007)

    Article  MathSciNet  Google Scholar 

  13. Casas, E.: Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems. Adv. Comput. Math. 26(1–3), 137–153 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Casas, E.: Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50(4), 2355–2372 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Casas, E., Chrysafinos, K.: A discontinuous Galerkin time-stepping scheme for the velocity tracking problem. SIAM J. Numer. Anal. 50(5), 2281–2306 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Casas, E., De los Reyes, J.C., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19(2), 616–643 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Casas, E., Dhamo, V.: Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations. Control Cybern. 40(2), 457–490 (2011)

    MathSciNet  Google Scholar 

  18. Casas, E., Herzog, R., Wachsmuth, G.: Approximation of sparse controls in semilinear equations by piecewise linear functions. Numer. Math. 122(4), 645–669 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Casas, E., Herzog, R., Wachsmuth, G.: Optimality conditions and error analysis of semilinear elliptic control problems with L 1 cost functional. SIAM J. Optim. 22(3), 795–820 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Casas, E., Mateos, M.: Second order sufficient optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40, 1431–1454 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Casas, E., Mateos, M.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 39, 265–295 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Casas, E., Mateos, M., Raymond, J.-P.: Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier-Stokes equations. SIAM J. Control Optim. 46(3), 952–982 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Casas, E., Mateos, M., Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31, 193–220 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Casas, E., Ryll, C., Tröltzsch, F.: Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation (2014, submitted)

  25. Casas, E., Ryll, C., Tröltzsch, F.: Sparse optimal control of the Schlögl and FitzHugh-Nagumo systems. Comput. Methods Appl. Math. 13, 415–442 (2014)

    Google Scholar 

  26. Casas, E., Tröltzsch, F.: Second-order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. Appl. Math. Optim. 39, 211–227 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Casas, E., Tröltzsch, F.: Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. 13, 406–431 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Casas, E., Tröltzsch, F.: First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations. SIAM J. Control Optim. 48, 688–718 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Casas, E., Tröltzsch, F.: Recent advances in the analysis of pointwise state-constrained elliptic control problems. ESAIM Control Optim. Calc. Var. 16, 581–600 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22, 261–279 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Casas, E., Tröltzsch, F.: Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls. SIAM J. Control Optim. 52, 1010–1033 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  32. Casas, E., Tröltzsch, F., Unger, A.: Second order sufficient optimality conditions for a nonlinear elliptic control problem. Z. Anal. Anwend. (ZAA) 15, 687–707 (1996)

    Article  MATH  Google Scholar 

  33. Casas, E., Tröltzsch, F., Unger, A.: Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38(5), 1369–1391 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  34. De Los Reyes, J.C., Griesse, R.: State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations. J. Math. Anal. Appl. 343(1), 257–272 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. De los Reyes, J.C., Merino, P., Rehberg, J., Tröltzsch, F.: Optimality conditions for state-constrained PDE control problems with time-dependent controls. Control Cybern. 37, 7–38 (2008)

    Google Scholar 

  36. Dontchev, A.L., Hager, W.W.: Lipschitz stability in nonlinear control and optimization. SIAM J. Control Optim. 31, 569–603 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  37. Dontchev, A.L., Hager, W.W., Poore, A.B., Yang, B.: Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim. 31, 297–326 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  38. Druet, P.E., Klein, O., Sprekels, J., Tröltzsch, F., Yousept, I.: Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects. SIAM J. Control Optim. 49(4), 1707–1736 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. Dunn, J.C.: On second order sufficient optimality conditions for structured nonlinear programs in infinite-dimensional function spaces. In: Fiacco, A. (ed.) Mathematical Programming with Data Perturbations, pp. 83–107. Marcel Dekker, New York (1998)

    Google Scholar 

  40. Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York-London-Sydney (1968)

    MATH  Google Scholar 

  41. Goldberg, H., Tröltzsch, F.: Second order sufficient optimality conditions for a class of non-linear parabolic boundary control problems. SIAM J. Control Optim. 31, 1007–1025 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  42. Goldberg, H., Tröltzsch, F.: On a Lagrange-Newton method for a nonlinear parabolic boundary control problem. Optim. Methods Softw. 8, 225–247 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  43. Griesse, R., Metla, N., Rösch, A.: Local quadratic convergence of SQP for elliptic optimal control problems with mixed control-state constraints. Control Cybern. 39(3), 717–738 (2010)

    MATH  Google Scholar 

  44. Griesse, R., Volkwein, S.: A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system. SIAM J. Control Optim. 44(2), 467–494 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  45. Haller-Dintelmann, R., Meyer, C., Rehberg, J., Schiela, A.: Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60(3), 397–428 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  46. Hintermüller, M., Tröltzsch, F., Yousept, I.: Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numer. Math. 108(4), 571–603 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  47. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. J. Comput. Optim. Appl. 30, 45–63 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  48. Hinze, M., Kunisch, K.: Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40(3), 925–946 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  49. Hinze, M., Kunisch, K.: Second order methods for boundary control of the instationary Navier-Stokes system. Z. Angew. Math. Mech. 84, 171–187 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  50. Hinze, M., Meyer, C.: Stability of semilinear elliptic optimal control problems with pointwise state constraints. Comput. Optim. Appl. 52(1), 87–114 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  51. Hinze, M., Rösch, A.: Discretization of optimal control problems. In: Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S. (eds.) Constrained Optimization and Optimal Control for Partial Differential Equations. ISNM, vol. 160, pp. 391–430. Birkhäuser, Basel (2012)

    Chapter  Google Scholar 

  52. Hinze, M., Tröltzsch, F.: Discrete concepts versus error analysis in PDE constrained optimization. GAMM-Mitt. 33, 148–162 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  53. Ioffe, A.D.: Necessary and sufficient conditions for a local minimum 3: second order conditions and augmented duality. SIAM J. Control Optim. 17, 266–288 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  54. Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

  55. Kammann, E., Tröltzsch, F., Volkwein, S.: A-posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD. ESAIM: Math. Mod. Numer. Anal. 47(2), 555–581 (2013)

    Article  MATH  Google Scholar 

  56. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  57. Krumbiegel, K., Neitzel, I., Rösch, A.: Sufficient optimality conditions for the Moreau-Yosida-type regularization concept applied to semilinear elliptic optimal control problems with pointwise state constraints. Ann. Acad. Rom. Sci. Ser. Math. Appl. 2(2), 222–246 (2010)

    MATH  MathSciNet  Google Scholar 

  58. Krumbiegel, K., Neitzel, I., Rösch, A.: Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints. Comput. Optim. Appl. 52(1), 181–207 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  59. Krumbiegel, K., Rehberg, J.: Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints. SIAM J. Control Optim. 51(1), 304–331 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  60. Lions, J.L.: Contrôle Optimal de Systèmes Gouvernès Par des Équations aux Dérivées Partielles. Dunod, Gauthier-Villars, Paris (1968)

    MATH  Google Scholar 

  61. Luenberger, D.G.: Linear and Nonlinear Programming. Addison Wesley, Reading (1984)

    MATH  Google Scholar 

  62. Malanowski, K.: Stability and sensitivity of solutions to optimal control problems for systems with control appearing linearly. Appl. Math. Optim. 16, 73–91 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  63. Malanowski, K., Tröltzsch, F.: Lipschitz stability of solutions to parametric optimal control for parabolic equations. J. Anal. Appl. (ZAA) 18, 469–489 (1999)

    MATH  Google Scholar 

  64. Malanowski, K., Tröltzsch, F.: Lipschitz stability of solutions to parametric optimal control for elliptic equations. Control Cybern. 29, 237–256 (2000)

    MATH  Google Scholar 

  65. Maurer, H.: First and second order sufficient optimality conditions in mathematical programming and optimal control. Math. Program. Stud. 14, 163–177 (1981)

    Article  MATH  Google Scholar 

  66. Maurer, H., Zowe, J.: First- and second-order conditions in infinite-dimensional programming problems. Math. Program. 16, 98–110 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  67. Meyer, C., Philip, P., Tröltzsch, F.: Optimal control of a semilinear PDE with nonlocal radiation interface conditions. SIAM J. Control Optim. 45, 699–721 (2006)

    Article  MathSciNet  Google Scholar 

  68. Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43, 970–985 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  69. Meyer, C., Yousept, I.: State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions. SIAM J. Control Optim. 48(2), 734–755 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  70. Mittelmann, H.D., Tröltzsch, F.: Sufficient optimality in a parabolic control problem. In: Siddiqi, A.H., Kocvara, M. (eds.) Trends in Industrial and Applied Mathematics, pp. 305–316. Kluwer Academic, Dordrecht (2002)

    Chapter  Google Scholar 

  71. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  72. Osmolovskii, N.P., Maurer, H.: Applications to Regular and Bang-Bang Control. Advances in Design and Control, vol. 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2012). Second-order necessary and sufficient optimality conditions in calculus of variations and optimal control.

    Book  MATH  Google Scholar 

  73. Pieper, K., Vexler, B.: A priori error analysis for discretization of sparse elliptic optimal control problems in measure space. SIAM J. Control Optim. 51(4), 2788–2808 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  74. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)

    MATH  Google Scholar 

  75. Raymond, J.-P., Tröltzsch, F.: Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst. 6, 431–450 (2000)

    Article  MATH  Google Scholar 

  76. Rösch, A.: Second order optimality conditions and stability estimates for the identification of nonlinear heat transfer laws. In: Vorau-Proceedings (1997)

    Google Scholar 

  77. Rösch, A., Tröltzsch, F.: Sufficient second order optimality conditions for a state-constrained optimal control problem of a weakly singular integral equation. Num. Funct. Anal. Appl. 23, 173–193 (2002)

    Article  MATH  Google Scholar 

  78. Rösch, A., Tröltzsch, F.: Sufficient second order optimality conditions for a parabolic optimal control problem with pointwise control-state constraints. SIAM J. Control Optim. 42(1), 138–154 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  79. Rösch, A., Tröltzsch, F.: Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17(3), 776–794 (2006) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  80. Rösch, A., Wachsmuth, D.: Numerical verification of optimality conditions. SIAM J. Control Optim. 47(5), 2557–2581 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  81. Rösch, A., Wachsmuth, D.: A-posteriori error estimates for optimal control problems with state and control constraints. Numer. Math. 120(4), 733–762 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  82. Roubíček, T., Tröltzsch, F.: Lipschitz stability of optimal controls for the steady state Navier-Stokes equations. Control Cybern. 32, 683–705 (2003)

    MATH  Google Scholar 

  83. Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications, vol. 112. Am. Math. Soc., Providence (2010)

    MATH  Google Scholar 

  84. Tröltzsch, F., Wachsmuth, D.: Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. ESAIM Control Optim. Calc. Var. 12, 93–119 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  85. Ulbrich, M.: Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces. MOS-SIAM Series on Optimization, vol. 11. Society for Industrial and Applied Mathematics (SIAM), Mathematical Optimization Society, Philadelphia (2011)

    Book  MATH  Google Scholar 

  86. Ulbrich, S., Ziems, J.C.: Adaptive multilevel inexact SQP methods for PDE-constrained optimization. SIAM J. Optim. 21, 1–40 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  87. Unger, A.: Hinreichende Optimalitätsbedingungen 2. Ordnung und Konvergenz des SQP–Verfahrens für semilineare elliptische Randsteuerprobleme. Ph.D. Thesis, Technische Universität Chemnitz (1997)

  88. Wachsmuth, D., Rösch, A.: How to check numerically the sufficient optimality conditions for infinite-dimensional optimization problems. In: Optimal Control of Coupled Systems of Partial Differential Equations. Internat. Ser. Numer. Math., vol. 158, pp. 297–317. Birkhäuser, Basel (2009)

    Chapter  Google Scholar 

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  1. Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005, Santander, Spain

    Eduardo Casas

  2. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Fredi Tröltzsch

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  1. Eduardo Casas
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Correspondence to Fredi Tröltzsch.

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The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711, the second author by DFG in the framework of the Collaborative Research Center SFB 910, project B6.

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Casas, E., Tröltzsch, F. Second Order Optimality Conditions and Their Role in PDE Control. Jahresber. Dtsch. Math. Ver. 117, 3–44 (2015). https://doi.org/10.1365/s13291-014-0109-3

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