Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations

A Correction to this article is available

This article has been updated

Abstract

In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. We then apply the results to the heat and wave equations.

This is a preview of subscription content, log in to check access.

Change history

  • 19 May 2018

    We make corrections to the paper “Optimal Control of Infinite Dimensional Bilinear Systems: Application to the Heat and Wave Equations”, by M.S. Aronna, J.F. Bonnans, and A. Kröner, published in Mathematical Programming 168-1, (2018), pp. 717–757.

References

  1. 1.

    Aronna, M.S., Bonnans, J.F., Kröner, A.: Optimal control of PDEs in a complex space setting; application to the Schrödinger equation. Research report, INRIA (2016)

  2. 2.

    Aronna, M.S., Bonnans, J.F., Dmitruk, A.V., Lotito, P.A.: Quadratic order conditions for bang-singular extremals. Numer. Algebra Control Optim. 2(3), 511–546 (2012). (AIMS Journal, special issue dedicated to Professor Helmut Maurer on the occasion of his 65th birthday)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Aronna, M.S., Bonnans, J.F., Goh, B.S.: Second order analysis of control-affine problems with scalar state constraint. Math. Progr. 160(1–2), 115–147 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Aubin, J.-P.: Un théorème de compacité. C. R. Acad. Sci. Paris 256, 5042–5044 (1963)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Ball, J.M.: Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Am. Math. Soc. 63(2), 370–373 (1977)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and Control of Infinite Dimensional Systems: Systems and Control: Foundations and Applications, vol. 2. Birkhäuser Boston, Inc., Boston (2007)

    Google Scholar 

  7. 7.

    Bergounioux, M., Tiba, D.: General optimality conditions for constrained convex control problems. SIAM J. Control Optim. 34(2), 698–711 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bonnans, J.F.: Optimal control of a semilinear parabolic equation with singular arcs. Optim. Methods Softw. 29(5), 964–978 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bonnans, J.F., Tiba, D.: Control problems with mixed constraints and application to an optimal investment problem. Math. Rep. (Bucur.) 11(61)(4), 293–306 (2009)

    MathSciNet  MATH  Google Scholar 

  10. 10.

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

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Casas, E., Ryll, C., Tröltzsch, F.: Second order and stability analysis for optimal sparse control of the FitzHugh–Nagumo equation. SIAM J. Control Optim. 53(4), 2168–2202 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Casas, E., Tröltzsch, F.: Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math. Ver. 117(1), 3–44 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Dmitruk, A.V.: Quadratic conditions for a weak minimum for singular regimes in optimal control problems. Soviet Math. Dokl. 18(2), 418–422 (1977)

    MATH  Google Scholar 

  14. 14.

    Dmitruk, A.V.: Quadratic conditions for the Pontryagin minimum in an optimal control problem linear with respect to control. II. Theorems on the relaxing of constraints on the equality. Izv. Akad. Nauk SSSR Ser. Mat. 51(4), 812–832 (1987). 911

    Google Scholar 

  15. 15.

    Fattorini, H.O.: Infinite Dimensional Linear Control Systems, North-Holland Mathematics Studies, vol. 201. Elsevier, Amsterdam (2005)

    Google Scholar 

  16. 16.

    Fattorini, H.O., Frankowska, H.: Necessary conditions for infinite-dimensional control problems. Math. Control Signals Syst. 4(1), 41–67 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Frankowska, H., Tonon, D.: The Goh necessary optimality conditions for the Mayer problem with control constraints. In: Decision and Control, pp. 538–543 (2013)

  18. 18.

    Goh, B.S.: The second variation for the singular Bolza problem. J. SIAM Control 4(2), 309–325 (1966)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Goldberg, H., Tröltzsch, F.: Second-order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J. Control Optim. 31(4), 1007–1025 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Hestenes, M.R.: Applications of the theory of quadratic forms in Hilbert space to the calculus of variations. Pac. J. Math. 1, 525–581 (1951)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Kelley, H.J.: A second variation test for singular extremals. AIAA J. 2, 1380–1382 (1964)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Li, X., Yao, Y.: Maximum principle of distributed parameter systems aith time lages, Die Grundlehren der mathematischen Wissenschaften, vol. 181. Springer, New York-Heidelberg, Lecture Notes in Control and Information Science (1985)

  23. 23.

    Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1. Dunod, Paris (1968)

    Google Scholar 

  24. 24.

    Li, X., Yong, J.: Necessary conditions for optimal control of distributed parameter systems. SIAM J. Control Optim. 29(4), 895–908 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Milyutin, A.A., Osmolovskiim, N.P.: 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. MR1641590

  26. 26.

    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Google Scholar 

  27. 27.

    Poggiolini, L., Stefani, G.: Sufficient optimality conditions for a bang-singular extremal in the minimum time problem. Control Cybern. 37(2), 469–490 (2008)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 15(2), 616–634 (electronic) (2004/2005)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to J. Frédéric Bonnans.

Additional information

Dedicated to Terry Rockafellar on the occasion of his 80th birthday.

The second and third author were supported by the project “Optimal control of partial differential equations using parameterizing manifolds, model reduction, and dynamic programming” funded by the Foundation Hadamard/Gaspard Monge Program for Optimization and Operations Research (PGMO).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aronna, M.S., Bonnans, J.F. & Kröner, A. Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations. Math. Program. 168, 717–757 (2018). https://doi.org/10.1007/s10107-016-1093-4

Download citation

Keywords

  • Optimal control
  • Partial differential equations
  • Second-order optimality conditions
  • Goh transform
  • Semigroup theory
  • Heat equation
  • Wave equation
  • Bilinear control systems

Mathematics Subject Classification

  • 49K20
  • 35K05
  • 35L05
  • 90C48