A HJB-POD feedback synthesis approach for the wave equation

Abstract

We propose a computational approach for the solution of an optimal control problem governed by the wave equation. We aim at obtaining approximate feedback laws by means of the application of the dynamic programming principle. Since this methodology is only applicable for low-dimensional dynamical systems, we first introduce a reduced-order model for the wave equation by means of Proper Orthogonal Decomposition. The coupling between the reduced-order model and the related dynamic programming equation allows to obtain the desired approximation of the feedback law. We discuss numerical aspects of the feedback synthesis and providenumerical tests illustrating this approach.

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References

  1. [1]

    A. Alla and M. Falcone. An adaptive POD approximation method for the control of advection-diffusion equations, in K. Kunisch, K. Bredies, C. Clason, G. von Winckel, (eds.) Control and Optimization with PDE Constraints, International Series of Numerical Mathematics, 164, Birkhäuser, Basel, (2013), 1–17.

    Google Scholar 

  2. [2]

    A. Alla and M. Falcone. A Time-Adaptive POD Method for Optimal Control Problems, in the Proceedings of the 1st IFACWorkshop on Control of SystemsModeled by Partial Differential Equations, 1 (2013), 245–250.

    Google Scholar 

  3. [3]

    A. Alla, M. Falcone and D. Kalise. An efficient policy iteration algorithm for dynamic programming equations. SIAM J. Sci. Comput., 37(1) (2015), 181–200.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    A. Alla and M. Hinze. HJB-POD feedback control for Navier-Stokes equations, preprint, available at: http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2014-09.pdf.

  5. [5]

    J.A. Atwell and B.B. King. Proper OrthogonalDecomposition for Reduced Basis Feedback Controllers for Parabolic Equations. Matematical and Computer Modelling, 33 (2001), 1–19.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    S. Ervedoza and E. Zuazua. The wave equation: control and numerics. Control of partial differential equations, Lecture Notes in Math., 2048, Springer, Heidelberg, (2012), 245–339.

    Google Scholar 

  7. [7]

    M. Falcone. Some remarks on the synthesis of feedback controls via numerical methods, in J.L. Menaldi, E. Rofman, A. Sulem (eds), Optimal Control and Partial Differential Equations, IOS Press, (2001), 456–465.

    Google Scholar 

  8. [8]

    M. Falcone and R. Ferretti. Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, (2014).

    Google Scholar 

  9. [9]

    M. Gubisch and S. Volkwein. Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control, preprint, available at: http://kops.ub.uni-konstanz.de/handle/urn:nbn:de:bsz:352-250378.

  10. [10]

    E. Hernández, D. Kalise and E. Otárola. Numerical approximation of the LQR problem in a strongly damped wave equation. Computational Optimization and Applications, 47(1) (2010), 161–178.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    D. Kalise and A. Kröner. Reduced-order minimum time control of advectionreaction-diffusion systems via dynamic programming. Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, (2014), 1196–1202.

    Google Scholar 

  12. [12]

    D. Kalise, A. Kröner and K. Kunisch. Local minimization algorithms for dynamic programming equations, preprint, available at: http://www.ricam.oeaw.ac.at/ publications/reports/15/rep15-04.pdf.

  13. [13]

    A. Kröner, K. Kunisch and H. Zidani. Optimal feedback control of undamped wave equations by solving a HJB equation, preprint, available at: http://www.cmap. polytechnique.fr/kroener/KroenerKunischZidani2013.pdf.

  14. [14]

    K. Kunisch, S. Volkwein and L. Xie. HJB-POD Based Feedback Design for the Optimal Control of Evolution Problems. SIAMJ. on AppliedDynamical Systems, 4 (2004), 701–722.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    K. Kunisch and L. Xie. POD-Based Feedback Control of Burgers Equation by Solving the Evolutionary HJB Equation. Computers andMathematics with Applications, 49 (2005), 1113–1126.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    I. Lasiecka and R. Triggiani. Global exact controllability of semilinearwave equations by a double compactness/uniqueness argument. Discrete Contin. Dyn. Syst., (2005), 556–565.

    Google Scholar 

  17. [17]

    Y. Privat, E. Trélat and E. Zuazua. Optimal location of controllers for the onedimensionalwave equation.Ann. Inst. H. Poincar’éAnal. NonLinéaire, 30 (2013), 1097–1126.

    Article  MATH  Google Scholar 

  18. [18]

    L. Sirovich. Turbulence and the dynamics of coherent structures. Parts I-II. Quarterly of AppliedMathematics, XVL (1987), 561–590.

    Google Scholar 

  19. [19]

    F. Tröltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Application. American Mathematical Society (2010).

    Google Scholar 

  20. [20]

    X. Zhang, C. Zheng and E. Zuazua. Exact controllability of the time discrete wave equation: a multiplier approach. Comput. Methods Appl. Sci., 15 (2010), 229–245.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Alessandro Alla.

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Alla, A., Falcone, M. & Kalise, D. A HJB-POD feedback synthesis approach for the wave equation. Bull Braz Math Soc, New Series 47, 51–64 (2016). https://doi.org/10.1007/s00574-016-0121-6

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Keywords

  • optimal control
  • feedback control
  • dynamic programming
  • Hamilton-Jacobi-Bellman equation
  • Proper Orthogonal Decomposition
  • wave equation

Mathematical subject classification

  • Primary: 49J20, 49N35, 78M34
  • Secondary: 49L20, 93B52