Advertisement

Model Order Reduction Approaches for Infinite Horizon Optimal Control Problems via the HJB Equation

  • Alessandro Alla
  • Andreas Schmidt
  • Bernard Haasdonk
Chapter
Part of the MS&A book series (MS&A, volume 17)

Abstract

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods.

Notes

Acknowledgements

The first author is supported by US Department of Energy grant number DE-SC0009324. The second and third authors thank the Baden Württemberg Stiftung gGmbH and the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.

References

  1. 1.
    Alla, A., Falcone, M.: 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, vol. 164, pp. 1–17. Birkhäuser, Basel (2013)Google Scholar
  2. 2.
    Alla, A., Falcone, M., Kalise, D.: An efficient policy iteration algorithm for dynamic programming equations,. SIAM J. Sci. Comput. 37, 181–200 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alla, A., Falcone, M., Volkwein, S.: Error Analysis for POD approximations of infinite horizon problems via the dynamic programming principle. SIAM J. Control Optim. (to appear)Google Scholar
  4. 4.
    Alla, A., Falcone, M., Kalise, D.: A HJB-POD feedback synthesis approach for wave equation. Bull. Braz. Math. Soc. New Ser. 47, 51–64 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2005)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Basel (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Benner, P., Saak, J.: Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey. GAMM-Mitteilungen 36, 32–52 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Burns, J., Kang, S.: A control problem for Burgers’ equation with bounded input/output. Nonlinear Dyn. 2, 235–262 (1991)CrossRefGoogle Scholar
  9. 9.
    Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Curtain, R.F., Zwart, H.J.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34, 937–969 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi equations. Society for Industrial and Applied Mathematics, Philadelphia (2014)zbMATHGoogle Scholar
  13. 13.
    Garcke, J., Kröner, A.: Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids. J. Sci. Comput. 70(1), 1–28 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grüne, L., Panneck, J.: Nonlinear Model Predictive Control: Theory and Applications. Springer, New York (2011)CrossRefGoogle Scholar
  15. 15.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, Cham (2009)Google Scholar
  16. 16.
    Kalise, D., Kröner, A.: Reduced-order minimum time control of advection-reaction-diffusion systems via dynamic programming. In: Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, pp. 1196–1202 (2014)Google Scholar
  17. 17.
    Kunisch, K., Xie, L.: POD-based feedback control of Burgers equation by solving the evolutionary HJB equation. Comput. Math. Appl. 49, 1113–1126 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kunisch, K., Volkwein, S., Xie, L.: HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 4, 701–722 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Scherpen, J.: Balancing for nonlinear systems. Syst. Control Lett. 21, 143–153 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schmidt, A., Haasdonk, B.: Reduced Basis Approximation of Large Scale Algebraic Riccati Equations. ESAIM: Control, optimisation and Calculus of Variations. EDP Sciences (2017)Google Scholar
  21. 21.
    Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I-II. Q. Appl. Math. XVL, 561–590 (1987)Google Scholar
  22. 22.
    Studinger, A., Volkwein, S.: Numerical analysis of POD a-posteriori error estimation for optimal control. In: Kunisch, K., Bredies, K., Clason, C., von Winckel, G. (eds.) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol. 164, pp. 137–158. Birkhäuser, Basel (2013)CrossRefGoogle Scholar
  23. 23.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Application. American Mathematical Society, Providence (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture Notes, University of Konstanz (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alessandro Alla
    • 1
  • Andreas Schmidt
    • 2
  • Bernard Haasdonk
    • 2
  1. 1.Department of Scientific ComputingFlorida State UniversityTallahasseeUSA
  2. 2.Institute for Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany

Personalised recommendations