POD-Based Economic Optimal Control of Heat-Convection Phenomena

  • Luca MechelliEmail author
  • Stefan Volkwein
Part of the Springer INdAM Series book series (SINDAMS, volume 29)


In the setting of energy efficient building operation, an optimal boundary control problem governed by the heat equation with a convection term is considered together with bilateral control and state constraints. The aim is to keep the temperature in a prescribed range with the least possible heating cost. In order to gain regular Lagrange multipliers a Lavrentiev regularization for the state constraints is utilized. The regularized optimal control problem is solved by a primal-dual active set strategy (PDASS) which can be interpreted as a semismooth Newton method and, therefore, has a superlinear rate of convergence. To speed up the PDASS a reduced-order approach based on proper orthogonal decomposition (POD) is applied. An a-posteriori error analysis ensures that the computed (suboptimal) POD solutions are sufficiently accurate. Numerical test illustrates the efficiency of the proposed strategy.


Convection-diffusion equation Optimal control State constraints Primal-dual active set strategy Model order reduction 



The authors gratefully acknowledge support by the German Science Fund DFG grant VO 1658/4-1 Reduced-Order Methods for Nonlinear Model Predictive Control.


  1. 1.
    Afanasiev, K., Hinze, M.: Adaptive control of a wake flow using proper orthogonal decomposition. In: Shape Optimization and Optimal Design. Lecture Notes in Pure and Applied Mathematics, vol. 216, pp. 317–332. Marcel Dekker, New York (2001)Google Scholar
  2. 2.
    Arian, E., Fahl, M., Sachs, E.W.: Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE (2000)Google Scholar
  3. 3.
    Balay, S., Gropp, W.D., Curfman McInnes, L., Smith, B.F.: Efficient management of parallelism in object oriented numerical software libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (eds.) Modern Software Tools in Scientific Computing, pp. 163–202. Birkhäuser Press, Basel (1997)CrossRefGoogle Scholar
  4. 4.
    Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., Curfman McInnes, L., Rupp, K., Smith, B.F., Zampini, S., Zhang, H.: PETSc Users Manual. ANL-95/11 - Revision 3.7. Argonne National Laboratory, Argonne (2016)Google Scholar
  5. 5.
    Banholzer, S., Beermann, D., Volkwein, S.: POD-based error control for reduced-order bicriterial PDE-constrained optimization. Annu. Rev. Control 44, 226–237 (2017)CrossRefGoogle Scholar
  6. 6.
    Berkooz, G., Holmes, P., Lumley, J.L.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  7. 7.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I. Springer, Berlin (2000)CrossRefGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grimm, E., Gubisch, M., Volkwein, S.: Numerical analysis of optimality-system POD for constrained optimal control. In: Recent Trends in Computational Engineering - CE2014: Optimization, Uncertainty, Parallel Algorithms, Coupled and Complex Problems. Lecture Notes in Computational Science and Engineering, vol. 105, pp. 297–317. Springer, Cham (2015)Google Scholar
  10. 10.
    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control: Theory and Algorithms, 2nd edn. Springer, London (2017)CrossRefGoogle Scholar
  11. 11.
    Gubisch, M.: Model order reduction techniques for the optimal control of parabolic partial differential equations with control and state constraints. Ph.D thesis, Department of Mathematics and Statistics, University of Konstanz. (2017)
  12. 12.
    Gubisch, M., Volkwein, S.: POD a-posteriori error analysis for optimal control problems with mixed control-state constraints. Comput. Optim. Appl. 58, 619–644 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gubisch, M., Volkwein, S.: Proper orthogonal decomposition for linear-quadratic optimal control. In: Ohlberger, M., Benner, P., Cohen, A., Willcox, K. (eds.) Model Reduction and Approximation: Theory and Algorithms, pp. 5–66. SIAM, Philadelphia (2017)Google Scholar
  14. 14.
    Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31(3), 351–362 (2005). MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hintermüller, M., Kopacka, I., Volkwein, S.: Mesh-independence and preconditioning for solving control problems with mixed control-state constraints. ESAIM: COCV 15, 626–652 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hinze, M. , Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2009)zbMATHGoogle Scholar
  18. 18.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)CrossRefGoogle Scholar
  19. 19.
    Krumbiegel, K., Rösch, A.: A virtual control concept for state constrained optimal control problems. Comput. Optim. Appl. 43, 213–233 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kunisch, K., Volkwein, S.: Proper orthogonal decomposition for optimality systems. ESAIM: M2AN 42, 1–23 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefGoogle Scholar
  22. 22.
    Mechelli, L., Volkwein, S.: POD-based economic model predictive control for heat convection phenomena. In: Radu, F.A., Kumar, K., Berre, I., Nordbotten, J.M., Pop, I.S. (eds.) Numerical Mathematics and Advanced Applications ENUMATH 2017. Springer (2018)Google Scholar
  23. 23.
    Roman, J.E., Campos, C., Romero, E., Tomas, A.: SLEPc Users Manual. DSIC-II/24/02 – Revision 3.7. D. Sistemes Informàtics i Computació, Universitat Politècnica de València (2016)Google Scholar
  24. 24.
    Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 22, 616–635 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  26. 26.
    Tröltzsch, F., Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44, 83–115 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, Philadelphia (2011)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of KonstanzDepartment of Mathematics and StatisticsKonstanzGermany

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