Model Order Reduction Approaches for Infinite Horizon Optimal Control Problems via the HJB Equation
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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.
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.
- 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
- 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
- 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.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
- 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.Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I-II. Q. Appl. Math. XVL, 561–590 (1987)Google Scholar
- 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
- 24.Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture Notes, University of Konstanz (2013)Google Scholar