We present an algorithm for the approximation of a finite horizon optimal control problem for advection-diffusion equations. The method is based on the coupling between an adaptive POD representation of the solution and a Dynamic Programming approximation scheme for the corresponding evolutive Hamilton–Jacobi equation. We discuss several features regarding the adaptivity of the method, the role of error estimate indicators to choose a time subdivision of the problem and the computation of the basis functions. Some test problems are presented to illustrate the method.
Bitte loggen Sie sich ein, um Zugang zu diesem Inhalt zu erhalten
M. Bardi, I. Capuzzo Dolcetta,
Optimal Control and Viscosity Solutions of Hamilton–Jacobi-Bellman Equations (Birkhäuser, Basel, 1997)
S. Cacace, E. Cristiani, M. Falcone, A. Picarelli, A patchy dynamic programming scheme for a class of Hamilton–Jacobi-Bellman equations. SIAM J. Sci. Comput.
34, 2625–2649 (2012)
E. Carlini, M. Falcone, R. Ferretti, An efficient algorithm for Hamilton–Jacobi equations in high dimension. Comput. Vis. Sci.
7(1), 15–29 (2004)
M. Falcone, Numerical solution of dynamic programming equations, Appendix in
Optimal Control and Viscosity Solutions of Hamilton–Jacobi-Bellman Equations, ed. by M. Bardi, I. Capuzzo Dolcetta (Birkhäuser, Boston, 1997), pp. 471–504
M. Falcone, R. Ferretti,
Semi-Lagrangian Approximation Schemes for Linear and Hamilton–Jacobi Equations (SIAM, Philadelphia, to appear)
M. Falcone, T. Giorgi, An approximation scheme for evolutive Hamilton–Jacobi equations, in
Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, ed. by W.M. McEneaney, G. Yin, Q. Zhang (Birkhäuser, Basel, 1999), pp. 289–303
K. Kunisch, S. Volkwein, Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl.
102, 345–371 (1999)
K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math.
90, 117–148 (2001)
K. Kunisch, S. Volkwein, Optimal snapshot location for computing POD basis functions. ESAIM: M2AN
44, 509–529 (2010)
K. Kunisch, S. Volkwein, L. Xie, HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst.
4, 701–722 (2004)
K. Kunisch, L. Xie, POD-based feedback control of burgers equation by solving the evolutionary HJB equation. Comput. Math. Appl.
49, 1113–1126 (2005)
A.T. Patera, G. Rozza,
Reduced Basis Approximation and a Posteriori Error Estimation for Parameterized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering (2006)
M.L. Rapun, J.M. Vega, Reduced order models based on local POD plus Galerkin projection. J. Comput. Phys.
229, 3046–3063 (2010)
Optimal Control of Partial Differential Equations: Theory, Methods and Applications (AMS, Providence, 2010)