An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton−Jacobi Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.
DOI: 10.1051/cocv/2014033
Keywords: Optimal control, wave equation, Hamilton−Jacobi Bellman equation, spectral elements
@article{COCV_2015__21_2_442_0, author = {Kr\"oner, Axel and Kunisch, Karl and Zidani, Hasnaa}, title = {Optimal feedback control for undamped wave equations by solving a {HJB} equation}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {442--464}, publisher = {EDP-Sciences}, volume = {21}, number = {2}, year = {2015}, doi = {10.1051/cocv/2014033}, zbl = {1318.49069}, mrnumber = {3348407}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014033/} }
TY - JOUR AU - Kröner, Axel AU - Kunisch, Karl AU - Zidani, Hasnaa TI - Optimal feedback control for undamped wave equations by solving a HJB equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 442 EP - 464 VL - 21 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014033/ DO - 10.1051/cocv/2014033 LA - en ID - COCV_2015__21_2_442_0 ER -
%0 Journal Article %A Kröner, Axel %A Kunisch, Karl %A Zidani, Hasnaa %T Optimal feedback control for undamped wave equations by solving a HJB equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 442-464 %V 21 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014033/ %R 10.1051/cocv/2014033 %G en %F COCV_2015__21_2_442_0
Kröner, Axel; Kunisch, Karl; Zidani, Hasnaa. Optimal feedback control for undamped wave equations by solving a HJB equation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 442-464. doi : 10.1051/cocv/2014033. http://archive.numdam.org/articles/10.1051/cocv/2014033/
A. Alla and M. Falcone, An adaptive POD approximation method for the control of advection-diffusion equations. Control and Optimization with PDE Constraints. Edited by C. Kunisch, K. von Winckel, G. Bredies, K. Clason. In vol. 164 of Int. Ser. Numer. Math. Springer (2013). | MR | Zbl
A general Hamilton−Jacobi framework for non-linear state-constrained control problems. ESAIM: COCV 19 (2013) 337–357. | Numdam | MR | Zbl
, and ,The linear regulator problem for parabolic systems. SIAM J. Control. Optim. 22 (1984) 499–515. | DOI | MR | Zbl
and ,E. Bänsch and P. Benner, Stabilization of incompressible flow problems by Riccati-based feedback. Constrained Optimization and Optimal Control for Partial Differential Equations. Edited by G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich and S. Ulbrich. Birkhäuser (2012) 5–20. | MR
Internal exponential stabilization to a nonstationary solution for 3d Navier–Stokes equation, SIAM J. Control Optim. 49 (2011) 1454–1478. | DOI | MR | Zbl
, and ,M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton−Jacobi-Bellman Equations. Birkhäuser, Boston (2008). | Zbl
O. Bokanowski, Y. Cheng and C.-W. Shu, A discontinuous Galerkin scheme for front propagation with obstacles. Numer. Math. (2013) 1–31. | MR | Zbl
An efficient data structure and accurate scheme to solve front propagation problems. J. Sci. Comput. 42 (2010) 251–273. | DOI | MR | Zbl
, and ,O. Bokanowski, A. Desilles and H. Zidani, ROC-HJ-Solver. A C++ library for solving HJ equations (2013).
A general Hamilton-Jacobi framework for non-linear state constrained control problems. SIAM J. Control Optim. 48 (2010) 4292–4316.
, and ,An adaptive sparse grid semi-Lagrangian scheme for first order Hamilton-Jacobi Bellman Equations. J. Sci. Comput. 55 (2012) 575–605. | DOI | MR | Zbl
, , and ,Convergence of a non-monotone scheme for Hamilton−Jacobi-Bellman equations with discontinuous initial data. Numer. Math. 115 (2010) 1–44. | DOI | MR | Zbl
, and ,Nonlinear feedback stabilization of a two dimensional Burgers equation. ESAIM: COCV 16 (2010) 929–955. | Numdam | MR | Zbl
, and ,C. Canuto, M.Y. Hussaini., A. Quarteroni and T.A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Scientific Computation. Springer-Verlag, Berlin, Heidelberg (2007). | MR
M. Falcone and R. Ferreti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton−Jacobi Equations. Appl. Math. Society for Industrial and Applied Mathematics (2014). | MR
Internal approximation schemes for optimal control problems in Hilbert spaces J. Math. Systems 7 (1997) 1–25. | MR
,L. Gaudio and A. Quarteroni, Spectral element discretization of optimal control problems. Spectral and High Order Methods for Partial Differential Equations. Edited by J.S. Hesthaven and E.M. Rønquist. In vol. 76 of Lect. Notes in Comput. Sci. Engrg. Springer Berlin Heidelberg (2011) 393–401. | MR | Zbl
Numerical optimal control of the wave equation: Optimal boundary control of a string to rest in finite time. Math. Comput. Simul. 79 (2008) 1020–1032. | DOI | MR | Zbl
, and ,An analysis of optimal modal regulation: convergence and stability. SIAM J. Control. Optim. 19 (1981) 686–707. | DOI | MR | Zbl
,A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems. SIAM J. Control Optim. 29 (1991) 499–515. | DOI | MR | Zbl
,Hamilton−Jacobi equations for control problems of parabolic equations. ESAIM: COCV 12 (2006) 311–349. | Numdam | MR | Zbl
and ,D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications. Capital City Press, Montpelier, Vermont, USA (1991).
L-norm minimal control of the wave equation: on the weakness of the bang-bang principle. ESAIM: COCV 14 (2008) 254–283. | Numdam | MR | Zbl
and ,Numerical approximation of the LQR problem in a strongly damped wave equation. Comput. Optim. Appl. 47 (2010) 161–178. | DOI | MR | Zbl
, and ,A locking-free scheme for the LQR control of a Timoshenko beam. Comput. Appl. Math. 235 (2011) 1383–1393. | DOI | MR | Zbl
, and ,Uniqueness of unbounded viscosity solution of Hamilton−Jacobi equations. Indiana Univ. Math. J. 33 (1984) 721–748. | DOI | MR | Zbl
,On the regularity of solutions of an operator Riccati equation arising in linear quadratic optimal control problems for hereditary differential systems. J. Math. Anal. Appl. 140 (1989) 396–406. | DOI | MR | Zbl
,On the convergence of finite element methods for Hamilton–Jacobi–Bellman Equations. SIAM J. Numer. Anal. 51 (2013) 137–162. | DOI | MR | Zbl
and ,A minimum effort optimal control problem for the wave equation. Comput. Optim. Appl. 57 (2014) 241–270. | DOI | MR | Zbl
and ,Semi-smooth Newton methods for optimal control of the wave equation with control constraints. SIAM J. Control Optim. 49 (2011) 830–858. | DOI | MR | Zbl
, and ,HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Systems 4 (2004) 701–722. | DOI | MR | Zbl
, and ,On time optimal control of the wave equation, its regularization and optimality system. ESAIM: COCV 19 (2013) 317–336. | Numdam | MR | Zbl
and ,POD-based feedback control of the Burgers equation by solving the evolutionary HJB equation. Comput. Math. Appl. 49 (2005) 1113–1126. | DOI | MR | Zbl
and ,I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. In vol. 164. Springer-Verlag (1991) 160. | MR | Zbl
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. In vol. I. Springer-Verlag, Berlin (1972). | MR | Zbl
Explicit solutions for a Riccati equation from transport theory. SIAM J. Matrix Anal. Appl. 30 (2008) 1339–1357. | DOI | MR | Zbl
and ,Essentially nonoscillatory schemes for Hamilton−Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907–922. | DOI | MR | Zbl
and ,A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54 (1984) 468–488. | DOI | Zbl
,Feedback boundary stabilization of the two-dimensional Navier–Stokes equations. SIAM J. Control Optim. 45 (2005) 790–828. | DOI | MR | Zbl
,Fast marching methods. SIAM Reviews 41 (1999) 119–235. | DOI | MR | Zbl
,Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. Henri Poincaré, Section C 10 (1993) 109–129. | DOI | Numdam | MR | Zbl
,Cited by Sources: