A HJB - POD feedback synthesis approach for the wave equation
نویسندگان
چکیده
We propose a computational approach for the solution of an optimal control problem governed by the wave equation. We aim at obtaining approximate feedback laws by means of the application of the dynamic programming principle. Since this methodology is only applicable for low-dimensional dynamical systems, we first introduce a reduced-order model for the wave equation by means of Proper Orthogonal Decomposition. The coupling between the reducedorder model and the related dynamic programming equation allows to obtain the desired approximation of the feedback law. We discuss numerical aspects of the feedback synthesis and provide numerical tests illustrating this approach.
منابع مشابه
Coupling MPC and HJB for the Computation of POD-based Feedback Laws
In this paper we use a reference trajectory computed by a model predictive method to shrink the computational domain where we set the Hamilton-Jacobi Bellman (HJB) equation. Via a reduced-order approach based on proper orthogonal decomposition(POD), this procedure allows for an efficient computation of feedback laws for systems driven by parabolic equations. Some numerical examples illustrate t...
متن کاملOptimal Feedback Control for Undamped Wave Equations by Solving a Hjb Equation
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 ...
متن کاملHJB-POD-Based Feedback Design for the Optimal Control of Evolution Problems
The numerical realization of closed loop control for distributed parameter systems is still a significant challenge and in fact infeasible unless specific structural techniques are employed. In this paper we propose the combination of model reduction techniques based on proper orthogonal decomposition (POD) with the numerical treatment of the Hamilton–Jacobi–Bellman (HJB) equation for infinite ...
متن کاملError Analysis for POD Approximations of Infinite Horizon Problems via the Dynamic Programming Approach
In this paper infinite horizon optimal control problems for nonlinear high-dimensional dynamical systems are studied. Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation which stems from the dynamic programming approach. However, the bottleneck is mainly due to the curse of dime...
متن کاملPolynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs
A procedure for the numerical approximation of high-dimensional Hamilton-JacobiBellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is kn...
متن کامل