Boundary Control of 2-D Burgers' PDE: An Adaptive Dynamic Programming Approach


In this paper, an adaptive dynamic programming-based near optimal boundary controller is developed for partial differential equations (PDEs) modeled by the uncertain Burgers' equation under Neumann boundary condition in 2-D. Initially, Hamilton-Jacobi-Bellman equation is derived in infinite-dimensional space. Subsequently, a novel neural network (NN) identifier is introduced to approximate the nonlinear dynamics in the 2-D PDE. The optimal control input is derived by online estimation of the value function through an additional NN-based forward-in-time estimation and approximated dynamic model. Novel update laws are developed for estimation of the identifier and value function online. The designed control policy can be applied using a finite number of actuators at the boundaries. Local ultimate boundedness of the closed-loop system is studied in detail using Lyapunov theory. Simulation results confirm the optimizing performance of the proposed controller on an unstable 2-D Burgers' equation.


Electrical and Computer Engineering

Second Department

Mathematics and Statistics


This work was supported in part by NSF under Grant ECCS1128281 and in part by the Intelligent Systems Center.

Keywords and Phrases

Actuators; Boundary conditions; Closed loop systems; Controllers; Dynamical systems; Estimation; Mathematical models; Neural networks; Nonlinear dynamical systems; Nonlinear equations; Optimal control systems; Partial differential equations; Approximate dynamic programming; Boundary controls; Burgers' equations; Optimal controls; Partial Differential Equations (PDEs); Reduced order systems; Stability analysis; Dynamic programming; 2-D partial differential equations (PDEs); Burgers' equation; PDE boundary control

International Standard Serial Number (ISSN)

2162-237X; 2162-2388

Document Type

Article - Journal

Document Version


File Type





© 2018 Institute of Electrical and Electronics Engineers (IEEE), All rights reserved.

Publication Date

01 Aug 2018