On global stabilization of Burgers' equation by boundary control

Although often referred to as the one-dimensional "cartoon" of Navier-Stokes equation because it does not exhibit turbulence, the Burgers' equation is a natural first step towards developing methods for control of hows. Recent references include Burns and Kang [1], Choi, Temam, Moin, and Kim [3], a series of papers by Byrnes, Gilliam, and Shubov including [2], and Ly, Mease, and Titi [7]. Byrnes et al. [2] show that a linear boundary controller achieves local exponential stability (the initial condition needs to be small in L-2). Ly et al. [7] improve this result (they extend it to L-infinity) but remain local. Achieving a global result for the Burgers' equation is non-trivial because for large initial conditions the quadratic (convective) term-which is negligible in a linear/local analysis-dominates the dynamics. We derive nonlinear boundary control laws that achieve global asymptotic stability (in a very strong sense). We consider both the viscous and the inviscid Burgers' equation, using both Neumann and Dirichlet boundary control. We also study the case where the viscosity parameter is uncertain, as well as the case of stochastic Burgers' equation. For some of the control laws that would require the measurement in the interior of the domain, we develop the observer-based versions. The full paper can be downloaded from the web page www-ames.ucsd.edu/research/krstic/papers/burgers.