Burgers ’ Equation with Nonlinear Boundary Feedback " H Stability , WelI-Posedness and Simulation

We consider the viscous Burgers' equation under recently proposed nonlinear boundary conditions and show that it guarantees global asymptotic stabilization and semiglobal exponential stabilization in H1 sense. Our result is global in time and allows arbitrary size of initial data. It strengthens recent results by Byrnes, Gilliam, and Shubov, Ly, Mease, and Titi, and Ito and Yan. The global existence and uniqueness of classical solutions follows from the general theory of quasi-linear parabolic equations. We include a numerical result which illustrates the performance of the boundary controller.


INTRODUCTION
Burgers' equation is a natural first step towards developing methods for control of flows.Recent references by Burns and Kang [1], Byrnes et al. [3,4], Ly et al. [12], and Ito and Yan [8] achieve progress in local stabilization and global analysis of attractors.The problem of global exponential stabilization in L 2 norm was first addressed by Krsti6 [9].This problem is non-trivial because for large initial conditions the qua- dratic (convective) term which is negligible in a linear/local analysis dominates the dynamics.Linear boundary conditions do not always ensure global exponential stability [4] or prevent finite blow-up [5] in the case of nonlinear reaction-diffusion equations.Nonlinear bound- ary conditions might cause finite blow-up [11], even for the simple heat equation [7].
With the introduction of cubic Neumann boundary feedback con- trol we obtain a closed loop system which is globally asymptotically stable and semi-globally exponentially stable in H norm and, hence in maximum norm whenever the initial data is compatible with the equa- tion and the boundary conditions.
For clarity, our treatment does not include external forcing as in [3,4,8,12].External forcing would preclude equilibrium stability but one could still establish appropriate forms of disturbance attenuation and regularity of solution.
The choice of Wx at the boundary as the control input is motivated by physical considerations.For example, in thermal problems one cannot actuate the temperature w, but only the heat flux Wx.This makes the stabilization problem non-trivial because, as Byrnes et al.
[3] argue, homogeneous Neumann boundary conditions make any con- stant profile an equilibrium solution, thus preventing not only global but even local asymptotic stability.Even mixed linear boundary condi- tions can introduce multiple stationary solutions [2].

DEFINITION
The zero solution of a dynamical system is said to be globally asymptotically stable in an E spatial norm if IIw(t)ll t), vt 0, ( where 13(., .) is a class 1C.function, i.e., a function with the properties that forfixed t, (r, t) is a monotonically increasing continuous function of r such that 3(0, t) =_ 0; forfixed r, (r, t) is a monotonically decreasing continuous function of such that limt+oo (r, t) 0.
(2.13) Since 7-/n-C n for n integer, the theorem assumes initial data smoother than C 2 (but not necessarily as smooth as C3).Specifically, the initial data need to satisfy sup Iwg(x)wg(y)l < (2.14) x,yE[0,1l

GLOBAL ASYMPTOTIC STABILITY
While irrelevant for finite-dimensional systems where all vector norms are equivalent, for PDEs, the question of the type of norm with respect to which one wants to establish stability is a delicate one.Any meaningful stability claim should imply boundedness of solutions.We first establish global exponential stability in L q for any q E [2, ), which does not guarantee boundedness.Then we show global asymp- totic (plus local exponential) stability in an Hi-like sense which, by com- bining Agmon's and Poincar6's inequalities, guarantees boundedness.
(3.1) Thus the solution w(x, t)=_ 0 is globally exponentially stable in an L q sense for any q E [2, ).Letting p z in (3.5), we get ess sup Iw(x, t)l _< ess sup Iw(x,O)l, This result is not particularly useful for two reasons: (1) The above estimate does not guarantee convergence to zero (it guarantees stability but not asymptotic stability).
(2) Without additional effort to establish continuity, with ess sup we cannot guarantee boundedness for all (but only for almost all) xE[O, 1].
The existence of classical solutions follows from Theorem 7.4 in [10], Chapter V.This Theorem establishes, for a more general quasi-linear parabolic boundary value problem, the existence of a unique solution in the H61der space of functions -[2+l'1+l/2([0, 1] x [0, T]) for some />0.Since 7-/2+l'1+t/2([0, 1] x [0, T])c C2'1([0, 1] x [0, T]), we obtain the existence of classical solutions for time intervals [0, T], where T > 0 is arbitrarily large.The proof in [10] is based on linearization of the sys- tem, and on application ofthe Leray-Schauder theorem on fixed points.It is important to note that a crucial step in the proof is establishing uniform a priori estimates for the system.These estimates are for the H61der norms of solutions and hence are different from our Sobolev type energy estimates.The H61der estimates establish boundedness of solutions, while our energy estimates establish stability.The existence of strong (but not necessarily classical) solutions was proved in [8] using a different method.

SIMULATION EXAMPLE
It is well known (see, e.g.[2,6]) that nonlinear problems, especially fluid dynamical problems, require extremely careful numerical analy- sis.Typically there is a trade-off between convergence, accuracy and numerical oscillation.This is the case in particular when the initial data is large relative to the viscosity coefficient e in Burgers' equation.
Higher order methods are preferred to lower order methods only when the time and/or spatial step sizes are sufficiently small, where the small- ness is a delicate question.It is not the purpose of our paper to find the best approximation scheme for our problem, simply to demonstrate our theoretical results.Our numerical simulation is based on an uncon- ditionally stable, fully implicit scheme of second order accuracy, using three time level quadratic approximation in time and central difference scheme in space.The simulations were carried out on various platforms using several different numerical packages (OCTAVE, SCILAB, MATLAB), and they show grid independence for sufficiently small time and spatial grid.
We consider first Burgers' equation (2.1) with zero Neumann bound- ary condition (uncontrolled system) and then the regulation error system (2.4)-(2.7)with e 0.1 and with initial data Wo(X) Wo(x) Wa, where Wa= 3 and W0(x)= 20(0.5-x)3.The uncontrolled system is shown in Fig. l(a).The solution seems to converge to a nonzero "equilibrium" profile, although it eventually approaches zero, which could be seen only for >> (in fact, for some initial data, the numer- ical solution gets trapped into this profile and never converges to zero [2]).This unsatisfactory behavior is remedied by applying boundary feedback, as shown in Fig. (b). ) ) < llw(t)ll + kll w(t)I[ A(t).
The estimation of the various terms follows: e W 2 2