On the Global Dynamics of a Controlled Viscous Burgers' Equation

In this paper we consider a boundary control problem for a forced Burgers' equation in a Hilbert state space consisting of square integrable functions on a finite interval. Our first main result consists in proving the global in time existence of solutions of the closed loop boundary control system for arbitrary L2 initial data and quite general forcing terms (disturbances). For the unforced problem, our main interest is in stability of equilibria, while for the forced problem we are interested in steady state behavior of solutions. We note that the uncontrolled (open loop), unforced problem is not asymptotically stable on L2. However, for positive gains we show that the unforced closed loop system is globally Lyapunov stable and locally exponentially stable. In addition, for nonzero stationary forcing, we show that there is a local absorbing ball and that the corresponding nonlinear semigroup is compact for all t > 0. Using this fact, we are able to deduce several consequences regarding the existence and properties of local attractors. Our method of proof is based on a systematic investigation of the smoothing properties of the controlled dynamics. In particular we establish various regularity results for the dynamics of the closed loop system.

[1]  H. Bateman,et al.  SOME RECENT RESEARCHES ON THE MOTION OF FLUIDS , 1915 .

[2]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[3]  E. Hopf.  The partial differential equation ut + uux = μxx , 1950 .

[4]  川口 光年,et al.  O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Sci. Pub. New York-London, 1963, 184頁, 15×23cm, 3,400円. , 1964 .

[5]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[6]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[7]  O. Ladyzhenskaya,et al.  A dynamical system generated by the Navier-Stokes equations , 1975 .

[8]  R. Temam.  Navier-Stokes Equations , 1977 .

[9]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[10]  T. Mueller,et al.  Experimental Studies of Separation on a Two-Dimensional Airfoil at Low Reynolds Numbers , 1982 .

[11]  M. Vishik,et al.  Attractors of partial differential evolution equations and estimates of their dimension , 1983 .

[12]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[13]  R. Temam,et al.  Attractors Representing Turbulent Flows , 1985 .

[14]  O. Ladyzhenskaya.  On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations , 1987 .

[15]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[16]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[17]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[18]  A. Isidori,et al.  Output regulation of nonlinear systems , 1990 .

[19]  D. Gilliam,et al.  Stability of certain distributed parameter systems by low dimensional controllers: a root locus approach , 1990, 29th IEEE Conference on Decision and Control.

[20]  M. Vishik,et al.  Attractors of partial differential evolution equations in an unbounded domain , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  Christopher I. Byrnes,et al.  Boundary feedback design for nonlinear distributed parameter systems , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[22]  A. Cain,et al.  Evaluation of shear layer cavity resonance mechanisms by numerical simulation , 1992 .

[23]  C. Byrnes.  Root-Locus and Boundary Feedback Design for a Class of Distributed Parameter Systems , 1994 .

[24]  Christopher I. Byrnes,et al.  High Gain Limits of Trajectories and Attractors for a Boundary Controlled Viscous Burgers' Equation , 1998 .

[25]  Y. Yan,et al.  Viscous Scalar Conservation Law with Nonlinear Flux Feedback and Global Attractors , 1998 .

[26]  C. Byrnes,et al.  Semiglobal stabilization of a boundary controlled viscous Burgers' equation , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).