Coupling estimation and control for a two dimensional Burgers type equation

Buchot, Jean-Marie; Raymond, Jean-Pierre; Tiago, Jorge

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 535-560.

The aim of this paper is to study the boundary feedback stabilization of a two dimensional Burgers type equation with a Dirichlet boundary control and boundary measurements. Thus we have to deal with highly unbounded control and observation operators. We study the well posedness of the infinite dimensional system obtained by coupling a linear estimator with a linear feedback control law for the corresponding linearized parabolic system in a neighborhood of an unstable stationary solution. We prove the local stabilization of the system obtained by applying to the nonlinear equation the linear feedback control coupled with the linear compensator. Numerical experiments confirm the theoretical results.

M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier–Stokes equations. SIAM J. Control Optim. 48 (2009) 1797–1830. | DOI | Zbl

M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier–Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete Contin. Dyn. Syst. 32 (2012) 1169–1208. | DOI | Zbl

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: application to the Navier–Stokes system. SIAM J. Control Optim. 49 (2011) 420–463. | DOI | Zbl

H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures, modeling, estimation and control. Masson/John Wiley, Paris/Chichester (1996). | Zbl

A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Vol. 1. Birkhäuser, Boston (1992). | Zbl

A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Vol. 2. Birkhäuser, Boston (1992). | Zbl

J. Burns and H. Marrekchi, Optimal fixed-finite-dimensional compensator for Burgers’ equation with unbounded input/output operators, Computation and Control III. Progress System Control Theory. Vol. 15. Birkhauser, Boston, MA (1993) 83–104. | Zbl

J. Burns, A. Balogh, D.S. Gilliam and V.I. Shubov, Numerical Stationary Solutions for a Viscous Burgers’ Equation. J. Math. Syst. Estim. Control 8 (1998) 1–16. | Zbl

C.M. Chen and V. Thomée, The lumped mass finite element method for a parabolic problem. J. Austral. Math. Soc. Ser. B 26 (1985) 329–354. | DOI | Zbl

J.-M. Coron, Control and nonlinearity. American Mathematical Society. Providence, RI (2007). | Zbl

R. Curtain, Finite dimensional compensators for parabolic distributed systems with unbounded control and observation. SIAM J. Control Optim. 22 (1984) 255–276. | DOI | Zbl

R. Curtain, A comparison of finite-dimensional controller designs for distributed parameter systems, Control Theory Adv. Technol. 9 (1993) 609–628.

R. Curtain and D. Salamon, Finite dimensional compensators for infinite dimensional systems with unbounded input operators. SIAM J. Control Optim. 24 (1986) 797–816. | DOI | Zbl

R. Curtain and H.J. Zwart, An introduction to infinite dimensional linear systems theory. Springer-Verlag, New York (1995). | Zbl

H.C. Elman, D.J. Silvester and A.J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, New York (2005). | Zbl

A.V. Fursikov, Optimal control of distributed systems. Theory and Applications. AMS Providence, Rhode Island (2000). | Zbl

P. Grisvard, Caractérisation de quekques espaces d’interpolation. Arch. Rat. Mech. An. 25 (1967) 40–63. | DOI | Zbl

P. Grisvard, Elliptic problems in nonsmooth domains. Pitman Monogr. Studies in Mathematics. Vol. 24. Advanced Publishing Program, Boston, MA (1985). | Zbl

P. Grisvard, Singularities in boundary value problems. Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Vol. 22. Masson, Paris; Springer-Verlag, Berlin (1992). | Zbl

G. Ji and I. Lasiecka, Partially observed analytic systems with fully unbounded actuators and sensors-FEM algorithms. Comput. Optim. Appl. 11 (1998) 111–136. | DOI | Zbl

S. Jund and S. Salmon, Arbitrary high-order finite element schemes and high-order mass lumping. Int. J. Appl. Math. Comput. Sci. 17 (2007) 375–393. | DOI | Zbl

T. Kato, Perturbation theory for linear operators, Reprint of the 1980 Edition. Springer-Verlag (1995). | Zbl

S. Kesavan and J.P. Raymond, On a degenerate Riccati equation. Control Cybern. 38 (2009) 1393–1410. | Zbl

I. Lasiecka, Galerkin approximations of infinite dimensional compensators for flexible structures with unbounded control action. Acta Appl. Math. 28 (1992) 101–133. | DOI | Zbl

I. Lasiecka, Finite element approximations of compensator design for analytic generators with fully unbound controls/observations. SIAM J. Control Optim. 33 (1995) 67–88. | DOI | Zbl

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximations Theories. Vol. I. Cambridge University Press, Cambridge (2000). | Zbl

A.J. Laub, A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control 24 (1979) 913–925. | DOI | Zbl

J.-L. Lions, Contrôle optimal des équations aux dérivées partielles. Dunod, Paris (1968). | Zbl

J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. 2. Springer, Berlin-Heidelberg-New York (1972). | Zbl

A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New-York (1983). | Zbl

J.P. Raymond, Boundary feedback stabilization of the two dimensional Navier–Stokes equations. SIAM J. Control Optim. 45 (2006) 790–828. | DOI | Zbl

J.P. Raymond, Feedback boundary stabilization of the three dimensional incompressible Navier–Stokes equations. J. Math. Pures Appl. 87 (2007) 627–669. | DOI | Zbl

J.P. Raymond, Stabilizability of infinite dimensional systems by finite dimensional controls, submitted.

J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier–Stokes equations with finite dimensional controllers. Discrete Contin. Dyn. Syst. 27 (2010) 1159–1187. | DOI | Zbl

L.F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003). | Zbl

J.M. Schumacher, A direct approach to compensator design for distributed parameter systems. SIAM J. Control and Optim. 21 (1983) 823–836. | DOI | Zbl

L. Thevenet, Lois de feedback pour le contrôle d’écoulements. Ph.D. Thesis, Université de Toulouse (2009).

L. Thevenet, J.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two-dimensional Burgers equation. ESAIM: COCV 16 (2010) 929–955. | Numdam | Zbl

M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhuser Verlag, Basel (2009). | Zbl