Systems & Control Letters 39 (2000) 283–294www.elsevier.com/locate/sysconleGlobal stabilization of the Kuramoto–Sivashinsky equation viadistributed output feedback controlPanagiotis D. Christodes∗, Antonios ArmaouDepartment of Chemical Engineering, School of Engineering & Applied Sciences, University of California, Los Angeles,405 Hilgard Avenue, Box 951592, CA 90095, USAReceived 6 February 1999; received in revised form 22 October 1999AbstractThis work addresses the problem of global exponential stabilization of the Kuramoto–Sivashinsky equation (KSE) subjectto periodic boundary conditions via distributed static output feedback control. Under the assumption that the number ofmeasurements is equal to the total number of unstable and critically stable eigenvalues of the KSE and a necessary andsucient stability condition is satised, linear static output feedback controllers are designed that globally exponentiallystabilize the zero solution of the KSE. The controllers are designed on the basis of nite-dimensional approximations ofthe KSE which are obtained through Galerkin’s method. The theoretical results are conrmed by computer simulations ofthe closed-loop system.c 2000 Elsevier Science B.V. All rights reserved.Keywords: Kuramoto–Sivashinsky equation; Distributed control; Global stabilization; Wave suppression1. IntroductionThe KSE is a nonlinear dissipative fourth-order partial dierential equation (PDE) of the form@x@t= −@4x@z4−@2x@z2− x@x@z; (1)where ¿0 is the so-called instability parameter, which describes incipient instabilities in a variety of physicaland chemical systems. Examples include falling liquid lms [8], unstable ame fronts [21,19,22], Belouzov–Zabotinskii reaction patterns [16,17] and interfacial instabilities between two viscous uids [14]. Analyticaland numerical studies of the dynamics of Eq. (1) with periodic boundary conditions (e.g., [23,8,13,15])have revealed the existence of steady and periodic wave solutions, as well as chaotic behavior for very smallvalues of .In addition to the existence of complex solution patterns, the above studies have revealed that the dominantdynamics of the KSE can be adequately characterized by a small number of degrees of freedom (e.g., [23]).Motivated by this, we recently addressed [2,1] the design of linear/nonlinear nite-dimensional output feedback∗Corresponding author. Tel.: 000-310-794-1015; fax: 000-310-206-4107.E-mail address: [email protected] (P.D. Christodes)0167-6911/00/$ - see front matterc 2000 Elsevier Science B.V. All rights reserved.PII: S0167-6911(99)00108-5284 P.D. Christodes, A. Armaou / Systems & Control Letters 39 (2000) 283–294controllers for stabilization of the zero solution of the KSE on the basis of ordinary dierential equation (ODE)approximations, obtained through linear/nonlinear Galerkin’s method, that accurately describe the dominantdynamics. However, even though these control algorithms achieve stabilization of the zero solution of theKSE for any value of the instability parameter , their application is limited to local (i.e., for sucientlysmall initial conditions) stabilization. This limitation motivates the study of the problem of global stabilizationof the KSE. In this direction, a nonlinear boundary feedback controller was proposed in [18] that enhancesthe rate of convergence to the spatially uniform steady state of the KSE for values of  for which this steadystate is naturally stable.In this paper, we consider the problem of global exponential stabilization of the zero solution, x(z; t)=0, ofthe KSE with periodic boundary conditions, for any value of the instability parameter , via distributed staticoutput feedback control. Under the assumptions that the number of measurements is equal to the total numberof unstable and critically stable eigenvalues of the KSE and a necessary and sucient stability conditionis satised, linear static output feedback controllers are designed that achieve global (i.e., for every initialcondition) stabilization of the x(z; t) = 0 solution of Eq. (1). The proposed output feedback controllers aredesigned on the basis of ODE approximations of the KSE obtained through Galerkin’s method. Numericalsimulations of the closed-loop system, for dierent values of the instability parameter, conrm the theoreticalresults.2. PreliminariesWe consider the integrated form of the controlled Kuramoto–Sivashinsky equation@x@t= −@4x@z4−@2x@z2− x@x@z+mXi=1biui(t);ym=Z−s(z)x d z;  =1;:::;p; (2)subject to the periodic boundary conditions@jx@zj(−;t)=@jx@zj(+;t);j=0;:::;3; (3)and the initial conditionx(z; 0) = x0(z); (4)where x ∈ L2p(−; ) is the state of the system, L2p(−; ) denotes the space of square integrable func-tions that satisfy the boundary conditions of Eq. (3) (i.e., L2p(−; )={x ∈L2(−;): (@jx=@zj)(−;t)=(@jx=@zj)(+;t);j=0;:::;3});zis the spatial coordinate,  is the instability parameter which is assumed to beknown, t is the time and 2 is the length of the spatial domain, x0(z) ∈ L2p([ −; ]) is the initial condition, mis the number of manipulated inputs (i.e., variables that can be manipulated externally to modify the dynamicsof Eq. (2) in a desired fashion), ui(t)istheith manipulated input, bi(z) is the actuator distribution function(i.e., bi(z) determines how the control action computed by the ith control actuator, ui(t), is distributed (e.g.,point or distributed actuation) in the spatial interval [ −; ]);ym∈Rdenotes a measured output, and s(z)isa known smooth function of z which is determined by the location and type of the measurement sensors (e.g.,point/distributed sensing). We note that in the case of point actuation (sensing) which inuences (measures)the system at z0(i.e., bi(z)ors(z) is equal to (z − z0) where (·) is the standard Dirac function), weapproximate the function (z − z0) by the nite value 1=2 in the interval [z0− ; z0+ ] (where  is a smallpositive real number) and zero elsewhere in [ − ; ]. Finally, in L2p([ − ; ]), we dene the inner productand norm: (!1;!2)=R−!1(z)!2(z)dz; k!1k2=(!1;!1)1=2where !1;!2are two elements of L2p([ −; ]).P.D. Christodes, A. Armaou / Systems & Control Letters 39 (2000) 283–294 285The dynamics of Eq. (2) depend heavily on the value of . To