# American Institute of Mathematical Sciences

September  2021, 20(9): 3113-3128. doi: 10.3934/cpaa.2021098

## Boundary stabilization of non-diagonal systems by proportional feedback forms

 Alexandru Ioan Cuza University, Department of Mathematics, and Octav Mayer Institute of Mathematics (Romanian Academy), Carol I No. 11, 700506 Iaşi, Romania

Received  January 2021 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: This work was supported by a grant of the Romanian Ministry of Research and Innovation, CNCS – UEFISCDI, project number PN-III-P1-1.1-TE-2019-0348, within PNCDI III

In this work, we are concerned with the problem of boundary exponential stabilization, in a Hilbert space $H$, of parabolic type equations, namely equations for which their linear parts generate analytic $C_0-$semigroups. We consider the case where the projection of the linear leading operator, on a given Riesz basis of $H$, is non-diagonal. We do not assume that the linear operator has compact resolvent. Therefore, the Riesz basis is not necessarily an eigenbasis. The boundary stabilizer is given in a simple linear feedback form, of finite-dimensional structure, involving only the Riesz basis. To illustrate the results, at the end of the paper, we provide an example of stabilization of a fourth-order evolution equation on the half-axis.

Citation: Ionuţ Munteanu. Boundary stabilization of non-diagonal systems by proportional feedback forms. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3113-3128. doi: 10.3934/cpaa.2021098
##### References:
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##### References:
 [1] L. Arnold, H. Crauel and V. Wihstutz, Stabilization of linear systems by noise, SIAM J. Control Optim., 21 (1983), 451-461.  doi: 10.1137/0321027. [2] 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: 10.1137/090778146. [3] V. Barbu, Boundary Stabilization of Equilibrium Solutions to Parabolic Equations, IEEE Trans. Autom. Control, 58 (2013), 2416-2420.  doi: 10.1109/TAC.2013.2254013. [4] V. Barbu, Controllability and Stabilization of Parabolic Equations, Birkhäuser-Springer, Basel, 2018. doi: 10.1007/978-3-319-76666-9. [5] V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high-and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746.  doi: 10.1016/j.na.2005.09.012. [6] M. G. Burgos and L. Teresa, Some results on controllability for linear and nonlinear heat equations in unbounded domains, Adv. Differ. Equ., 12 (2007), 1201-1240. [7] A. V. Filinovskii, Stabilization of the solutions of the wave equation in unbounded domains, Mat. Sb., 187 (1996), 131-160.  doi: 10.1070/SM1996v187n06ABEH000141. [8] A. V. Gorshkov, Stabilization of the one-dimensional heat equation on a semibounded rod, Russ. Math. Surv., 56 (2001), 409-410.  doi: 10.1070/RM2001v056n02ABEH000388. [9] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs (Advances in Design and Control), SIAM, 2008. doi: 10.1137/1.9780898718607. [10] I. Lasiecka and R. Triggiani, Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlin. Anal. Theory Meth. Appl., 121 (2015), 424-446.  doi: 10.1016/j.na.2015.03.012. [11] I. Lasiecka and R. Triggiani, Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations, SIAM J. Control Optim., 21 (1983), 766-803.  doi: 10.1137/0321047. [12] H. Lian, J. Zhao and R. P. Agarwal, Upper and lower solution method for nth-order BVPs on an infinite interval, Bound. Value Probl. (2014), 17 pp. doi: 10.1186/1687-2770-2014-100. [13] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\mathbb{R}^n$, Studia Math., 128 (1998), 171-198.  doi: 10.4064/sm-128-2-171-198. [14] S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. Am. Math. Soc., 353 (2001), 1635-1659.  doi: 10.1090/S0002-9947-00-02665-9. [15] F. Kh. Mukminov and I. M. Bikkulov, Stabilization of the norm of the solution of a mixed problem in an unbounded domain for parabolic equations of orders 4 and 6, Mat. Sb., 195 (2004), 115-142.  doi: 10.1070/SM2004v195n03ABEH000810. [16] I. Munteanu, Boundary Stabilization of Parabolic Equations, Birkhauser-Springer, Basel, 2019. doi: 10.1007/978-3-030-11099-4. [17] I. Munteanu, Exponential stabilization of the semilinear heat equation with nonlocal boundary conditions, J. Math. Analysis Appl., 492 (2020), art. no. 124512. doi: 10.1016/j.jmaa.2020.124512. [18] A. Preumont, Collocated versus non-collocated control. In: Vibration Control of Active Structures. Solid Mechanics and Its Applications, Springer, Dordrecht, 1997. doi: 10.1007/978-94-011-5654-7. [19] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies Appl. Math., 52 (1973), 189-211.  doi: 10.1002/sapm1973523189. [20] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095. [21] R. Triggiani, Boundary feedback stabilization of parabolic equations, Appl. Math. Optim., 6 (1980), 201-220.  doi: 10.1007/BF01442895.
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