# American Institute of Mathematical Sciences

June  2022, 11(3): 939-973. doi: 10.3934/eect.2021032

## Local stabilization of viscous Burgers equation with memory

 Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai - 400076, India

* Corresponding author: Debanjana Mitra

Received  August 2020 Revised  March 2021 Published  June 2022 Early access  July 2021

Fund Project: The second author acknowledges the support by an Inspire Faculty Fellowship, RD/0118- DSTIN40-001

In this article, we study the local stabilization of the viscous Burgers equation with memory around the steady state zero using localized interior controls. We first consider the linearized equation around zero which corresponds to a system coupled between a parabolic equation and an ODE. We show the feedback stabilization of the system with any exponential decay $-\omega$, where $\omega\in (0, \omega_0)$, for some $\omega_0>0$, using a finite dimensional localized interior control. The control is obtained from the solution of a suitable degenerate Riccati equation. We do an explicit analysis of the spectrum of the corresponding linearized operator. In fact, $\omega_0$ is the unique accumulation point of the spectrum of the operator. We also show that the system is not stabilizable with exponential decay $-\omega$, where $\omega>\omega_0$, using any $L^2$-control. Finally, we obtain the local stabilization result for the nonlinear system by means of the feedback control stabilizing the linearized system using the Banach fixed point theorem.

Citation: Wasim Akram, Debanjana Mitra. Local stabilization of viscous Burgers equation with memory. Evolution Equations and Control Theory, 2022, 11 (3) : 939-973. doi: 10.3934/eect.2021032
##### References:
 [1] 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. [2] V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Differential Integral Equations, 13 (2000), 1393-1412. [3] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128 pp. doi: 10.1090/memo/0852. [4] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.  doi: 10.1512/iumj.2004.53.2445. [5] A. Bensoussan, G. D. Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional System., 2$^{nd}$ edition, Birkhäuser Boston, 2007. doi: 10.1007/978-0-8176-4581-6. [6] T. Breiten and K. Kunisch, Riccati-based feedback control of the monodomain equations with the Fitzhugh-Nagumo model, SICON, 52 (2014), 4057-4081.  doi: 10.1137/140964552. [7] T. Breiten and K. Kunisch, Compensator design for the monodomain equations with the Fitzhugh-Nagumo model, ESAIM: COCV, 23 (2017), 241-262.  doi: 10.1051/cocv/2015047. [8] J.-M. Buchot, J.-P. Raymond and J. Tiago, Coupling estimation and control for a two dimensional Burgers type equation, ESAIM Control Optim. Calc. Var., 21 (2015), 535-560.  doi: 10.1051/cocv/2014037. [9] F. W. Chaves-Silva, X. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM J. Control Optim., 55 (2017), 2437-2459.  doi: 10.1137/151004239. [10] B. Coleman and M. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912. [11] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 2, Functional and Variational Methods, Translated from the French by Ian N. Sneddon, Springer-Verlag, 1988. [12] S. Guerrero and O. Yu. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013. [13] M. Gurtin and A. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Ration. Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373. [14] A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems Control Lett., 61 (2012), 999-1002.  doi: 10.1016/j.sysconle.2012.07.002. [15] S. Ivanov and L. Pandolfi, Heat equation with memory: lack of controllability to rest, J. Math. Anal. Appl., 355 (2009), 1-11.  doi: 10.1016/j.jmaa.2009.01.008. [16] S. Kesavan, Topics in Functional Analysis and Applications, , 2$^{nd}$ edition, New Age International Publishers, 2015. [17] S. Kesavan and J. P. Raymond, On a degenerate Riccati equation, Control Cybernet., 38 (2009), 1393-1410. [18] M. Krstic, On global stabilization of Burgers' equation by boundary control, Systems Control Lett., 37 (1999), 123-141.  doi: 10.1016/S0167-6911(99)00013-4. [19] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I., Abstract parabolic systems. Encyclopedia of Mathematics and its Applications, 74., Cambridge University Press, Cambridge, 2000. [20] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II., Abstract hyperbolic-like systems over a finite time horizon. Encyclopedia of Mathematics and its Applications, 75., Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511574801.002. [21] L. Li, X. Zhou and H. Gao, The stability and exponential stabilization of the heat equation with memory, J. Math. Anal. Appl., 466 (2018), 199-214.  doi: 10.1016/j.jmaa.2018.05.078. [22] I. Munteanu, Stabilization of semilinear heat equations, with fading memory, by boundary feedbacks, J. DifferentialEquations, 259 (2015), 454-472.  doi: 10.1016/j.jde.2015.02.010. [23] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828.  doi: 10.1137/050628726. [24] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Second edition. Texts in Applied Mathematics, 13. Springer-Verlag, New York, 2004. [25] M. Renardy, Mathematical analysis of viscoelastic fluids, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 229–265. doi: 10.1016/S1874-5717(08)00005-4. [26] L. Thevenet, J.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two dimensional Burgers equation, ESAIM: COCV., 16 (2010), 929-955.  doi: 10.1051/cocv/2009028. [27] R. Triggiani, On the stabilizability problem in Banach space, J.Math. Anal. Appl., 52 (1975), 383-403.  doi: 10.1016/0022-247X(75)90067-0. [28] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, , Birkhäuser Advanced Texts, 2009. doi: 10.1007/978-3-7643-8994-9. [29] X. Zhou and H. Gao, Interior approximate and null controllability of the heat equation with memory, Comput. Math. Appl., 67 (2014), 602-613.  doi: 10.1016/j.camwa.2013.12.005. [30] J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition, Modern Birkhäuser Classics, Birkäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

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##### References:
 [1] 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. [2] V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Differential Integral Equations, 13 (2000), 1393-1412. [3] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), x+128 pp. doi: 10.1090/memo/0852. [4] V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.  doi: 10.1512/iumj.2004.53.2445. [5] A. Bensoussan, G. D. Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional System., 2$^{nd}$ edition, Birkhäuser Boston, 2007. doi: 10.1007/978-0-8176-4581-6. [6] T. Breiten and K. Kunisch, Riccati-based feedback control of the monodomain equations with the Fitzhugh-Nagumo model, SICON, 52 (2014), 4057-4081.  doi: 10.1137/140964552. [7] T. Breiten and K. Kunisch, Compensator design for the monodomain equations with the Fitzhugh-Nagumo model, ESAIM: COCV, 23 (2017), 241-262.  doi: 10.1051/cocv/2015047. [8] J.-M. Buchot, J.-P. Raymond and J. Tiago, Coupling estimation and control for a two dimensional Burgers type equation, ESAIM Control Optim. Calc. Var., 21 (2015), 535-560.  doi: 10.1051/cocv/2014037. [9] F. W. Chaves-Silva, X. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM J. Control Optim., 55 (2017), 2437-2459.  doi: 10.1137/151004239. [10] B. Coleman and M. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912. [11] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 2, Functional and Variational Methods, Translated from the French by Ian N. Sneddon, Springer-Verlag, 1988. [12] S. Guerrero and O. Yu. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013. [13] M. Gurtin and A. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Ration. Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373. [14] A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems Control Lett., 61 (2012), 999-1002.  doi: 10.1016/j.sysconle.2012.07.002. [15] S. Ivanov and L. Pandolfi, Heat equation with memory: lack of controllability to rest, J. Math. Anal. Appl., 355 (2009), 1-11.  doi: 10.1016/j.jmaa.2009.01.008. [16] S. Kesavan, Topics in Functional Analysis and Applications, , 2$^{nd}$ edition, New Age International Publishers, 2015. [17] S. Kesavan and J. P. Raymond, On a degenerate Riccati equation, Control Cybernet., 38 (2009), 1393-1410. [18] M. Krstic, On global stabilization of Burgers' equation by boundary control, Systems Control Lett., 37 (1999), 123-141.  doi: 10.1016/S0167-6911(99)00013-4. [19] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I., Abstract parabolic systems. Encyclopedia of Mathematics and its Applications, 74., Cambridge University Press, Cambridge, 2000. [20] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II., Abstract hyperbolic-like systems over a finite time horizon. Encyclopedia of Mathematics and its Applications, 75., Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511574801.002. [21] L. Li, X. Zhou and H. Gao, The stability and exponential stabilization of the heat equation with memory, J. Math. Anal. Appl., 466 (2018), 199-214.  doi: 10.1016/j.jmaa.2018.05.078. [22] I. Munteanu, Stabilization of semilinear heat equations, with fading memory, by boundary feedbacks, J. DifferentialEquations, 259 (2015), 454-472.  doi: 10.1016/j.jde.2015.02.010. [23] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828.  doi: 10.1137/050628726. [24] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Second edition. Texts in Applied Mathematics, 13. Springer-Verlag, New York, 2004. [25] M. Renardy, Mathematical analysis of viscoelastic fluids, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 229–265. doi: 10.1016/S1874-5717(08)00005-4. [26] L. Thevenet, J.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two dimensional Burgers equation, ESAIM: COCV., 16 (2010), 929-955.  doi: 10.1051/cocv/2009028. [27] R. Triggiani, On the stabilizability problem in Banach space, J.Math. Anal. Appl., 52 (1975), 383-403.  doi: 10.1016/0022-247X(75)90067-0. [28] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, , Birkhäuser Advanced Texts, 2009. doi: 10.1007/978-3-7643-8994-9. [29] X. Zhou and H. Gao, Interior approximate and null controllability of the heat equation with memory, Comput. Math. Appl., 67 (2014), 602-613.  doi: 10.1016/j.camwa.2013.12.005. [30] J. Zabczyk, Mathematical Control Theory: An Introduction, Reprint of the 1995 edition, Modern Birkhäuser Classics, Birkäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.
Spectrum of $A$ in one-dimension with $\Omega = (0, 10)$
Spectrum after feedback stabilization, i.e, Spectrum of $(A-BB^*P)$ in one dimension with $\Omega = (0, 10)$
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