Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

Christophe Zhang

doi:10.3934/mcrf.2021006

Previous Article
Optimal control of the transmission rate in compartmental epidemics
MCRF Home
This Issue
Next Article
Noise and stability in reaction-diffusion equations

March 2022, 12(1): 169-200. doi: 10.3934/mcrf.2021006

Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

Christophe Zhang ^,

Chair for Applied Analysis (Alexander von Humboldt Professorship), Friedrich-Alexander Universität Nürnberg, Cauerstr. 11, 91058 Erlangen

^* Corresponding author: Christophe Zhang

Received September 2020 Revised November 2020 Published March 2022 Early access March 2021

Fund Project: This work was partially supported by ANR project Finite4SoS (ANR-15-CE23-0007), the French Corps des Mines, and the Chair of Applied Analysis, Alexander von Humboldt Professorship, Friedrich-Alexander Universität Nürnberg

We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval $ (0,L) $, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than $ 1 $, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.

Keywords: Backstepping, system equivalence, Fredholm transformations, transport equation, stabilization, rapid stabilization, internal control.

Mathematics Subject Classification: 35L02, 93D15, 93B17, 93B30, 93D23.

Citation: Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2022, 12 (1) : 169-200. doi: 10.3934/mcrf.2021006

References:

[1]	D. M. Bosković, A. Balogh and M. Krstić, Backstepping in infinite dimension for a class of parabolic distributed parameter systems, Math. Control Signals Systems, 16 (2003), 44-75. doi: 10.1007/s00498-003-0128-6. Google Scholar
[2]	G. Bastin and J. -M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, vol. 88, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-32062-5. Google Scholar
[3]	A. Balogh and M. Krstić, Infinite dimensional backstepping–style feedback transformations for a heat equation with an arbitrary level of instability, European Journal of Control, 8 (2002), 165-175. doi: 10.3166/ejc.8.165-175. Google Scholar
[4]	S. Bialas, On the Lyapunov matrix equation, IEEE Trans. Automat. Control, 25 (1980), 813-814. doi: 10.1109/TAC.1980.1102438. Google Scholar
[5]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar
[6]	P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188. Google Scholar
[7]	O. Christensen, An Introduction to Frames and Riesz Bases, 2nd edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-25613-9. Google Scholar
[8]	J.-M. Coron and B. d'Andréa-Novel, Stabilization of a rotating body beam without damping, IEEE Trans. Automat. Control, 43 (1998), 608-618. doi: 10.1109/9.668828. Google Scholar
[9]	J.-M. Coron, L. Gagnon and M. Morancey, Rapid stabilization of a linearized bilinear 1-D Schrödinger equation, J. Math. Pures Appl. (9), 115 (2018), 24-73. doi: 10.1016/j.matpur.2017.10.006. Google Scholar
[10]	J.-M. Coron, L. Hu and G. Olive, Stabilization and controllability of first-order integro-differential hyperbolic equations, J. Funct. Anal., 271 (2016), 3554-3587. doi: 10.1016/j.jfa.2016.08.018. Google Scholar
[11]	J.-M. Coron, L. Hu and G. Olive, Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation, Automatica J. IFAC, 84 (2017), 95-100. doi: 10.1016/j.automatica.2017.05.013. Google Scholar
[12]	J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl. (9), 102 (2014), 1080-1120. doi: 10.1016/j.matpur.2014.03.004. Google Scholar
[13]	J.-M. Coron and Q. Lü, Fredholm transform and local rapid stabilization for a Kuramoto-Sivashinsky equation, J. Differential Equations, 259 (2015), 3683-3729. doi: 10.1016/j.jde.2015.05.001. Google Scholar
[14]	J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach, Arch. Ration. Mech. Anal., 225 (2017), 993-1023. doi: 10.1007/s00205-017-1119-y. Google Scholar
[15]	J. -M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136. Google Scholar
[16]	J.-M. Coron, Stabilization of control systems and nonlinearities, In Proceedings of the 8th International Congress on Industrial and Applied Mathematics, Higher Ed. Press, Beijing, 2015. Google Scholar
[17]	J.-M. Coron, R. Vazquez, M. Krstić and G. Bastin, Local exponential $H^2$ stabilization of a $2\times 2$ quasilinear hyperbolic system using backstepping, SIAM J. Control Optim., 51 (2013), 2005-2035. doi: 10.1137/120875739. Google Scholar
[18]	R. Datko, A linear control problem in an abstract Hilbert space, J. Differential Equations, 9 (1971), 346-359. doi: 10.1016/0022-0396(71)90087-8. Google Scholar
[19]	F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, In 1999 European Control Conference (ECC), IEEE, (1999), 3232–3237. doi: 10.23919/ECC. 1999.7099825. Google Scholar
[20]	L. Grafakos, Classical Fourier Analysis, vol. 2, Springer, New York, 2008. Google Scholar
[21]	A. Hayat, Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions, preprint, October 2017. https://hal.archives-ouvertes.fr/hal-01613139 Google Scholar
[22]	A. Hayat, On boundary stability of inhomogeneous $2\times2$ 1-D hyperbolic systems for the $C^1$ norm, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 82, 31 pp. doi: 10.1051/cocv/2018059. Google Scholar
[23]	D. Kleinman, An easy way to stabilize a linear constant system, IEEE Transactions on Automatic Control, 15 (1970), 692-692. Google Scholar
[24]	M. Krstić, B.-Z. Guo, A. Balogh and A. Smyshlyaev, Output-feedback stabilization of an unstable wave equation, Automatica J. IFAC, 44 (2008), 63-74. doi: 10.1016/j.automatica.2007.05.012. Google Scholar
[25]	V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613. doi: 10.1137/S0363012996301609. Google Scholar
[26]	M. Krstić and A. Smyshlyaev, Boundary Control of PDEs, Advances in Design and Control, vol. 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607. Google Scholar
[27]	M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., New York, NY, 1995. Google Scholar
[28]	W. H. Kwon and A. E. Pearson, A note on the algebraic matrix Riccati equation, IEEE Trans. Automatic Control, AC-22 (1977), 143-144. doi: 10.1109/tac.1977.1101441. Google Scholar
[29]	I. Lasiecka and R. Triggiani, Algebraic Riccati equations arising in boundary/point control: A review of theoretical and numerical results. I. Continuous case, In Perspectives in Control Theory (Sielpia, 1988), Progr. Systems Control Theory, volume 2, Birkhäuser Boston, Boston, MA, 1990, 175–210. Google Scholar
[30]	W. J. Liu and M. Krstić, Backstepping boundary control of Burgers' equation with actuator dynamics, Systems Control Lett., 41 (2000), 291-303. doi: 10.1016/S0167-6911(00)00068-2. Google Scholar
[31]	W. J. Liu and M. Krstić, Boundary feedback stabilization of homogeneous equilibria in unstable fluid mixtures, Internat. J. Control, 80 (2007), 982-989. doi: 10.1080/00207170701280895. Google Scholar
[32]	Y. Orlov and D. Dochain, Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor, IEEE Trans. Automat. Control, 47 (2002), 1293-1304. doi: 10.1109/TAC.2002.800737. Google Scholar
[33]	D. L. Lukes, Stabilizability and optimal control, Funkcial. Ekvac., 11 (1968), 39-50. Google Scholar
[34]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar
[35]	J. L. Pitarch, M. Rakhshan, M. Mardani, M. Sadeghi and C. Prada, Distributed nonlinear control of a plug-flow reactor under saturation, IFAC-PapersOnLine, 49 (2016), 87-92. doi: 10.1016/j.ifacol.2016.10.760. Google Scholar
[36]	R. Rebarber, Spectral assignability for distributed parameter systems with unbounded scalar control, SIAM J. Control Optim., 27 (1989), 148-169. doi: 10.1137/0327009. Google Scholar
[37]	D. L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl., 40 (1972), 336-368. doi: 10.1016/0022-247X(72)90055-8. Google Scholar
[38]	D. L. Russell, Canonical forms and spectral determination for a class of hyperbolic distributed parameter control systems, J. Math. Anal. Appl., 62 (1978), 186-225. doi: 10.1016/0022-247X(78)90229-9. Google Scholar
[39]	A. Smyshlyaev, E. Cerpa and M. Krstić, Boundary stabilization of a 1-D wave equation with in-domain antidamping, SIAM J. Control Optim., 48 (2010), 4014-4031. doi: 10.1137/080742646. Google Scholar
[40]	E. D. Sontag, Mathematical Control Theory, Texts in Applied Mathematics, vol. 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7. Google Scholar
[41]	S. H. Sun, On spectrum distribution of completely controllable linear systems, SIAM J. Control Optim., 19 (1981), 730-743. doi: 10.1137/0319048. Google Scholar
[42]	D. Tsubakino, M. Krstić, and S. Hara, Backstepping control for parabolic pdes with in-domain actuation, 2012 American Control Conference (ACC), (2012), 2226–2231. doi: 10.1109/ACC. 2012.6315358. Google Scholar
[43]	J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., 43 (2005), 2233-2244. doi: 10.1137/S0363012901388452. Google Scholar
[44]	A. Vest, Rapid stabilization in a semigroup framework, SIAM J. Control Optim., 51 (2013), 4169-4188. doi: 10.1137/130906994. Google Scholar
[45]	F. Woittennek, S. Q. Wang and T. Knüppel, Backstepping design for parabolic systems with in-domain actuation and Robin boundary conditions, IFAC Proceedings Volumes, 19th IFAC World Congress, 47 (2014), 5175–5180. doi: 10.3182/20140824-6-ZA-1003.02285. Google Scholar
[46]	S. Q. Xiang, Null controllability of a linearized Korteweg–de Vries equation by backstepping approach, SIAM J. Control Optim., 57 (2019), 1493-1515. doi: 10.1137/17M1115253. Google Scholar
[47]	S. Q. Xiang, Small-time local stabilization for a Korteweg–de Vries equation, Systems Control Lett., 111 (2018), 64-69. doi: 10.1016/j.sysconle.2017.11.003. Google Scholar
[48]	X. Yu, C. Xu, H. C. Jiang, A. Ganesan and G. J. Zheng., Backstepping synthesis for feedback control of first-order hyperbolic PDEs with spatial-temporal actuation, Abstr. Appl. Anal., (2014), Art. ID 643640, 13 pp. doi: 10.1155/2014/643640. Google Scholar
[49]	J. Zabczyk, Mathematical Control Theory, Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9. Google Scholar

show all references

References:

[1]	D. M. Bosković, A. Balogh and M. Krstić, Backstepping in infinite dimension for a class of parabolic distributed parameter systems, Math. Control Signals Systems, 16 (2003), 44-75. doi: 10.1007/s00498-003-0128-6. Google Scholar
[2]	G. Bastin and J. -M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, vol. 88, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-32062-5. Google Scholar
[3]	A. Balogh and M. Krstić, Infinite dimensional backstepping–style feedback transformations for a heat equation with an arbitrary level of instability, European Journal of Control, 8 (2002), 165-175. doi: 10.3166/ejc.8.165-175. Google Scholar
[4]	S. Bialas, On the Lyapunov matrix equation, IEEE Trans. Automat. Control, 25 (1980), 813-814. doi: 10.1109/TAC.1980.1102438. Google Scholar
[5]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar
[6]	P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188. Google Scholar
[7]	O. Christensen, An Introduction to Frames and Riesz Bases, 2nd edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-25613-9. Google Scholar
[8]	J.-M. Coron and B. d'Andréa-Novel, Stabilization of a rotating body beam without damping, IEEE Trans. Automat. Control, 43 (1998), 608-618. doi: 10.1109/9.668828. Google Scholar
[9]	J.-M. Coron, L. Gagnon and M. Morancey, Rapid stabilization of a linearized bilinear 1-D Schrödinger equation, J. Math. Pures Appl. (9), 115 (2018), 24-73. doi: 10.1016/j.matpur.2017.10.006. Google Scholar
[10]	J.-M. Coron, L. Hu and G. Olive, Stabilization and controllability of first-order integro-differential hyperbolic equations, J. Funct. Anal., 271 (2016), 3554-3587. doi: 10.1016/j.jfa.2016.08.018. Google Scholar
[11]	J.-M. Coron, L. Hu and G. Olive, Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation, Automatica J. IFAC, 84 (2017), 95-100. doi: 10.1016/j.automatica.2017.05.013. Google Scholar
[12]	J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl. (9), 102 (2014), 1080-1120. doi: 10.1016/j.matpur.2014.03.004. Google Scholar
[13]	J.-M. Coron and Q. Lü, Fredholm transform and local rapid stabilization for a Kuramoto-Sivashinsky equation, J. Differential Equations, 259 (2015), 3683-3729. doi: 10.1016/j.jde.2015.05.001. Google Scholar
[14]	J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach, Arch. Ration. Mech. Anal., 225 (2017), 993-1023. doi: 10.1007/s00205-017-1119-y. Google Scholar
[15]	J. -M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136. Google Scholar
[16]	J.-M. Coron, Stabilization of control systems and nonlinearities, In Proceedings of the 8th International Congress on Industrial and Applied Mathematics, Higher Ed. Press, Beijing, 2015. Google Scholar
[17]	J.-M. Coron, R. Vazquez, M. Krstić and G. Bastin, Local exponential $H^2$ stabilization of a $2\times 2$ quasilinear hyperbolic system using backstepping, SIAM J. Control Optim., 51 (2013), 2005-2035. doi: 10.1137/120875739. Google Scholar
[18]	R. Datko, A linear control problem in an abstract Hilbert space, J. Differential Equations, 9 (1971), 346-359. doi: 10.1016/0022-0396(71)90087-8. Google Scholar
[19]	F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, In 1999 European Control Conference (ECC), IEEE, (1999), 3232–3237. doi: 10.23919/ECC. 1999.7099825. Google Scholar
[20]	L. Grafakos, Classical Fourier Analysis, vol. 2, Springer, New York, 2008. Google Scholar
[21]	A. Hayat, Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions, preprint, October 2017. https://hal.archives-ouvertes.fr/hal-01613139 Google Scholar
[22]	A. Hayat, On boundary stability of inhomogeneous $2\times2$ 1-D hyperbolic systems for the $C^1$ norm, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 82, 31 pp. doi: 10.1051/cocv/2018059. Google Scholar
[23]	D. Kleinman, An easy way to stabilize a linear constant system, IEEE Transactions on Automatic Control, 15 (1970), 692-692. Google Scholar
[24]	M. Krstić, B.-Z. Guo, A. Balogh and A. Smyshlyaev, Output-feedback stabilization of an unstable wave equation, Automatica J. IFAC, 44 (2008), 63-74. doi: 10.1016/j.automatica.2007.05.012. Google Scholar
[25]	V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613. doi: 10.1137/S0363012996301609. Google Scholar
[26]	M. Krstić and A. Smyshlyaev, Boundary Control of PDEs, Advances in Design and Control, vol. 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607. Google Scholar
[27]	M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., New York, NY, 1995. Google Scholar
[28]	W. H. Kwon and A. E. Pearson, A note on the algebraic matrix Riccati equation, IEEE Trans. Automatic Control, AC-22 (1977), 143-144. doi: 10.1109/tac.1977.1101441. Google Scholar
[29]	I. Lasiecka and R. Triggiani, Algebraic Riccati equations arising in boundary/point control: A review of theoretical and numerical results. I. Continuous case, In Perspectives in Control Theory (Sielpia, 1988), Progr. Systems Control Theory, volume 2, Birkhäuser Boston, Boston, MA, 1990, 175–210. Google Scholar
[30]	W. J. Liu and M. Krstić, Backstepping boundary control of Burgers' equation with actuator dynamics, Systems Control Lett., 41 (2000), 291-303. doi: 10.1016/S0167-6911(00)00068-2. Google Scholar
[31]	W. J. Liu and M. Krstić, Boundary feedback stabilization of homogeneous equilibria in unstable fluid mixtures, Internat. J. Control, 80 (2007), 982-989. doi: 10.1080/00207170701280895. Google Scholar
[32]	Y. Orlov and D. Dochain, Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor, IEEE Trans. Automat. Control, 47 (2002), 1293-1304. doi: 10.1109/TAC.2002.800737. Google Scholar
[33]	D. L. Lukes, Stabilizability and optimal control, Funkcial. Ekvac., 11 (1968), 39-50. Google Scholar
[34]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar
[35]	J. L. Pitarch, M. Rakhshan, M. Mardani, M. Sadeghi and C. Prada, Distributed nonlinear control of a plug-flow reactor under saturation, IFAC-PapersOnLine, 49 (2016), 87-92. doi: 10.1016/j.ifacol.2016.10.760. Google Scholar
[36]	R. Rebarber, Spectral assignability for distributed parameter systems with unbounded scalar control, SIAM J. Control Optim., 27 (1989), 148-169. doi: 10.1137/0327009. Google Scholar
[37]	D. L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl., 40 (1972), 336-368. doi: 10.1016/0022-247X(72)90055-8. Google Scholar
[38]	D. L. Russell, Canonical forms and spectral determination for a class of hyperbolic distributed parameter control systems, J. Math. Anal. Appl., 62 (1978), 186-225. doi: 10.1016/0022-247X(78)90229-9. Google Scholar
[39]	A. Smyshlyaev, E. Cerpa and M. Krstić, Boundary stabilization of a 1-D wave equation with in-domain antidamping, SIAM J. Control Optim., 48 (2010), 4014-4031. doi: 10.1137/080742646. Google Scholar
[40]	E. D. Sontag, Mathematical Control Theory, Texts in Applied Mathematics, vol. 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7. Google Scholar
[41]	S. H. Sun, On spectrum distribution of completely controllable linear systems, SIAM J. Control Optim., 19 (1981), 730-743. doi: 10.1137/0319048. Google Scholar
[42]	D. Tsubakino, M. Krstić, and S. Hara, Backstepping control for parabolic pdes with in-domain actuation, 2012 American Control Conference (ACC), (2012), 2226–2231. doi: 10.1109/ACC. 2012.6315358. Google Scholar
[43]	J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., 43 (2005), 2233-2244. doi: 10.1137/S0363012901388452. Google Scholar
[44]	A. Vest, Rapid stabilization in a semigroup framework, SIAM J. Control Optim., 51 (2013), 4169-4188. doi: 10.1137/130906994. Google Scholar
[45]	F. Woittennek, S. Q. Wang and T. Knüppel, Backstepping design for parabolic systems with in-domain actuation and Robin boundary conditions, IFAC Proceedings Volumes, 19th IFAC World Congress, 47 (2014), 5175–5180. doi: 10.3182/20140824-6-ZA-1003.02285. Google Scholar
[46]	S. Q. Xiang, Null controllability of a linearized Korteweg–de Vries equation by backstepping approach, SIAM J. Control Optim., 57 (2019), 1493-1515. doi: 10.1137/17M1115253. Google Scholar
[47]	S. Q. Xiang, Small-time local stabilization for a Korteweg–de Vries equation, Systems Control Lett., 111 (2018), 64-69. doi: 10.1016/j.sysconle.2017.11.003. Google Scholar
[48]	X. Yu, C. Xu, H. C. Jiang, A. Ganesan and G. J. Zheng., Backstepping synthesis for feedback control of first-order hyperbolic PDEs with spatial-temporal actuation, Abstr. Appl. Anal., (2014), Art. ID 643640, 13 pp. doi: 10.1155/2014/643640. Google Scholar
[49]	J. Zabczyk, Mathematical Control Theory, Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9. Google Scholar

[1]	Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[2]	Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6359-6376. doi: 10.3934/dcdsb.2021022
[3]	Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050
[4]	Jean-Michel Coron. Phantom tracking method, homogeneity and rapid stabilization. Mathematical Control & Related Fields, 2013, 3 (3) : 303-322. doi: 10.3934/mcrf.2013.3.303
[5]	Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1439-1467. doi: 10.3934/dcdsb.2019235
[6]	Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021
[7]	Abdallah Benabdallah, Mohsen Dlala. Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021092
[8]	B. E. Ainseba, Sebastian Aniţa. Internal nonnegative stabilization for some parabolic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 491-512. doi: 10.3934/cpaa.2008.7.491
[9]	Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805
[10]	José R. Quintero, Alex M. Montes. Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021021
[11]	Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1
[12]	Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1803-1811. doi: 10.3934/dcdss.2020106
[13]	Xiaorui Wang, Genqi Xu. Uniform stabilization of a wave equation with partial Dirichlet delayed control. Evolution Equations & Control Theory, 2020, 9 (2) : 509-533. doi: 10.3934/eect.2020022
[14]	Louis Tcheugoue Tebou. Equivalence between observability and stabilization for a class of second order semilinear evolution. Conference Publications, 2009, 2009 (Special) : 744-752. doi: 10.3934/proc.2009.2009.744
[15]	Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273
[16]	Elena-Alexandra Melnig. Internal feedback stabilization for parabolic systems coupled in zero or first order terms. Evolution Equations & Control Theory, 2021, 10 (2) : 333-351. doi: 10.3934/eect.2020069
[17]	Abdelkarim Kelleche, Nasser-Eddine Tatar. Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback. Evolution Equations & Control Theory, 2018, 7 (4) : 599-616. doi: 10.3934/eect.2018029
[18]	Serge Nicaise. Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks. Mathematical Control & Related Fields, 2011, 1 (3) : 331-352. doi: 10.3934/mcrf.2011.1.331
[19]	Andrei Fursikov. Stabilization of the simplest normal parabolic equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1815-1854. doi: 10.3934/cpaa.2014.13.1815
[20]	Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161

American Institute of Mathematical Sciences

Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors