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

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.

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

D. M. BoskovićA. 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.

[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.

[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.

[4]

S. Bialas, On the Lyapunov matrix equation, IEEE Trans. Automat. Control, 25 (1980), 813-814.  doi: 10.1109/TAC.1980.1102438.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[6]

P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188. 

[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.

[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.

[9]

J.-M. CoronL. 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.

[10]

J.-M. CoronL. 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.

[11]

J.-M. CoronL. 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.

[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.

[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.

[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.

[15]

J. -M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[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. 
[17]

J.-M. CoronR. VazquezM. 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.

[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.

[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.

[20]

L. Grafakos, Classical Fourier Analysis, vol. 2, Springer, New York, 2008.

[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

[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.

[23]

D. Kleinman, An easy way to stabilize a linear constant system, IEEE Transactions on Automatic Control, 15 (1970), 692-692. 

[24]

M. KrstićB.-Z. GuoA. 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.

[25]

V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613.  doi: 10.1137/S0363012996301609.

[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.

[27]

M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., New York, NY, 1995.

[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.

[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.

[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.

[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.

[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.

[33]

D. L. Lukes, Stabilizability and optimal control, Funkcial. Ekvac., 11 (1968), 39-50. 

[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.

[35]

J. L. PitarchM. RakhshanM. MardaniM. 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.

[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.

[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.

[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.

[39]

A. SmyshlyaevE. 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.

[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.

[41]

S. H. Sun, On spectrum distribution of completely controllable linear systems, SIAM J. Control Optim., 19 (1981), 730-743.  doi: 10.1137/0319048.

[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.

[43]

J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., 43 (2005), 2233-2244.  doi: 10.1137/S0363012901388452.

[44]

A. Vest, Rapid stabilization in a semigroup framework, SIAM J. Control Optim., 51 (2013), 4169-4188.  doi: 10.1137/130906994.

[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.

[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.

[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.

[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.

[49]

J. Zabczyk, Mathematical Control Theory, Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

show all references

References:
[1]

D. M. BoskovićA. 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.

[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.

[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.

[4]

S. Bialas, On the Lyapunov matrix equation, IEEE Trans. Automat. Control, 25 (1980), 813-814.  doi: 10.1109/TAC.1980.1102438.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[6]

P. Brunovský, A classification of linear controllable systems, Kybernetika (Prague), 6 (1970), 173-188. 

[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.

[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.

[9]

J.-M. CoronL. 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.

[10]

J.-M. CoronL. 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.

[11]

J.-M. CoronL. 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.

[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.

[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.

[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.

[15]

J. -M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[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. 
[17]

J.-M. CoronR. VazquezM. 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.

[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.

[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.

[20]

L. Grafakos, Classical Fourier Analysis, vol. 2, Springer, New York, 2008.

[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

[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.

[23]

D. Kleinman, An easy way to stabilize a linear constant system, IEEE Transactions on Automatic Control, 15 (1970), 692-692. 

[24]

M. KrstićB.-Z. GuoA. 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.

[25]

V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613.  doi: 10.1137/S0363012996301609.

[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.

[27]

M. Krstic, P. V. Kokotovic and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., New York, NY, 1995.

[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.

[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.

[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.

[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.

[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.

[33]

D. L. Lukes, Stabilizability and optimal control, Funkcial. Ekvac., 11 (1968), 39-50. 

[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.

[35]

J. L. PitarchM. RakhshanM. MardaniM. 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.

[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.

[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.

[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.

[39]

A. SmyshlyaevE. 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.

[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.

[41]

S. H. Sun, On spectrum distribution of completely controllable linear systems, SIAM J. Control Optim., 19 (1981), 730-743.  doi: 10.1137/0319048.

[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.

[43]

J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., 43 (2005), 2233-2244.  doi: 10.1137/S0363012901388452.

[44]

A. Vest, Rapid stabilization in a semigroup framework, SIAM J. Control Optim., 51 (2013), 4169-4188.  doi: 10.1137/130906994.

[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.

[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.

[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.

[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.

[49]

J. Zabczyk, Mathematical Control Theory, Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

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