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June  2022, 12(2): 405-420. doi: 10.3934/mcrf.2021027

Feedback stabilization of parabolic systems with input delay

Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

* Corresponding author: Julie Valein

Received  October 2020 Revised  February 2021 Published  June 2022 Early access  April 2021

Fund Project: The two first authors were partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010). The third author was partially supported by the ANR research projects ISDEEC (ANR-16-CE40-0013) and ANR ODISSE (ANR-19-CE48-0004-01)

This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the $ N $-dimensional linear reaction-convection-diffusion equation with $ N\geq 1 $ and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.

Citation: Imene Aicha Djebour, Takéo Takahashi, Julie Valein. Feedback stabilization of parabolic systems with input delay. Mathematical Control and Related Fields, 2022, 12 (2) : 405-420. doi: 10.3934/mcrf.2021027
References:
[1]

C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, In 1993 American Control Conference, IEEE, (1993), 3106–3107.

[2]

Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Trans. Automat. Control, 27 (1982), 869-879.  doi: 10.1109/TAC.1982.1103023.

[3]

A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters—a reaction-diffusion paradigm, Evol. Equ. Control Theory, 3 (2014), 579-594.  doi: 10.3934/eect.2014.3.579.

[4]

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.

[5]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.

[6]

V. Barbu, Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Automat. Control, 58 (2013), 2416-2420.  doi: 10.1109/TAC.2013.2254013.

[7]

V. Barbu and I. Lasiecka, The unique continuation property of eigenfunctions to Stokes-Oseen operator is generic with respect to the coefficients, Nonlinear Anal., 75 (2012), 4384-4397.  doi: 10.1016/j.na.2011.07.056.

[8]

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.

[9]

N. Bekiaris-Liberis and M. Krstic, Nonlinear Control under Nonconstant Delays, volume 25 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. doi: 10.1137/1.9781611972856.

[10]

N. Bekiaris-Liberis and M. Krstic, Predictor-feedback stabilization of multi-input nonlinear systems, IEEE Trans. Automat. Control, 62 (2017), 516-531.  doi: 10.1109/TAC.2016.2558293.

[11]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, second edition, 2007. doi: 10.1007/978-0-8176-4581-6.

[12]

D. Bresch-Pietri and M. Krstic, Delay-adaptive control for nonlinear systems, IEEE Trans. Automat. Control, 59 (2014), 1203-1218.  doi: 10.1109/TAC.2014.2298711.

[13]

D. Bresch-Pietri and M. Krstic, Output-feedback adaptive control of a wave PDE with boundary anti-damping, Automatica J. IFAC, 50 (2014), 1407-1415.  doi: 10.1016/j.automatica.2014.02.040.

[14]

D. Bresch-PietriC. Prieur and E. Trélat, New formulation of predictors for finite-dimensional linear control systems with input delay, Systems Control Lett., 113 (2018), 9-16.  doi: 10.1016/j.sysconle.2017.12.007.

[15]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.

[16]

R. DatkoJ. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.

[17]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.

[18]

C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596.  doi: 10.1080/03605309608821198.

[19]

H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.  doi: 10.1137/0304048.

[20]

M. L. J. Hautus, Controllability and observability conditions of linear autonomous systems, Nederl. Akad. Wetensch. Proc. Ser., 72 (1969), 443-448. 

[21]

L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976.

[22]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 1966.

[23]

M. Krstic, Control of an unstable reaction-diffusion PDE with long input delay, Systems Control Lett., 58 (2009), 773-782.  doi: 10.1016/j.sysconle.2009.08.006.

[24]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 67, 38 pp. doi: 10.1051/cocv/2018054.

[25]

H. Lhachemi and C. Prieur, Feedback stabilization of a class of diagonal infinite-dimensional systems with delay boundary control, IEEE Trans. Automat. Control, 66 (2021), 105-120.  doi: 10.1109/TAC.2020.2975003.

[26]

H. Lhachemi and R. Shorten, Boundary input-to-state stabilization of a damped Euler-Bernoulli beam in the presence of a state-delay, https://arXiv.org/pdf/1912.01117.pdf, 2019.

[27]

H. Lhachemi, R. Shorten and C. Prieur, Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control, Systems Control Lett., 138 (2020), 104651, 10 pp. doi: 10.1016/j.sysconle.2020.104651.

[28]

A. Z. Manitius and A. W. Olbrot, Finite spectrum assignment problem for systems with delays, IEEE Trans. Automat. Control, 24 (1979), 541-553.  doi: 10.1109/TAC.1979.1102124.

[29]

M. T. Nihtilä, Input delay systems: Adaptive control and lumped approximations, In Mathematics of the Analysis and Design of Process Control (Dublin, 1991/Lille, 1991), 503–512. North-Holland, Amsterdam, 1992.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

C. Prieur and E. Trélat, Feedback stabilization of a 1-D linear reaction-diffusion equation with delay boundary control, IEEE Trans. Automat. Control, 64 (2019), 1415-1425.  doi: 10.1109/TAC.2018.2849560.

[32]

J.-P. Raymond, Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1537-1564.  doi: 10.3934/dcdsb.2010.14.1537.

[33]

J.-P. Raymond, Stabilizability of infinite-dimensional systems by finite-dimensional controls, Comput. Methods Appl. Math., 19 (2019), 797-811.  doi: 10.1515/cmam-2018-0031.

[34]

J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete Contin. Dyn. Syst., 27 (2010), 1159-1187.  doi: 10.3934/dcds.2010.27.1159.

[35]

R. Temam, Navier-Stokes Equations, volume 2 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, revised edition, 1979. Theory and numerical analysis, With an appendix by F. Thomasset.

[36]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, volume 18 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978.

[37]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[38]

K. Yosida, Lectures on Differential and Integral Equations, Pure and Applied Mathematics, Vol. X. Interscience Publishers, New York-London, 1960.

show all references

References:
[1]

C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, In 1993 American Control Conference, IEEE, (1993), 3106–3107.

[2]

Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Trans. Automat. Control, 27 (1982), 869-879.  doi: 10.1109/TAC.1982.1103023.

[3]

A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters—a reaction-diffusion paradigm, Evol. Equ. Control Theory, 3 (2014), 579-594.  doi: 10.3934/eect.2014.3.579.

[4]

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.

[5]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.

[6]

V. Barbu, Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Automat. Control, 58 (2013), 2416-2420.  doi: 10.1109/TAC.2013.2254013.

[7]

V. Barbu and I. Lasiecka, The unique continuation property of eigenfunctions to Stokes-Oseen operator is generic with respect to the coefficients, Nonlinear Anal., 75 (2012), 4384-4397.  doi: 10.1016/j.na.2011.07.056.

[8]

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.

[9]

N. Bekiaris-Liberis and M. Krstic, Nonlinear Control under Nonconstant Delays, volume 25 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. doi: 10.1137/1.9781611972856.

[10]

N. Bekiaris-Liberis and M. Krstic, Predictor-feedback stabilization of multi-input nonlinear systems, IEEE Trans. Automat. Control, 62 (2017), 516-531.  doi: 10.1109/TAC.2016.2558293.

[11]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, second edition, 2007. doi: 10.1007/978-0-8176-4581-6.

[12]

D. Bresch-Pietri and M. Krstic, Delay-adaptive control for nonlinear systems, IEEE Trans. Automat. Control, 59 (2014), 1203-1218.  doi: 10.1109/TAC.2014.2298711.

[13]

D. Bresch-Pietri and M. Krstic, Output-feedback adaptive control of a wave PDE with boundary anti-damping, Automatica J. IFAC, 50 (2014), 1407-1415.  doi: 10.1016/j.automatica.2014.02.040.

[14]

D. Bresch-PietriC. Prieur and E. Trélat, New formulation of predictors for finite-dimensional linear control systems with input delay, Systems Control Lett., 113 (2018), 9-16.  doi: 10.1016/j.sysconle.2017.12.007.

[15]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.

[16]

R. DatkoJ. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.

[17]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.

[18]

C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596.  doi: 10.1080/03605309608821198.

[19]

H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.  doi: 10.1137/0304048.

[20]

M. L. J. Hautus, Controllability and observability conditions of linear autonomous systems, Nederl. Akad. Wetensch. Proc. Ser., 72 (1969), 443-448. 

[21]

L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976.

[22]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 1966.

[23]

M. Krstic, Control of an unstable reaction-diffusion PDE with long input delay, Systems Control Lett., 58 (2009), 773-782.  doi: 10.1016/j.sysconle.2009.08.006.

[24]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 67, 38 pp. doi: 10.1051/cocv/2018054.

[25]

H. Lhachemi and C. Prieur, Feedback stabilization of a class of diagonal infinite-dimensional systems with delay boundary control, IEEE Trans. Automat. Control, 66 (2021), 105-120.  doi: 10.1109/TAC.2020.2975003.

[26]

H. Lhachemi and R. Shorten, Boundary input-to-state stabilization of a damped Euler-Bernoulli beam in the presence of a state-delay, https://arXiv.org/pdf/1912.01117.pdf, 2019.

[27]

H. Lhachemi, R. Shorten and C. Prieur, Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control, Systems Control Lett., 138 (2020), 104651, 10 pp. doi: 10.1016/j.sysconle.2020.104651.

[28]

A. Z. Manitius and A. W. Olbrot, Finite spectrum assignment problem for systems with delays, IEEE Trans. Automat. Control, 24 (1979), 541-553.  doi: 10.1109/TAC.1979.1102124.

[29]

M. T. Nihtilä, Input delay systems: Adaptive control and lumped approximations, In Mathematics of the Analysis and Design of Process Control (Dublin, 1991/Lille, 1991), 503–512. North-Holland, Amsterdam, 1992.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

C. Prieur and E. Trélat, Feedback stabilization of a 1-D linear reaction-diffusion equation with delay boundary control, IEEE Trans. Automat. Control, 64 (2019), 1415-1425.  doi: 10.1109/TAC.2018.2849560.

[32]

J.-P. Raymond, Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1537-1564.  doi: 10.3934/dcdsb.2010.14.1537.

[33]

J.-P. Raymond, Stabilizability of infinite-dimensional systems by finite-dimensional controls, Comput. Methods Appl. Math., 19 (2019), 797-811.  doi: 10.1515/cmam-2018-0031.

[34]

J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete Contin. Dyn. Syst., 27 (2010), 1159-1187.  doi: 10.3934/dcds.2010.27.1159.

[35]

R. Temam, Navier-Stokes Equations, volume 2 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, revised edition, 1979. Theory and numerical analysis, With an appendix by F. Thomasset.

[36]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, volume 18 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978.

[37]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[38]

K. Yosida, Lectures on Differential and Integral Equations, Pure and Applied Mathematics, Vol. X. Interscience Publishers, New York-London, 1960.

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