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September  2020, 9(3): 795-816. doi: 10.3934/eect.2020034

Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise

Ionuţ Munteanu

Alexandru Ioan Cuza University, Department of Mathematics and Octav Mayer Institute of Mathematics (Romanian Academy), 700506 Iaşi, România

Received  August 2019 Revised  September 2019 Published  December 2019

Fund Project: This work was supported by a grant of the "Alexandru Ioan Cuza" University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2018-03

Here we study the problem of boundary feedback stabilization to unbounded trajectories for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. The simple-form feedback allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument its stability is proved. The approach requires the initial data to be a random variable implying the fact that the solution of the random equation is not adapted. Thus, one cannot recover the solution of the initial stochastic equation from the random one. Hence, the designed feedback controller stabilizes the associated random equation and not the original stochastic equation. Anyway, it stabilizes its random version.

Keywords: Feedback stabilization, boundary controls, spectrum, stochastic parabolic equations.
Mathematics Subject Classification: Primary: 93D15, 93B52, 35R60; Secondary: 60H15.
Citation: Ionuţ Munteanu. Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise. Evolution Equations & Control Theory, 2020, 9 (3) : 795-816. doi: 10.3934/eect.2020034
References:
[1]

A. Balogh and M. Krstic, Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability, Eur. J. Control, 8 (2002), 165-176.  doi: 10.3166/ejc.8.165-175.  Google Scholar

[2]

V. Barbu and M. Rockner, Global solutions to random 3D vorticity equations for small initial data, J. Diff. Equations, 263 (2017), 5395-5411.  doi: 10.1016/j.jde.2017.06.020.  Google Scholar

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V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier–Stokes equation, SIAM J. Control Optim., 49 (2012), 1-20.  doi: 10.1137/09077607X.  Google Scholar

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V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations, SIAM J. Control. Optim., 49 (2011), 1454-1478.  doi: 10.1137/100785739.  Google Scholar

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D. M. Boskovic, M. Krstic and W. Liu, Boundary control of an unstable heat equation via measurement of domain-averaged temperature, IEEE Tran. Autom. Control, 46 (2001), 2022-2028.  doi: 10.1109/9.975513.  Google Scholar

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H. Brezis, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk, 57 (2002), 59-74; translation in Russian Math. Surveys, 57 (2002), 693-708. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar

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T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., New York, 2006.  Google Scholar

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T. Caraballo, H. Crauel and J. A. Langa, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.  doi: 10.1090/S0002-9939-06-08593-5.  Google Scholar

[10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
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R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.  doi: 10.1007/BF02477753.  Google Scholar

[12] G. W. Griffiths and W. E. Schiesser, Traveling Wave Analysis of Partial Differential Equations, Elsevier/Academic Press, Amsterdam, 2012.   Google Scholar
[13]

M. Krstic, On global stabilization of Burgers equation by boundary control, Syst. Control Lett., 37 (1999), 123-141.  doi: 10.1016/S0167-6911(99)00013-4.  Google Scholar

[14] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximations Theoreis, Cambrige, U.K.: Cambrige Univ. Press, 2000.   Google Scholar
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J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.  doi: 10.1016/j.jmaa.2007.11.019.  Google Scholar

[16]

I. Munteanu, Boundary Stabilization of Parabolic Equations, Springer, 2019. doi: 10.1007/978-3-030-11099-4.  Google Scholar

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I. Munteanu, Boundary stabilization of the stochastic heat equation by proportional feedbacks, Automatica, 87 (2018), 152-158.  doi: 10.1016/j.automatica.2017.10.003.  Google Scholar

[18]

I. Munteanu, Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback, Discrete Contin. Dyn. Syst., 39 (2019), 2173-2185.  doi: 10.3934/dcds.2019091.  Google Scholar

[19]

I. Munteanu, Boundary stabilisation to non-stationary solutions for deterministic and stochastic parabolic-type equations, Int. J. Control, 92 (2019), 1720-1728.  doi: 10.1080/00207179.2017.1407878.  Google Scholar

[20]

I. Munteanu, Stabilisation of parabolic semilinear equations, Int. J. Control, 90 (2017), 1063-1076.  doi: 10.1080/00207179.2016.1200747.  Google Scholar

[21]

I. Munteanu, Boundary stabilization of the phase field system by finite-dimensional feedback controllers, J. Math. Anal. Appl., 412 (2014), 964-975.  doi: 10.1016/j.jmaa.2013.11.018.  Google Scholar

[22]

I. Munteanu, Boundary stabilization of the Navier - Stokes equation with fading memory, Int. J. Control, 88 (2015), 531-542.  doi: 10.1080/00207179.2014.964780.  Google Scholar

[23]

I. Munteanu, Stabilization of semilinear heat equations, with fading memory, by boundary feedbacks, J. Diff. Equations, 259 (2015), 454-472.  doi: 10.1016/j.jde.2015.02.010.  Google Scholar

[24]

I. Munteanu, Boundary stabilization of a 2-D periodic MHD channel flow, by proportional feedbacks, ESAIM: COCV, 23 (2017), 1253-1266.  doi: 10.1051/cocv/2016025.  Google Scholar

[25]

I. Munteanu, Stabilization of a 3-D periodic channel flow by explicit normal boundary feedbacks, J. Dynam. Control Systems, 23 (2017), 387-403.  doi: 10.1007/s10883-016-9332-9.  Google Scholar

[26]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Syst., 30 (2018), Art. 11, 50 pp. doi: 10.1007/s00498-018-0218-0.  Google Scholar

[27]

S. Rodrigues, Feedback boundary stabilization to trajectories for 3D Navier–Stokes equations, Appl. Math. Optimization, 2018, 1–38. doi: 10.1007/s00245-017-9474-5.  Google Scholar

show all references

References:
[1]

A. Balogh and M. Krstic, Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability, Eur. J. Control, 8 (2002), 165-176.  doi: 10.3166/ejc.8.165-175.  Google Scholar

[2]

V. Barbu and M. Rockner, Global solutions to random 3D vorticity equations for small initial data, J. Diff. Equations, 263 (2017), 5395-5411.  doi: 10.1016/j.jde.2017.06.020.  Google Scholar

[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.  Google Scholar

[4]

V. Barbu and G. Da Prato, Internal stabilization by noise of the Navier–Stokes equation, SIAM J. Control Optim., 49 (2012), 1-20.  doi: 10.1137/09077607X.  Google Scholar

[5]

V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations, SIAM J. Control. Optim., 49 (2011), 1454-1478.  doi: 10.1137/100785739.  Google Scholar

[6]

D. M. Boskovic, M. Krstic and W. Liu, Boundary control of an unstable heat equation via measurement of domain-averaged temperature, IEEE Tran. Autom. Control, 46 (2001), 2022-2028.  doi: 10.1109/9.975513.  Google Scholar

[7]

H. Brezis, How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk, 57 (2002), 59-74; translation in Russian Math. Surveys, 57 (2002), 693-708. doi: 10.1070/RM2002v057n04ABEH000533.  Google Scholar

[8]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Inc., New York, 2006.  Google Scholar

[9]

T. Caraballo, H. Crauel and J. A. Langa, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.  doi: 10.1090/S0002-9939-06-08593-5.  Google Scholar

[10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[11]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.  doi: 10.1007/BF02477753.  Google Scholar

[12] G. W. Griffiths and W. E. Schiesser, Traveling Wave Analysis of Partial Differential Equations, Elsevier/Academic Press, Amsterdam, 2012.   Google Scholar
[13]

M. Krstic, On global stabilization of Burgers equation by boundary control, Syst. Control Lett., 37 (1999), 123-141.  doi: 10.1016/S0167-6911(99)00013-4.  Google Scholar

[14] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximations Theoreis, Cambrige, U.K.: Cambrige Univ. Press, 2000.   Google Scholar
[15]

J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.  doi: 10.1016/j.jmaa.2007.11.019.  Google Scholar

[16]

I. Munteanu, Boundary Stabilization of Parabolic Equations, Springer, 2019. doi: 10.1007/978-3-030-11099-4.  Google Scholar

[17]

I. Munteanu, Boundary stabilization of the stochastic heat equation by proportional feedbacks, Automatica, 87 (2018), 152-158.  doi: 10.1016/j.automatica.2017.10.003.  Google Scholar

[18]

I. Munteanu, Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback, Discrete Contin. Dyn. Syst., 39 (2019), 2173-2185.  doi: 10.3934/dcds.2019091.  Google Scholar

[19]

I. Munteanu, Boundary stabilisation to non-stationary solutions for deterministic and stochastic parabolic-type equations, Int. J. Control, 92 (2019), 1720-1728.  doi: 10.1080/00207179.2017.1407878.  Google Scholar

[20]

I. Munteanu, Stabilisation of parabolic semilinear equations, Int. J. Control, 90 (2017), 1063-1076.  doi: 10.1080/00207179.2016.1200747.  Google Scholar

[21]

I. Munteanu, Boundary stabilization of the phase field system by finite-dimensional feedback controllers, J. Math. Anal. Appl., 412 (2014), 964-975.  doi: 10.1016/j.jmaa.2013.11.018.  Google Scholar

[22]

I. Munteanu, Boundary stabilization of the Navier - Stokes equation with fading memory, Int. J. Control, 88 (2015), 531-542.  doi: 10.1080/00207179.2014.964780.  Google Scholar

[23]

I. Munteanu, Stabilization of semilinear heat equations, with fading memory, by boundary feedbacks, J. Diff. Equations, 259 (2015), 454-472.  doi: 10.1016/j.jde.2015.02.010.  Google Scholar

[24]

I. Munteanu, Boundary stabilization of a 2-D periodic MHD channel flow, by proportional feedbacks, ESAIM: COCV, 23 (2017), 1253-1266.  doi: 10.1051/cocv/2016025.  Google Scholar

[25]

I. Munteanu, Stabilization of a 3-D periodic channel flow by explicit normal boundary feedbacks, J. Dynam. Control Systems, 23 (2017), 387-403.  doi: 10.1007/s10883-016-9332-9.  Google Scholar

[26]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Syst., 30 (2018), Art. 11, 50 pp. doi: 10.1007/s00498-018-0218-0.  Google Scholar

[27]

S. Rodrigues, Feedback boundary stabilization to trajectories for 3D Navier–Stokes equations, Appl. Math. Optimization, 2018, 1–38. doi: 10.1007/s00245-017-9474-5.  Google Scholar

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