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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

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  September 2020 Early access  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.

Citation: Ionuţ Munteanu. Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise. Evolution Equations and 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.

[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.

[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.

[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.

[5]

V. BarbuS. 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.

[6]

D. M. BoskovicM. 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.

[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.

[8]

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

[9]

T. CaraballoH. 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.

[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.
[11]

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

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

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

[16]

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

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

V. BarbuS. 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.

[6]

D. M. BoskovicM. 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.

[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.

[8]

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

[9]

T. CaraballoH. 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.

[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.
[11]

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

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

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

[16]

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

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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