American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application
  • EECT Home
  • This Issue
  • Next Article
    Continuity with respect to fractional order of the time fractional diffusion-wave equation
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

[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

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

[1]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[2]

Yubiao Liu, Chunguo Zhang, Tehuan Chen. Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021006
[3]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004
[4]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434
[5]

Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1591-1605. doi: 10.3934/dcdss.2020413
[6]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[7]

Ruofeng Rao, Shouming Zhong. Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1375-1393. doi: 10.3934/dcdss.2020280
[8]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
[9]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052
[10]

Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263
[11]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
[12]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109
[13]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282
[14]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[15]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[16]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[17]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174
[18]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[19]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[20]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (93)
  • HTML views (360)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences