Global boundary stabilization to trajectories of the deterministic and stochastic porous-media equation

Ionuţ Munteanu

doi:10.3934/mcrf.2023037

Article Contents

2024, Volume 14, Issue 3: 1176-1192. Doi: 10.3934/mcrf.2023037

This issue Previous Article Conditions for uniform ensemble output controllability, and obstruction to uniform ensemble controllability Next Article Mixed Nash games and social optima for linear-quadratic forward-backward mean-field systems

Global boundary stabilization to trajectories of the deterministic and stochastic porous-media equation

Ionuţ Munteanu^{1, 2,}

1.
Faculty of Mathematics, Al. I. Cuza University, Bd. Carol I, 11, Iasi 700506, Romania
2.
O. Mayer Institute of Mathematics, Romanian Academy, Bd. Carol I, 8, Iasi 700505, Romania

Received: April 2023

Revised: September 2023

Early access: October 2023

Published: September 2024

The author is supported by CNRS International Research Network ECO-Math.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

Here we deal with the problem of boundary asymptotic exponential stabilization of flows through porous media. More exactly we study the porous media equation with general monotone porosity in a bounded domain of dimension $ d = 1,2,3 $. We construct an explicit, linear, of finite-dimensional structure feedback controller with Dirichlet part-boundary actuation, which stabilizes any trajectory of the system, for any given initial data. The form of the controller is based on the spectrum of the Dirichlet-Laplace operator and ensures exponential decay to zero of the fluctuation variable for any a priori prescribed decay rate. Also, we extend these results to the case of porous media equation perturbed by Itô Lipschitz noise.

Keywords:

Flows through porous media,
stochastic perturbation,
feedback stabilization,
stabilization to trajectories,
spectrum of the Dirichlet-Laplace operator.

Mathematics Subject Classification: Primary: 76S05, 93B52; Secondary: 47A10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	V. Barbu, G. Da Prato and M. Röckner, Existence of strong solutions for stochastic porous media equation under general monotonicity conditions, Ann. Probab., 37 (2009), 428-452. doi: 10.1214/08-AOP408.
[2]	I. Ciotir, D. Goreac and I. Munteanu, State-constrained porous media control systems with application to stabilization, J. Evol. Eqs., 23 (2023), Paper No. 25, 34 pp. doi: 10.1007/s00028-023-00874-2.
[3]	I. Ciotir, D. Goreac and I. Munteanu, On state-constrained porous-media systems with gradient-type multiplicative noise, Asian J. Control, 25 (2023), 2604-2616. doi: 10.1002/asjc.3013.
[4]	B. Cockburn, D. A. Jones and E. S. Titi, Degrés de liberté déterminants pour equations non linéaires dissipatives, C.R. Acad. Sci.-Paris I Math., 321 (1995), 563-568.
[5]	B. Cockburn, D. A. Jones and E. S. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Math. Comput., 97 (1997), 1073-1087. doi: 10.1090/S0025-5718-97-00850-8.
[6]	G. Da Prato, B. Rozovskii, M. Röckner and F.-Y. Wang, Strong solutions of stochastic generalized porous media equations: Existence, Commun. Partial Differ. Equ., 31 (2006), 277-291. doi: 10.1080/03605300500357998.
[7]	C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia Math. Appl., 83. Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754.
[8]	C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des equations de Navier-Stokes en dimension deux, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.
[9]	C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comput., 43 (1984), 117-133. doi: 10.1090/S0025-5718-1984-0744927-9.
[10]	B. H. Gilding, Stabilization of flows through porous media, SIAM J. Math. Anal., 10 (1979), 237-246. doi: 10.1137/0510022.
[11]	A. Hasan, B. Foss and S. Sagatun, Boundary control of fluid flow through porous media, AIP Conference Proceedings 2010, (2010).
[12]	A. Hasan, B. Foss and S. Sagatun, Flow control of fluids through porous media, Appl. Math. Comput., 219 (2012), 3323-3335. doi: 10.1016/j.amc.2011.07.001.
[13]	A. Ichikawa, Dynamic programming approach to stochastic evolution equations, SIAM J. Control Optim., 17 (1979), 152-174. doi: 10.1137/0317012.
[14]	M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Adv. Des. Control, 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607.
[15]	M. Krstic, On global stabilization of Burgers' equation by boundary control, Syst. Control Lett., 37 (1999), 123-142. doi: 10.1016/S0167-6911(99)00013-4.
[16]	O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 27 (1972), 91-114.
[17]	I. Lasiecka, B. Priyasad and R. Triggiani, Uniform stabilization of Boussinesq systems in critical Lq-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls, Discrete Contin. Dyn. Syst.- Serie B, 25 (2020), 4071-4117. doi: 10.3934/dcdsb.2020187.
[18]	I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/point Control Problems: Continuous Theory and Approximation Theory, Lect. Notes Control Inf. Sci., 164. Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0006880.
[19]	I. Munteanu, Boundary Stabilization of Parabolic Equations, Progr. Nonlinear Differential Equations Appl., 93. Subser. Control Birkhäuser/Springer, Cham, 2019. doi: 10.1007/978-3-030-11099-4.
[20]	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.
[21]	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., 1905. Springer, Berlin, 2007.
[22]	M. Ramaswamy, J. P. Raymond and A. Roy, Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions, J. Diff. Equ., 266 (2019), 4268-4304. doi: 10.1016/j.jde.2018.09.038.
[23]	M. Shinbrot, Lectures on Fluid Mechanics, Gordon and Breach 1973.
[24]	J. L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, USA, 2007.
[25]	J. B. Walsh, An introduction to stochastic partial differential equations, Lecture notes in Mathematics, Lecture Notes in Math., Springer- Berlin, 1180 (1986), 265-439. doi: 10.1007/BFb0074920.
[26]	F. Zhang, Matrix Theory, Basic Results and Techniques, Springer Second Edition, 2011. doi: 10.1007/978-1-4614-1099-7.