A note on boundary feedback stabilization for degenerate parabolic equations in multi-dimensional domains
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Ionuţ Munteanu
Abstract
In this paper, we are concerned with the problem of stabilization of a degenerate parabolic equation with a Dirichlet control, evolving in bounded domain 𝓞 ⊂ ℝd, d ≥ 2. We apply the proportional control design technique based on the spectrum of the linear operator which governs the evolution equation. The stabilizing feedback control, we design here, is linear, of finite-dimensional structure, easily manageable from the computational point of view.
This work was supported by a grant of the Romanian Ministery of Research and Innovation, CNCS UEFISCDI, project number PN-IV-P2-2.1-TE-2023-1983 within PNCDI IV
Communicated by Alberto Lastra
References
[1] Adolfsson, V.—Escauriaza, L.: C1,α domains and unique continuation at the boundary, Comm. Pure Appl. Math. 50 (1997), 935–969.10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-HSearch in Google Scholar
[2] Ali, H.: Control of a system governed by Dirichlet problem, Math. Slovaca 56 (2006), 361–369.Search in Google Scholar
[3] Barbu, V.: Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Aut. Control 58 (2013), 2416–2420.10.1109/TAC.2013.2254013Search in Google Scholar
[4] Biccari, U.—Hernandez-Santamaria, V.—Vancostenoble, J.: Existence and cost of boundary controls for a degenerate/singular parabolic equation, Math. Control Related Fields 12 (2022), 495–530.10.3934/mcrf.2021032Search in Google Scholar
[5] Fragnelli, G.—Mugnai, D.: Control of Degenerate and Singular Parabolic Equations: Carleman Estimates and Observability. Springer Briefs in Mathematics, 2021.10.1007/978-3-030-69349-7Search in Google Scholar
[6] Gagnon, L.—Lissy, P.—Marx, A.: A Fredholm transformation for the rapid stabilization of a degenerate parabolic equation, SIAM J. Control Optim. 59 (2021), 3828–3859.10.1137/20M1372603Search in Google Scholar
[7] Heinonen, J.—Kilpelainen, T.—Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, Oxford University Press, 1993.Search in Google Scholar
[8] Lasiecka, I.—Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Cambridge, Cambridge Univ. Press, 2000.10.1017/CBO9781107340848Search in Google Scholar
[9] Munteanu, I.: Stabilization of Parabolic Equations, Birkhauser-Springer, 2019.10.1007/978-3-030-11099-4Search in Google Scholar
[10] Serov, V.: Fundamental solution and Fourier series in eigenfunctions of degenerate elliptic operator, J. Math. Anal. Appl. 329 (2007), 132–144.10.1016/j.jmaa.2006.06.014Search in Google Scholar
[11] Tao, X.—Zhang, S.: Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials, J. Math. Anal. Appl. 339 (2008), 70–84.10.1016/j.jmaa.2007.06.042Search in Google Scholar
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Articles in the same Issue
- On the Paley graph of a quadratic character
- A topological duality for tense modal pseudocomplemented De Morgan algebras
- Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers
- A general formula in composition theory
- A nonlinear Filbert-like matrix with three free parameters: From linearity to nonlinearity
- On universality in short intervals for zeta-functions of certain cusp forms
- On asymptotics for lacunary partition functions
- Parallel surfaces of the non-lightlike solution of vortex filament equations
- New q-analogues of Van Hamme’s (F.2) supercongruence and of some related supercongruences
- Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients
- Approximation theorems via Pp-statistical convergence on weighted spaces
- Global existence and multiplicity of positive solutions for anisotropic eigenvalue problems
- Theoretical analysis of higher-order system of difference equations with generalized balancing numbers
- On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence
- Positive bases, cones, Helly-type theorems
- Digital Jordan surfaces arising from tetrahedral tiling
- Influence of ideals in compactifications
- On the functions ωf and Ωf
- On the 𝓐-generators of the polynomial algebra as a module over the Steenrod algebra, I
- A bivariate distribution with generalized exponential conditionals
- A note on boundary feedback stabilization for degenerate parabolic equations in multi-dimensional domains
