We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation associated with the fractional Laplace operator subject to the non-homogeneous Dirichlet type exterior condition. In the first part, we show that if 0 < s < 1, $$ (N ≥ 1) is a bounded Lipschitz domain and the parameter δ > 0, then there is no control function g such that the following system
| $$ |
is exact or null controllable at time T > 0. In the second part, we prove that for every δ ≥ 0 and 0 < s < 1, the system is indeed approximately controllable for any T > 0 and $$, where $$ is any non-empty open set.
Keywords: Fractional Laplace operator, wave equation, strong damping, exterior control, exact and null controllabilities, approximate controllability
@article{COCV_2020__26_1_A42_0,
author = {Warma, Mahamadi and Zamorano, Sebasti\'an},
title = {Analysis of the controllability from the exterior of strong damping nonlocal wave equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019028},
mrnumber = {4124319},
zbl = {1446.35258},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019028/}
}
TY - JOUR AU - Warma, Mahamadi AU - Zamorano, Sebastián TI - Analysis of the controllability from the exterior of strong damping nonlocal wave equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019028/ DO - 10.1051/cocv/2019028 LA - en ID - COCV_2020__26_1_A42_0 ER -
%0 Journal Article %A Warma, Mahamadi %A Zamorano, Sebastián %T Analysis of the controllability from the exterior of strong damping nonlocal wave equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019028/ %R 10.1051/cocv/2019028 %G en %F COCV_2020__26_1_A42_0
Warma, Mahamadi; Zamorano, Sebastián. Analysis of the controllability from the exterior of strong damping nonlocal wave equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 42. doi: 10.1051/cocv/2019028
[1] , and , External optimal control of nonlocal pdes. Inverse Problems, 2019. | MR | Zbl
[2] , and , Controlling the Kelvin force: basic strategies and applications to magnetic drug targeting. Optim. Eng. 19 (2018) 559–589. | MR | Zbl | DOI
[3] , and , Optimizing the Kelvin force in a moving target subdomain. Math. Models Methods Appl. Sci. 28 (2018) 95–130. | MR | Zbl | DOI
[4] , , and , Vector-valued Laplace transforms and Cauchy problems, Vol. 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2nd edn. (2011). | MR | Zbl
[5] , and , Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Comm. Partial Differ. Equ. 43 (2018) 1–24. | MR | DOI
[6] , Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator. Preprint (2014). | arXiv | MR
[7] and , Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects. IMA J. Math. Control Inf. 36 (2019) 1199–1235. | MR | Zbl | DOI
[8] and , The Poisson equation from non-local to local. Electr. J. Differ. Equ. 13 (2018) 145. | MR | Zbl
[9] , and , : Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud. 17 (2017) 837–839. | MR | Zbl | DOI
[10] , and , Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud. 17 (2017) 387–409. | MR | Zbl | DOI
[11] , and , Censored stable processes. Probab. Theory Related Fields 127 (2003) 89–152. | MR | Zbl | DOI
[12] , and , Stability of variational eigenvalues for the fractional p-Laplacian. Discrete Contin. Dyn. Syst. 36 (2016) 1813–1845. | MR | Zbl | DOI
[13] , and , Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12 (2010) 1151–1179. | MR | Zbl | DOI
[14] , and , Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. | MR | Zbl | DOI
[15] , and , Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33 (2017) 377–416. | MR | Zbl | DOI
[16] , and , Lévy flight superdiffusion: an introduction. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008) 2649–2672. | MR | Zbl | DOI
[17] and , Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272–292. | MR | Zbl | DOI
[18] and , Bounded solutions for nonlocal boundary value problems on Lipschitz manifolds with boundary. Adv. Nonlinear Stud. 16 (2016) 529–550. | MR | Zbl | DOI
[19] and , Fractional in time semilinear parabolic equations and applications. In Vol. 84 of Mathematiques et Applications. Springer (2020). | MR | Zbl | DOI
[20] and , Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces. Commun. Partial Differ. Equ. 42 (2017) 579–625. | MR | Zbl | DOI
[21] and , On some degenerate non-local parabolic equation associated with the fractional p-Laplacian. Dyn. Partial Differ. Equ. 14 (2017) 47–77. | MR | Zbl | DOI
[22] , and , The Calderón problem for the fractional Schrödinger equation. Preprint (2016). | arXiv | MR
[23] , and , Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fractals 34 (2007) 87–103. | MR | Zbl | DOI
[24] , Fractional Laplacians on domains, a development of Hörmander’s theory of μ-transmission pseudodifferential operators. Adv. Math. 268 (2015) 478–528. | MR | Zbl | DOI
[25] , and , Wave equations with strong damping in Hilbert spaces. J. Differ. Equ. 254 (2013) 3352–3368. | MR | Zbl | DOI
[26] and , On the interior approximate controllability for fractional wave equations. Discrete Contin. Dyn. Syst. 36 (2016) 3719–3739. | MR | Zbl | DOI
[27] and , Direct, near field acoustic testing. Technical report, SAE technical paper (1999).
[28] and , Approximate controllability from the exterior of space-time fractional wave equations. To appear in: Appl Math Optim (2018). | DOI | MR | Zbl
[29] and , On the lack of controllability of fractional in time ODE and PDE. Math. Control Signals Systems 28 (2016) Art. 10, 21. | MR | Zbl
[30] , , , , , , , , , et al., Clinical experiences with magnetic drug targeting: a phase i study with 4’-epidoxorubicin in 14 patients with advanced solid tumors. Cancer Res. 56 (1996) 4686–4693.
[31] , An introduction to mathematical models, in Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London (2010). | MR | Zbl | DOI
[32] and Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422–437. | MR | Zbl | DOI
[33] , and , Null controllability of the structurally damped wave equation with moving control. SIAM J. Control Optim. 51 (2013) 660–684. | MR | Zbl | DOI
[34] and , Electroencephalography: basic principles, clinical applications, and related fields. Lippincott Williams & Wilkins (2005).
[35] and , The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101 (2014) 275–302. | MR | Zbl | DOI
[36] and , The extremal solution for the fractional Laplacian. Calc. Var. Partial Differ. Equ. 50 (2014) 723–750. | MR | Zbl | DOI
[37] and , The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213 (2014) 587–628. | MR | Zbl | DOI
[38] and , On the controllability of a wave equation with structural damping. Int. J. Tomogr. Stat 5 (2007) 79–84. | MR
[39] , Grey noise. In Stochastic processes, physics and geometry (Ascona and Locarno, 1988). World Sci. Publ., Teaneck, NJ (1990) 676–681. | MR
[40] and , On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinburgh Sect. A 144 (2014) 831–855. | MR | Zbl | DOI
[41] , New developments in conventional hydrocarbon exploration with electromagnetic methods. CSEG Recorder 30 (2005) 34–38.
[42] , From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49 (2009) 33–44. | MR | Zbl
[43] , The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42 (2015) 499–547. | MR | Zbl | DOI
[44] , The fractional Neumann and Robin type boundary conditions for the regional fractional p-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 23 (2016) 1. | MR | Zbl | DOI
[45] , On the approximate controllability from the boundary for fractional wave equations. Appl. Anal. 96 (2017) 2291–2315. | MR | Zbl | DOI
[46] , Approximate controllabilty from the exterior of space-time fractional diffusive equations. SIAM J. Control Optim. 57 (2019) 2037–2063. | MR | Zbl | DOI
[47] and , Null controllability from the exterior of a one-dimensional nonlocal heat equation. Preprint (2018). | arXiv | MR
[48] , and , Fractional operators applied to geophysical electromagnetics. Preprint (2019). | arXiv
[49] , and , Electroencephalography (EEG) of human sleep: clinical applications. John Wiley & Sons (1974).
[50] and , Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22 (2006) 87–99. | MR | Zbl | DOI
[51] , Controllability of partial differential equations. 3ème cycle. Castro Urdiales, Espagne (2006).
Cité par Sources :
The work of the first author is partially supported by the Air Force Office of Scientific Research (AFOSR) under Award No: FA9550-18-1-0242. The second author is supported by the Fondecyt Postdoctoral Grant No: 3180322.
