American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Riesz systems and moment method in the study of viscoelasticity in one space dimension
  • DCDS-B Home
  • This Issue
  • Next Article
    Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure
November  2010, 14(4): 1465-1485. doi: 10.3934/dcdsb.2010.14.1465

A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups

Luc Miller 1,

1. 

MODAL’X, EA 3454, Bât. G, Université Paris Ouest Nanterre La Défense, 200 Av. de la République, 92001 Nanterre Cedex, France

Received  August 2009 Revised  February 2010 Published  August 2010

This paper generalizes and simplifies abstract results of Miller and Seidman on the cost of fast control/observation. It deduces final-observability of an evolution semigroup from a spectral inequality, i.e. some stationary observability property on some spaces associated to the generator, e.g. spectral subspaces when the semigroup has an integral representation via spectral measures. words Contrary to the original Lebeau-Robbiano strategy, it does not have recourse to null-controllability and it yields the optimal bound of the cost when applied to the heat equation, i.e. $c_0\exp(c/T)$, or to the heat diffusion in potential wells observed from cones, i.e. $c_0\exp(c/T^\beta)$ with optimal $\beta$. It also yields simple upper bounds for the cost rate $c$ in terms of the spectral rate.
   This paper also gives geometric lower bounds on the spectral and cost rates for heat, diffusion and Ginzburg-Landau semigroups, including on non-compact Riemannian manifolds, based on $L^2$ Gaussian estimates.
Keywords: observability, heat semigroup., null-controllability, fast control cost.
Mathematics Subject Classification: 93B07, 35B3.
Citation: Luc Miller. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1465-1485. doi: 10.3934/dcdsb.2010.14.1465
[1]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183
[2]

Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020011
[3]

Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019037
[4]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143
[5]

Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations & Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022
[6]

Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217
[7]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1
[8]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953
[9]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521
[10]

Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037
[11]

Sylvain Ervedoza, Enrique Zuazua. Observability of heat processes by transmutation without geometric restrictions. Mathematical Control & Related Fields, 2011, 1 (2) : 177-187. doi: 10.3934/mcrf.2011.1.177
[12]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695
[13]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441
[14]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[15]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001
[16]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019031
[17]

Debayan Maity. On the null controllability of the Lotka-Mckendrick system. Mathematical Control & Related Fields, 2019, 9 (4) : 719-728. doi: 10.3934/mcrf.2019048
[18]

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
[19]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
[20]

Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control & Related Fields, 2014, 4 (4) : 465-479. doi: 10.3934/mcrf.2014.4.465

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences