American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent
  • PROC Home
  • This Issue
  • Next Article
    Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity
2005, 2005(Special): 40-49. doi: 10.3934/proc.2005.2005.40

On the basis properties of the functions arising in the boundary control problem of a string with a variable tension

Sergei A. Avdonin 1, and  Boris P. Belinskiy 2,

1. 

University of Alaska Fairbanks, Fairbanks, AK 99775-6660, United States
2. 

Department of Mathematics, University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN 37403-2598, United States

Received  September 2004 Revised  March 2005 Published  September 2005

We consider the boundary control problem for a string. We say that the string is controllable if, by suitable manipulation of the exterior force, the string goes to rest. To prove our controllability results we apply the method of characteristics. Then, using the method of moments we establish a connection between the boundary control problem and the basis property of a system of functions that substitutes the system of nonharmonic exponential functions. The latter system regularly appears in the problems of controllability since the classical papers of H.O. Fattorini and D.L. Russell.
Keywords: Riesz basis, hyperbolic equation, Distributed parameter systems, characteristics., boundary control.
Mathematics Subject Classification: Primary: 93B05; Secondary: 35C20, 35L2.
Citation: Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40
[1]

Harry L. Johnson, David Russell. Transfer function approach to output specification in certain linear distributed parameter systems. Conference Publications, 2003, 2003 (Special) : 449-458. doi: 10.3934/proc.2003.2003.449
[2]

Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65
[3]

Scott W. Hansen, Rajeev Rajaram. Riesz basis property and related results for a Rao-Nakra sandwich beam. Conference Publications, 2005, 2005 (Special) : 365-375. doi: 10.3934/proc.2005.2005.365
[4]

Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187
[5]

Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77
[6]

Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2018, 14 (2) : 473-496. doi: 10.3934/jimo.2017056
[7]

Marc Puche, Timo Reis, Felix L. Schwenninger. Funnel control for boundary control systems. Evolution Equations & Control Theory, 2021, 10 (3) : 519-544. doi: 10.3934/eect.2020079
[8]

Domingo Tarzia, Carolina Bollo, Claudia Gariboldi. Convergence of simultaneous distributed-boundary parabolic optimal control problems. Evolution Equations & Control Theory, 2020, 9 (4) : 1187-1201. doi: 10.3934/eect.2020045
[9]

Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717
[10]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723
[11]

Ngoc Minh Trang Vu, Laurent Lefèvre. Finite rank distributed control for the resistive diffusion equation using damping assignment. Evolution Equations & Control Theory, 2015, 4 (2) : 205-220. doi: 10.3934/eect.2015.4.205
[12]

Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014
[13]

Barbara Kaltenbacher, William Rundell. On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021020
[14]

Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
[15]

Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control & Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1
[16]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006
[17]

C.P. Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 935-946. doi: 10.3934/dcds.2000.6.935
[18]

Gildas Besançon, Didier Georges, Zohra Benayache. Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks & Heterogeneous Media, 2009, 4 (2) : 211-221. doi: 10.3934/nhm.2009.4.211
[19]

Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks & Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393
[20]

Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014

 Impact Factor: 

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences