Exact Neumann boundary controllability for second order hyperbolic equations

Weijiu Liu; Graham Williams

Colloquium Mathematicae (1998)

  • Volume: 76, Issue: 1, page 117-142
  • ISSN: 0010-1354

Abstract

top
Using HUM, we study the problem of exact controllability with Neumann boundary conditions for second order hyperbolic equations. We prove that these systems are exactly controllable for all initial states in L 2 ( Ω ) × ( H 1 ( Ω ) ) ' and we derive estimates for the control time T.

How to cite

top

Liu, Weijiu, and Williams, Graham. "Exact Neumann boundary controllability for second order hyperbolic equations." Colloquium Mathematicae 76.1 (1998): 117-142. <http://eudml.org/doc/210545>.

@article{Liu1998,
abstract = {Using HUM, we study the problem of exact controllability with Neumann boundary conditions for second order hyperbolic equations. We prove that these systems are exactly controllable for all initial states in $L^2(\{\Omega \})\times (H^1(\{\Omega \}))^\{\prime \}$ and we derive estimates for the control time T.},
author = {Liu, Weijiu, Williams, Graham},
journal = {Colloquium Mathematicae},
keywords = {Neumann boundary condition; HUM; exact controllability; second order hyperbolic equation; Hilbert uniqueness method; second order hyperbolic equations; Neumann boundary control; exact boundary controllability; multiplier techniques; regularity of solutions; observability inequality},
language = {eng},
number = {1},
pages = {117-142},
title = {Exact Neumann boundary controllability for second order hyperbolic equations},
url = {http://eudml.org/doc/210545},
volume = {76},
year = {1998},
}

TY - JOUR
AU - Liu, Weijiu
AU - Williams, Graham
TI - Exact Neumann boundary controllability for second order hyperbolic equations
JO - Colloquium Mathematicae
PY - 1998
VL - 76
IS - 1
SP - 117
EP - 142
AB - Using HUM, we study the problem of exact controllability with Neumann boundary conditions for second order hyperbolic equations. We prove that these systems are exactly controllable for all initial states in $L^2({\Omega })\times (H^1({\Omega }))^{\prime }$ and we derive estimates for the control time T.
LA - eng
KW - Neumann boundary condition; HUM; exact controllability; second order hyperbolic equation; Hilbert uniqueness method; second order hyperbolic equations; Neumann boundary control; exact boundary controllability; multiplier techniques; regularity of solutions; observability inequality
UR - http://eudml.org/doc/210545
ER -

References

top
  1. [1] R. F. Apolaya, Exact controllability for temporally wave equations, Portugal. Math. 51 (1994), 475-488. Zbl0828.49008
  2. [2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), 1024-1065. Zbl0786.93009
  3. [3] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Evolution Problems I, Springer, Berlin, 1992. Zbl0755.35001
  4. [4] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969. Zbl0186.40901
  5. [5] V. Komornik, Exact controllability in short time for the wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 153-164. Zbl0672.49025
  6. [6] V. Komornik, Contrôlabilité exacte en un temps minimal, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 223-225. Zbl0611.49027
  7. [7] O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, New York, 1985. Zbl0588.35003
  8. [8] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. 65 (1986), 149-192. Zbl0631.35051
  9. [9] J. L. Lions, Contrôlabilité exacte , perturbations et stabilisation de systèmes distribués, Tome 1 , Contrôlabilité exacte, Masson, Paris, 1988. 
  10. [10] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vols. I and II, Springer, Berlin, 1972. Zbl0227.35001
  11. [11] M. M. Miranda, HUM and the wave equation with variable coefficients, Asymptotic Anal. 11 (1995), 317-341. Zbl0848.93031
  12. [12] J. E. Mu noz Rivera, Exact controllability: coefficient depending on the time, SIAM J. Control Optim. 28 (1990), 498-501. Zbl0695.93008
  13. [13] D. L. Russell, Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, in: Differential Games and Control Theory, E. O. Roxin, P. T. Liu, and R. L. Sternberg (eds.), Lecture Notes in Pure Appl. Math. 10, Marcel Dekker, New York, 1974, 291-319. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.