Journal of Dynamical and Control Systems

, Volume 18, Issue 3, pp 397–417 | Cite as

Optimal control problem for deflection plate with crack

  • J. A. D. ChuquipomaEmail author
  • C. A. Raposo
  • W. D. Bastos


We consider a control problem where the state variable is defined as the solution of a variational inequality. This system describes the vertical displacement of points of a thin plate with the presence of crack inside [7]. As the control we define the force that originates the deection of the plate. In order to get the system of optimality for the control problem we use a penalized problem [1] and its reformation as a Lagrangian problem. We prove the existence of a Lagrange multiplier to obtain a system of optimality to the exact problem via Lagrangian. Applying the method of bounded increments [19] we get the final result that characterizes the optimal state and control.

Key words and phrases

Optimal control optimality system penalty problem method of bounded increments 

2000 Mathematics Subject Classification

53C17 22E30 49J15 


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  1. 1.
    V. Barbu. Necessary Conditions for Distributed Control Problems Governed by Parabolic Variational Inequalites. SIAM J. Control and Optimization. 19 (1981), 64–86.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    ______, Necessary Conditions for Nonconvex Distributed Control Problems Governed by Elliptic Variational Inequalities. Journal de Mathematiques Pures et Appliqués 80 (1981), 566–598.MathSciNetGoogle Scholar
  3. 3.
    ______, Optimal Control of Variational Inequalities. Research Notes in Mathematics 100, Pitman Advanced Publishing Program, Iasi (1983).Google Scholar
  4. 4.
    A. Bermudez, and C. Saguez, Optimal Control of a Signorini Problem. SIAM J. Control and Optimization 25 (1987), 576–582.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris (1972).zbMATHGoogle Scholar
  6. 6.
    I. Ekeland and R. Teman, Analyse Convexe et Problèmes Variationelles. Paris, Dunod - Gauthier Villars (1973).Google Scholar
  7. 7.
    A. Khludnev and V. Kovtunenko, Analysis of Cracks in Solids. WIT Press, Southampton-Boston (2000).Google Scholar
  8. 8.
    A. Khludnev, A. Leontiev and J. Herskovits, Nonsmooth Domain Optimization for Elliptic Equations with Unilateral Constraints, Journal de Mathmatiques Pures et Appliques 82 (2003), 197–212.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    R. Glowinski, R. Trémolieres and J. L. Lions, Analyse Numerique des Inéquations Variationelles 1 and 2, Dunod (1976).Google Scholar
  10. 10.
    A. Leontiev, Necessary Optimality Conditions for the Control Problem of the Kirhgoff Plates, Dinamika Splochnoy Sredy. Novosibirsk 103 (1991), 88–89.Google Scholar
  11. 11.
    A. Leontiev, J. Herskovits and C. Eboli, Optimization Theory Application to Slitted Plate Bending Problems, International Journal of Solids and Structures 35 (1998), No. 20, 2679–2694.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. L. Lions, Contrôle Optimal des Systèmes Gouvernès par des Équations aux Deriveés Partielles. Dunod - Gauthiers - Villars, Paris (1968).zbMATHGoogle Scholar
  13. 13.
    ______, Function Spaces and Optimal Control of Distributed Systems. IM-UFRJ, Rio de Janeiro, Brasil (1980).Google Scholar
  14. 14.
    ______,Some aspects of the Optimal Control of Distributed Parameter Systems. SIAM, Philadelphia, Pennsylvania (1980).Google Scholar
  15. 15.
    ______, Sur Quelques Questions d’Analyse, de Mécanique et de Contrôle Optimal, Les Presses de l’Université de Montréal, Montréal (1976).zbMATHGoogle Scholar
  16. 16.
    J. L. Lions and G. Stampacchia, Variational Inequalities, Communications on Pure and Applied Mathematics XX (1967), 493–519.Google Scholar
  17. 17.
    F. Mignot, Contrôle dans les In_equations Variationelles Elliptiques. J. Funct. Anal. 22 (1976), 130–185.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    F. Mignot, J. P. Puel, Optimal Control in Some Variational Inequalities. SIAM J. Control and Optimization 22 (1984), 466–476.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Puel, J. P. Some Results on Optimal Control for Unilateral Problems. Lecture Notes in Control and Information Sciences 114 (1987), 225–235.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yu. N. Rabotnov, Mechanics of a defromed Solid Body. Nauka, Moscow (1979).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • J. A. D. Chuquipoma
    • 1
    Email author
  • C. A. Raposo
    • 1
  • W. D. Bastos
    • 2
  1. 1.Federal University of São João del-ReiDepartament of Mathematics UFSJ - BrazilSão João del-ReiBrazil
  2. 2.São Paulo State UniversityDepartament of Mathematics UNESP - BrazilSão PauloBrazil

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