Skip to content


  • Research Article
  • Open Access

Necessary Conditions of Optimality for Second-Order Nonlinear Impulsive Differential Equations

Advances in Difference Equations20072007:040160

  • Received: 2 February 2007
  • Accepted: 5 July 2007
  • Published:


We discuss the existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive differential equations in infinite dimensional spaces. We apply a direct approach to derive the maximum principle for the problem at hand. An example is also presented to demonstrate the theory.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation


Authors’ Affiliations

Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, China


  1. Pontryagin LS: The maximum principle in the theory of optimal processes. Proceedings of the 1st International Congress of the IFAC on Automatic Control, June-July 1960, Moscow, RussiaGoogle Scholar
  2. Ahmed NU: Optimal impulse control for impulsive systems in Banach spaces. International Journal of Differential Equations and Applications 2000,1(1):37–52.MATHMathSciNetGoogle Scholar
  3. Ahmed NU: Necessary conditions of optimality for impulsive systems on Banach spaces. Nonlinear Analysis 2002,51(3):409–424. 10.1016/S0362-546X(01)00837-9MATHMathSciNetView ArticleGoogle Scholar
  4. Butkovskiĭ AG: The maximum principle for optimum systems with distributed parameters. Avtomatika i Telemehanika 1961, 22: 1288–1301.Google Scholar
  5. Egorov AI: The maximum principle in the theory of optimal regulation. In Studies in Integro-Differential Equations in Kirghizia, No. 1 (Russian). Izdat. Akad. Nauk Kirgiz. SSR, Frunze, Russia; 1961:213–242.Google Scholar
  6. Fattorini HO: Infinite Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and Its Applications. Volume 62. Cambridge University Press, Cambridge, UK; 1999:xvi+798.View ArticleGoogle Scholar
  7. Li X, Yong J: Optimal Control Theory for Infinite Dimensional Systems, Systems & Control: Foundations & Applications. Birkhäuser, Boston, Mass, USA; 1995:xii+448.View ArticleGoogle Scholar
  8. Wei W, Xiang X: Optimal control for a class of strongly nonlinear impulsive equations in Banach spaces. Nonlinear Analysis 2005,63(5–7):e53-e63.MATHView ArticleGoogle Scholar
  9. Xiang X, Ahmed NU: Necessary conditions of optimality for differential inclusions on Banach space. Nonlinear Analysis 1997,30(8):5437–5445. 10.1016/S0362-546X(96)00222-2MATHMathSciNetView ArticleGoogle Scholar
  10. Xiang X, Wei W, Jiang Y: Strongly nonlinear impulsive system and necessary conditions of optimality. Dynamics of Continuous, Discrete & Impulsive Systems A 2005,12(6):811–824.MATHMathSciNetGoogle Scholar
  11. Peng Y, Xiang X: Second order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls. to appear in OptimizationGoogle Scholar
  12. Peng Y, Xiang X: Necessary conditions of optimality for second order nonlinear evolution equations on Banach spaces. Proceedings of the 4th International Conference on Impulsive and Hyprid Dynamical Systems, 2007, Nanning, China 433–437.Google Scholar
  13. Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Vol. I: Theory, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xvi+964.View ArticleGoogle Scholar
  14. Ahmed NU: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series. Volume 246. Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA; 1991:x+282.Google Scholar
  15. Xiang X, Kuang H: Delay systems and optimal control. Acta Mathematicae Applicatae Sinica 2000,16(1):27–35. 10.1007/BF02670961MATHMathSciNetView ArticleGoogle Scholar
  16. Xiang X, Peng Y, Wei W: A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces. Discrete and Continuous Dynamical Systems 2005,2005(supplement):911–919.MATHMathSciNetGoogle Scholar
  17. Yang T: Impulsive Control Theory, Lecture Notes in Control and Information Sciences. Volume 272. Springer, Berlin, Germany; 2001:xx+348.Google Scholar
  18. Balder EJ: Necessary and sufficient conditions for -strong-weak lower semicontinuity of integral functionals. Nonlinear Analysis 1987,11(12):1399–1404. 10.1016/0362-546X(87)90092-7MATHMathSciNetView ArticleGoogle Scholar


© Y. Peng et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.