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2004 | 24 | 1 | 49-72
Tytuł artykułu

Controllability of evolution equations and inclusions driven by vector measures

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EN
Abstrakty
EN
In this paper, we consider the question of controllability of a class of linear and semilinear evolution equations on Hilbert space with measures as controls. We present necessary and sufficient conditions for weak and exact (strong) controllability of a linear system. Using this result we prove that exact controllability of the linear system implies exact controllability of a perturbed semilinear system. Controllability problem for the semilinear system is formulated as a fixed point problem on the space of vector measures and is concluded controllability from the existence of a fixed point. Our results cover impulsive controls as well as regular controls.
Twórcy
autor
  • School of Information Technology and Engineering and Department of Mathematics, University of Ottawa, Ottawa, Ontario K1N6N5
Bibliografia
  • [1] H.O. Fattorini, Some Remarks on Complete Controllability, SICON 4 (1966), 686-694.
  • [2] H.O. Fattorini, On Complete Controllability of Linear Systems, JDE 3 (1967), 391-402.
  • [3] H.O. Fattorini, Local Controllability of a Nonlinear Wave Equation, Mathematical Systems Theory 9 (1975), 3-45.
  • [4] H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of mathematics and its applications, Cambridge University Press 1998.
  • [5] R. Triggiani, A Note on the Lack of Exact Controllability and Optimization, SICON 15 (1977), 407-411.
  • [6] D.L. Russell, Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, Siam Review 20 (1978), 639-739.
  • [7] N.U. Ahmed, Finite-Time Null Controllability for a Class of Linear Evolution Equations on a Banach Space with Control Constraints, J. Opt. Th. and Appl. 47 (2) (1985), 129-158.
  • [8] V. Lakshmikantham, D.D. Bainov and P.S. Simenov, Theory of Impulsive Differential Equations, World Scientific, 1989, Singapore, London.
  • [9] N.U. Ahmed, Systems Governed by Impulsive Differential Inclusions on Hilbert Spaces, Nonlinear Analysis: TMA 45 (2001), 693-706.
  • [10] N.U. Ahmed, Necessary Conditions of Optimality for Impulsive Systems on Banach Spaces, Nonlinear Analysis: TMA 51 (2002), 409-424.
  • [11] N.U. Ahmed, Impulsive Perturbation of C₀ Semigroups and Evolution Inclusions, Nonlinear Funct. Anal. & Appl. 7 (4) (2002), 555-580.
  • [12] N.U. Ahmed, Existence of Optimal Controls for a General Class of Impulsive Systems on Banach Spaces, SIAM, Journal on Contr. and Optim. 42 (2) (2003), 665-685.
  • [13] J. Diestel and J.J. Uhl, Jr., Vector Measures, AMS Mathematical Surveys 15 (1977), AMS, Providence, Rhode Island.
  • [14] K. Yosida, Functional Analysis, (second edition), Springer-Verlag New York inc. 1968.
  • [15] N.U. Ahmed, Semigroup Theory With Applications to Systems and Control, (1991), Pitman Research Notes in Mathematics Series, 246, Longman Scientific and Technical, U.K, Co-published with John Wiley, New York, USA.
  • [16] S. Hu and N.S. Papageorgiou, Hand Book of Multivalued Analysis, Vol.1, Theory, Kluwer Academic Publishers, Dordrecht, Boston, London 1997.
  • [17] M. Kisielewicz, M. Michta and J. Motyl, Set Valued Approach to Stochastic Control (I): Existence and Regularity Properties, Dynamic Systems and Applications 12 (2003), 405-432.
  • [18] M. Kisielewicz, M. Michta and J. Motyl, Set Valued Approach to Stochastic Control (II): Viability and Semimartingale Issues, Dynamic Systems and Applications 12 (2003), 433-466.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1052
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