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2008 | 28 | 1 | 165-189
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Optimal control of systems determined by strongly nonlinear operator valued measures

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In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪ H ↪ V* with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V*. The system is given by
dx + A(dt,x) = f(t,x)γ(dt) + B(t)u(dt), x(0) = ξ, t ∈ I ≡ [0,T]
where A is a strongly nonlinear operator valued measure mapping Σ × V to V* with Σ denoting the sigma algebra of subsets of the set I and f is a nonlinear operator mapping I × H to H, γ is a countably additive bounded positive measure and the control u is a suitable vector measure. We present existence, uniqueness and regularity properties of weak solutions and then prove the existence of optimal controls (vector valued measures) for a class of control problems.
  • School of Information Technology and Engineering, University of Ottawa, Ottawa, Canada
  • [1] N.U. Ahmed, Differential inclusions, operator valued measures and optimal control, Special Issue of Dynamic Systems and Applications, Set-Valued Methods in Dynamic Systems, Guest Editors: M. Michta and J. Motyl 16 (2007), 13-36.
  • [2] N.U. Ahmed, Evolution equations determined by operator valued measures and optimal control, Nonlinear Analysis 67 (2007), 3199-3216.
  • [3] N.U. Ahmed, Vector and operator valued measures as controls for infinite dimensional systems: optimal control, Discuss. Math. Differential Inclusions Control and Optimization 28 (2008), 95-131.
  • [4] N.U. Ahmed, A class of semilinear parabolic and hyperbolic systems determined by operator valued measures, DCDIS 14 (4) (2007).
  • [5] N.U. Ahmed, Parabolic systems determined by strongly nonlinear operator valued measures, Nonlinear Analysis, Special Issue (Felicitation of Professor V. Lakshmikantham on his 85th birth date).
  • [6] N.U. Ahmed, Optimization and Identification of Systems Governed by Evolution Equations on Banach Spaces, Pitman Research Notes in Mathematics Series 184, Longman Scientific and Technical, U.K. and co-publisher John Wiley, New York, 1988.
  • [7] N.U. Ahmed, K.L. Teo and S.H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis 54 (2003), 907-925.
  • [8] N.U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach spaces, DCDIS 8 (2001), 261-274.
  • [9] N.U. Ahmed, Impulsive perturbation of C₀-semigroups by operator valued measures, Nonlinear Functional Analysis & Applications 9 (1) (2004), 127-147.
  • [10] N.U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control. Optim. 42 (2) (2003), 669-685.
  • [11] J. Diestel and J.J. Uhl, Jr., Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, Rhode Island, 1977.
  • [12] N. Dunford and J.T. Schwartz, Linear Operators, Part 1: General Theory, Interscience Publishers, Inc., New York, London, 1958, 1964.
  • [13] H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, Vol. 62, Cambridge University, Cambridge, U.K., 1999.
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