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2008 | 28 | 1 | 95-131
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Vector and operator valued measures as controls for infinite dimensional systems: optimal control

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In this paper we consider a general class of systems determined by operator valued measures which are assumed to be countably additive in the strong operator topology. This replaces our previous assumption of countable additivity in the uniform operator topology by the weaker assumption. Under the relaxed assumption plus an additional assumption requiring the existence of a dominating measure, we prove some results on existence of solutions and their regularity properties both for linear and semilinear systems. Also presented are results on continuous dependence of solutions on operator and vector valued measures, and other parameters determining the system which are then used to prove some results on control theory including existence and necessary conditions of optimality. Here the operator valued measures are treated as structural controls. The paper is concluded with some examples from classical and quantum mechanics and a remark on future direction.
  • School of Information Technology and Engineering, Department of Mathematics, 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, DSA 16 (2007), 13-36.
  • [2] N.U. Ahmed, Evolution equations determined by operator valued measures and optimal control, Nonlinear Analsis: TMA Series A 67 (11) (2007), 3199-3216.
  • [3] N.U. Ahmed, Impulsive perturbation of C₀-semigroups by operator valued measures, Nonlinear Func. Anal. and Appl. 9 (1) (2004), 127-147.
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  • [6] N.U. Ahmed, Controllability of evolution equations and inclusions driven by vector measures, Discuss. Math. Differential Inclusions, Control and Optimization 24 (2004), 49-72.
  • [7] 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.
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  • [15] N.U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach spaces, DCDIS 8 (2001), 261-174.
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  • [18] N.U. Ahmed, A class of semilinear parabolic and hyperbolic systems determined by operator valued measures, DCDIS, Series A, Math. Anal. 14 (2007), 465-485.
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