Abstract
In this paper we present a brief review of some important results on weak compactness in the space of vector valued measures. We also review some recent results of the author on weak compactness of any set of operator valued measures. These results are then applied to optimal structural feedback control for deterministic systems on infinite dimensional spaces.
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Keywords
- Space of Operator valued measures
- Countably additive operator valued measures
- Weak compactness
- Semigroups of bounded linear operators
- Optimal Structural control
AMS(MOS) Subject Classification
References
Diestel, J., Uhl Jr., J.J.: Vector Measures. American Mathematical Society, Providence (1977)
Dunford, N., Schwartz, J.T.: Linear Operators, Part 1, General Theory, Second Printing (1964)
Brooks, J.K.: Weak Compactness in the Space of Vector Measures. Bulletin of the American Mathematical Society 78(2), 284–287 (1972)
Kuo, T.: Weak Convergence of Vector Measures on F-Spaces. Math. Z. 143, 175–180 (1975)
Dobrakov, I.: On integration in Banach spaces I. Czechoslovak Mathematical Journal 20(95), 511–536 (1970)
Dobrakov, I.: On Integration in Banach Spaces IV. Czechoslovak Mathematical Journal 30(105), 259–279 (1980)
Brooks, J.K., Lewis, P.W.: Linear Operators and Vector Measures. Trans. American Math. Soc. 192, 139–162 (1974)
Ahmed, N.U.: Vector and Operator Valued Measures as Controls for Infinite Dimensional Systems: Optimal Control. Differential Inclusions, Control and Optimization 28, 95–131 (2008)
Ahmed, N.U.: Impulsive Perturbation of C 0-Semigroups by Operator Valued Measures. Nonlinear Funct. Anal. & Appl. 9(1), 127–147 (2004)
Ahmed, N.U.: Weak Compactness in the Space of Operator Valued Measures. Publicationes Mathematicae, Debrechen (PMD) 77(3-4), 399–413 (2010)
Ahmed, N.U.: Weak Compactness in the Space of Operator Valued Measures \(M_{ba}(\Sigma,{\cal L}(X,Y))\) and its Applications. Differential Inclusions, Control and Optimization 31, 231–247 (2011)
Ahmed, N.U.: Some Remarks on the Dynamics of Impulsive Systems in Banach Spaces. Dynamics of Continuous, Discrete and Impulsive Systems 8, 261–274 (2001)
Ahmed, N.U.: Existence of Optimal Controls for a General Class of Impulsive Systems on Banach Spaces. SIAM J. Control. Optim. 42(2), 669–685 (2003)
Ahmed, N.U.: Dynamics of Hybrid systems Induced by Operator Valued Measures. Nonlinear Analysis: Hybrid Systems 2, 359–367 (2008)
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Ahmed, N. (2013). Weak Compactness in the Space of Operator Valued Measures and Optimal Control. In: Hömberg, D., Tröltzsch, F. (eds) System Modeling and Optimization. CSMO 2011. IFIP Advances in Information and Communication Technology, vol 391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36062-6_5
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DOI: https://doi.org/10.1007/978-3-642-36062-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36061-9
Online ISBN: 978-3-642-36062-6
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