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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1097,
author = {N.U. Ahmed},
title = {Vector and operator valued measures as controls for infinite dimensional systems: optimal control},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {28},
year = {2008},
pages = {95-131},
zbl = {1181.28013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1097}
}
N.U. Ahmed. Vector and operator valued measures as controls for infinite dimensional systems: optimal control. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 28 (2008) pp. 95-131. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1097/
[000] [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.
[001] [2] N.U. Ahmed, Evolution equations determined by operator valued measures and optimal control, Nonlinear Analsis: TMA Series A 67 (11) (2007), 3199-3216. | Zbl 1119.49004
[002] [3] N.U. Ahmed, Impulsive perturbation of C₀-semigroups by operator valued measures, Nonlinear Func. Anal. and Appl. 9 (1) (2004), 127-147.
[003] [4] N.U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series 246 (1991), Longman Scientific and Technical, U.K. and John Wiley, New York.
[004] [5] N.U. Ahmed, Optimization and Identification of Systems Governed by Evolution Equations on Banach Spaces, Pitman Research Notes in Mathematics Series 184 (1988), Longman Scientific and Technical, U.K. and John Wiley, New York.
[005] [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. | Zbl 1076.93007
[006] [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. | Zbl 1037.49036
[007] [8] J. Diestel and J.J. Uhl, Jr. Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, Rhode Island, 1977.
[008] [9] I. Dobrakov, On integration in Banach spaces I, Czechoslav Math. J. 20 (95) (1970), 511-536. | Zbl 0215.20103
[009] [10] I. Dobrakov, On Integration in Banach Spaces IV, Czechoslav Math. J. 30 (105) (1980), 259-279. | Zbl 0452.28006
[010] [11] N. Dunford and J.T. Schwartz, Linear Operators, Part 1, Interscience Publishers, Inc., New York, London, second printing, 1964.
[011] [12] T.V. Panchapagesan, On the distinguishing features of the Dobrakov integral, Divulgaciones Matematicas, Maracaibo, Venezuela 3 (1) (1995), 79-114. | Zbl 0883.28011
[012] [13] W.V. Smith and D.H. Tucker, Weak integral convergence theorems and operator measures, Pacific J. Math. 111 (1) (1984), 243-256. | Zbl 0569.46021
[013] [14] M.E. Munroe, Introduction to Measure and Integration, Addison-Wesley Publishing Company, Inc. Reading, Massachusetts, USA, 1953.
[014] [15] N.U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach spaces, DCDIS 8 (2001), 261-174.
[015] [16] J.K. Brooks and P.W. Lewis, Linear operators and vector measures, Trans. American Mathematical Society 192 (1974), 139-162. | Zbl 0331.46035
[016] [17] A.D. Bandrauk, M.C. Delfour and C.L. Bris (Eds), Quantum Control: Mathematical and Numerical Challenges, CRM Proceedings & Lecture Notes, AMS, Vol. 33, (2002/2003), Providence, Rhode Island USA.
[017] [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.