# Topological dual of ${B}_{\infty}(I,\u2081(X,Y))$ with application to stochastic systems on Hilbert space

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2009)

- Volume: 29, Issue: 1, page 67-90
- ISSN: 1509-9407

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topN.U. Ahmed. "Topological dual of $B_∞(I, ₁(X,Y))$ with application to stochastic systems on Hilbert space." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29.1 (2009): 67-90. <http://eudml.org/doc/271144>.

@article{N2009,

abstract = {In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to a stochastic structural control problem. An optimal operator-valued measure, considered as the structural control, is to be chosen so as to minimize fluctuation (volatility). Both existence of optimal policy and necessary conditions of optimality are presented including a conceptual algorithm.},

author = {N.U. Ahmed},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {representation theory; topological dual; finitely additive operator-valued measures; polish space; Hilbert space; stochastic systems; structural control; uncertainty abatement; stochastic system},

language = {eng},

number = {1},

pages = {67-90},

title = {Topological dual of $B_∞(I, ₁(X,Y))$ with application to stochastic systems on Hilbert space},

url = {http://eudml.org/doc/271144},

volume = {29},

year = {2009},

}

TY - JOUR

AU - N.U. Ahmed

TI - Topological dual of $B_∞(I, ₁(X,Y))$ with application to stochastic systems on Hilbert space

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2009

VL - 29

IS - 1

SP - 67

EP - 90

AB - In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to a stochastic structural control problem. An optimal operator-valued measure, considered as the structural control, is to be chosen so as to minimize fluctuation (volatility). Both existence of optimal policy and necessary conditions of optimality are presented including a conceptual algorithm.

LA - eng

KW - representation theory; topological dual; finitely additive operator-valued measures; polish space; Hilbert space; stochastic systems; structural control; uncertainty abatement; stochastic system

UR - http://eudml.org/doc/271144

ER -

## References

top- [1] J. Diestel and J.J. Uhl. Jr, Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977.
- [2] N. Dunford and J.T. Schwartz, Linear Operators, Part 1, General Theory, Second Printing, 1964.
- [3] N.U. Ahmed, Dynamics of Hybrid systems induced by operator-valued measures, Nonlinear Analysis, Hybrid Systems 2 (2008), 359-367. Zbl1170.47025
- [4] N.U. Ahmed, Vector and operator-valued measures as controls for infinite dimensional systems: optimal control, Differential Inclusions, Control and Optimization 28 (2008), 165-189.
- [5] N.U. Ahmed, Impulsive perturbation of C₀-semigroups by operator-valued measures, Nonlinear Funct. Anal. & Appl. 9 (1), (2004), 127-147.
- [6] N.U. Ahmed, State dependent vector measures as feedback controls for impulsive systems in Banach spaces, Dynamics of Continuous, Discrete and Impulsive Systems (B) 8 (2001), 251-261. Zbl0990.34056