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

• Autorzy: N.U. Ahmed
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 46A32: Spaces of linear operators; topological tensor products; approximation properties
• 49N25: Impulsive optimal control problems
• 47A62: Equations involving linear operators, with operator unknowns
• 93E20: Optimal stochastic control
• 49J27: Problems in abstract spaces
• 46B25: Classical Banach spaces in the general theory
• 46G10: Vector-valued measures and integration

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2009
• Tom: 29
• Numer: 1
• Strony: 67-90

Daty

wydano

2009

otrzymano

2009-05-18

Twórcy

autor

N.U. Ahmed

• SITE, University of Ottawa, Ottawa, Canada

Bibliografia

• [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.
• [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.

Identyfikatory

DOI

10.7151/dmdico.1105