IJPAM: Volume 51, No. 2 (2009)

Invited Lecture Delivered at
Fifth International Conference of Applied Mathematics
and Computing (Plovdiv, Bulgaria, August 12-18, 2008)


LOWER-SEMICONTINUITY AND OPTIMIZATION
OF CONVEX FUNCTIONALS

L.A.O. Fernandes$^1$, R. Arbach$^2$
$^{1,2}$Department of Mathematics
UNESP-Ilha Solteira, Alameda Rio de Janeiro
266, Zip Code 15385-000, Ilha Solteira, SP, BRASIL
$^1$e-mails: [email protected]
$^2$e-mail: [email protected]


Abstract.The result that we treat in this article allows to the utilization of classic tools of convex analysis in the study of optimality conditions in the optimal control convex process for a Volterra-Stietjes linear integral equation in the Banach space $ G([a,b],X) $ of the regulated functions in $ [a,b] $, that is, the functions $ f: [a,b] \to X $ that have only descontinuity of first kind, in Dushnik (or interior) sense, and with an equality linear restriction. In this work we introduce a convex functional $ L_{\beta,f}(x) $ of Nemytskii type, and we present conditions for its lower-semicontinuity. As consequence, Weierstrass Theorem garantees (under compacity conditions) the existence of solution to the problem $ \min \{ L_{\beta,f}(x) \} $.

Received: August 14, 2008

AMS Subject Classification: 45D05

Key Words and Phrases: Volterra-Stietjes linear integral equations, convex optimization, regulated functions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 51
Issue: 2