CUBO A Mathematical Journal Vol.13, Nº 02, (73–84). June 2011

Closure of Pointed Cones and Maximum Principle in Hilbert Spaces

Paolo d'Alessandro
Math. Department, Third University of Rome, Roma, email: dalex@mat.uniroma3.it

ABSTRACT

We prove, in a Hilbert space setting, that all targets of the minimum norm optimal control problems reachable with inputs of minimum norm p are support points for the the set reachable by inputs with norm bounded by p. This amount to say that the Maximum Principle always holds in Hilbert Spaces.

Keywords and phrases: Linear Control Systems in Hilbert Spaces, Norm Optimal Control, Maximum Principle

RESUMEN

En este artículo se demuestra que, para el problema de control óptimo a un nivel mínimo en los espacios de Hilbert, todos los estados alcanzables con un nivel mínimo de entrada de p son puntos de apoyo para el conjunto de estados alcanzables por la norma de entrada inferior o igual a p. Esto es equivalente a decir que el Principio Máximo siempre es válido en los espacios de Hilbert.

Referencias

[1] H.O. Fattorini, Infinite Dimensional Linear Control Systems, Elsevier, Amsterdam 2005.         [ Links ]

[2] A.V. Balakrishnan, Applied Functional Analysis,Springer-Verlag Berlin-Heidelberg-New York 1976.        [ Links ]

[3] P.R. Halmos, A Hilbert Space Problem Book Van Nostrand, New York, 1967.        [ Links ]

[4] J.L. Kelley and I. Namioka, Linear Topological Spaces, Springer, New York, 1963        [ Links ]

[5] J.L. Kelley, General Topology, Springer, New York, 1955.        [ Links ]

[6] H.O. Fattorini Smoothness of the Costate and the Target in the Time and Norm Optimal Problems Optimization, Vol.55, No.2, 2006, 19-36        [ Links ]

[7] H.O. Fattorini Regular and Strongly Regular Time and Norm Optimal Controls to appear        [ Links ]

Received: October 2009. Revised: January 2010.