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Optimal control of delay-differential inclusions with multivalued initial conditions in infinite dimensions

Mordukhovich B. S., Wang D., Wang L.

Control and Cybernetics

2008

Vol. 37, no 2

393-428

This paper is devoted to the study of a general class of optimal control problems described by delay-differential inclusions with infinite-dimensional state spaces, endpoints constraints, and multivalued initial conditions. To the best of our knowledge, problems of this type have not been considered in the literature, except for some particular cases when either the state space is finite-dimensional or there is no delay in the dynamics. We develop the method of discrete approximations to derive necessary optimality conditions in the extended Euler-Lagrange form by using advanced tools of variational analysis and generalized differentiation in infinite dimensions. This method consists of the three major parts: (a) constructing a well-posed sequence of discrete-time problems that approximate in an appropriate sense the original continuous-time problem of dynamic optimization; (b) deriving necessary optimality conditions for the approximating discrete-time problems by reducing them to infinite-dimensional problems of mathematical programming and employing then generalized differential calculus; (c) passing finally to the limit in the obtained results for discrete approximations to establish necessary conditions for the given optimal solutions to the original problem. This method is fully realized in the delay-differential systems under consideration.

Calculus of second-order subdifferentials in infinite dimensions

Morduhovic B.

Control and Cybernetics

2002

Vol. 31, no 3

557-573

We study two types of second-order subdifferentials of extended-real-valued functions on Banach spaces that are important for applications in variational analysis, especially to sensitivity issues and second-order optimality conditions. The main concern of the paper is to derive extended sum and chain rules for these subdifferentials in the case of Asplund and general Banach spaces, which provide the basis for further theory and applications.

Mixed coderivatives of set-valued mappings in variational analysis

Mordukhovich B. S., Shao Y.

Journal of Applied Analysis

1998

Vol. 4, nr 2

269-294

We consider a refined coderivative construction for nonsmooth and set-valued mappings between Banach spaces. This limiting mixed coderivative is different from "normal" coderiva-tives generated by normal cones/subdifferentials and turns out to be useful for studying some basic propertiers in variational analysis particularly related to Lipschitzian stability. We develop a strong calculus for this coderivative important for various applications.