Variation formulas of solutions and optimal control problems for differential equations with retarded argument |
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Authors: | G L Kharatishvili T A Tadumadze |
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Institution: | (1) Institute of Cybernetics, Georgian Academy of Science, Georgia;(2) I. N. Vekua Institute of Applied Mathematics, Tbilisi State University, Georgia |
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Abstract: | The authors prove a theorem on the continuous dependence of solutions of nonlinear systems of differential equations with
variable delay on the perturbations of initial data (initial instant, initial function, and initial value of the trajectory)
and the right-hand side in the case where these perturbations are small in the Euclidean and integral topology, respectively.
The variation formulas of solutions of a differential equation with discontinuous and continuous initial condition are deduced;
as compared with those known earlier, these formulas take into account the variation of the initial instant and the discontinuity
and continuity of the initial data. A necessary condition for criticality of mappings defined on a finitely locally convex
set is obtained. The quasiconvexity of filters in studying optimal problems with delays in controls is proved. Necessary optimality
conditions and existence theorems are proved for optimal problems with variable delays in phase coordinates and controls having
a nonfixed initial instant, a discontinuous and a continuous initial condition, and functional and boundary conditions of
general form. Necessary optimality conditions are obtained for optimal problems with variable structure and delays.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 25, Optimal
Control, 2005. |
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