# American Institute of Mathematical Sciences

September  2015, 35(9): 4323-4343. doi: 10.3934/dcds.2015.35.4323

## Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval

 1 Russian Academy of Sciences, Central Economics and Mathematics Institute and Lomonosov Moscow State University, Russia 117418, Moscow, Nakhimovskii prospekt, 47, Russian Federation 2 Systems Research Institute, Polish Academy of Sciences, Warszawa, Moscow State University of Civil Engineering, Russian Federation

Received  April 2014 Revised  October 2014 Published  April 2015

We study an optimal control problem with Volterra-type integral equation, considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the Euler--Lagrange equation and transversality conditions for the Lagrange problem in the classical calculus of variations with ordinary differential equations.
Citation: Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323
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