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

Andrei V. Dmitruk; Nikolai P. Osmolovski

doi:10.3934/mcrf.2017019

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2017, Volume 7, Issue 4: 507-535. Doi: 10.3934/mcrf.2017019

This issue Previous Article Next Article Controllability of fractional dynamical systems: A functional analytic approach

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

Andrei V. Dmitruk^1,2, and
Nikolai P. Osmolovski^3,4,

1.
Russian Academy of Sciences, Central Economics and Mathematics Institute, Lomonosov Moscow State University, Moscow, Russia
2.
Moscow Institute of Physics and Technology, Dolgoprudny, Russia, Russia 117418, Moscow, Nakhimovskii prospekt, 47, Russia
3.
Systems Research Institute, Polish Academy of Sciences, Warszawa, Moscow State University of Civil Engineering, Russia
4.
University of Technology and Humanities in Radom, Poland, 26-600 Radom, ul. Malczewskiego 20A, Poland

Received: November 30, 2016

Revised: May 31, 2017

Published: November 2017

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Abstract

We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. 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 corresponding ones for problems with ordinary differential equations.

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Mathematics Subject Classification: Primary: 49K21; Secondary: 49K15, 90C30.

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References

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