Open AccessCompactness of the Set of Trajectories of the Control System Described by a Urysohn Type Integral Equation
Author(s) -
Nesir Hüseyin
Publication year - 2016
Publication title -
an international journal of optimization and control: theories and applications/e-an international journal of optimization and control: theories and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.287
H-Index - 6
eISSN - 2146-5703
pISSN - 2146-0957
DOI - 10.11121/ijocta.01.2017.00299
Subject(s) - compact space , mathematics , ball (mathematics) , type (biology) , set (abstract data type) , affine transformation , nonlinear system , mathematical analysis , closed set , pure mathematics , physics , computer science , ecology , quantum mechanics , biology , programming language
The control system with integralconstraint on the controls is studied, where the behavior of the system by a Urysohn type integral equation is described. It is assumed thatthe system is nonlinear with respect to the state vector, affine with respect to the control vector. The closed ball ofthe space $L_p(E;\mathbb{R}^m)$ $(p>1)$ with radius $r$ and centered at theorigin, is chosen as the set of admissible control functions, where $E\subset \mathbb{R}^k$ is a compact set. Itis proved that the set of trajectories generated by all admissible control functions is a compact subset of the space of continuous functions.