scholarly journals PRECOMPACTNESS OF THE SET OF TRAJECTORIES OF THE CONTROLLABLE SYSTEM DESCRIBED BY A NONLINEAR VOLTERRA INTEGRAL EQUATION

2012 ◽  
Vol 17 (5) ◽  
pp. 686-695 ◽  
Author(s):  
Anar Huseyin ◽  
Nesir Huseyin
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Admissible Control ◽  
Continuous Functions ◽  
Closed Ball ◽  
Controllable System ◽  
Space Of Continuous Functions ◽  
Nonlinear Volterra Integral Equation ◽  
Control Functions

In this paper the controllable system whose behaviour is described by a nonlinear Volterra integral equation, is studied. The set of admissible control functions is the closed ball of the space L p (p > 1) with radius µ 0 and centered at the origin. It is shown that the set of trajectories of the system is a bounded and precompact subset of the space of continuous functions.

scholarly journals Compactness of the Set of Trajectories of the Control System Described by a Urysohn Type Integral Equation

2016 ◽  
Vol 7 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Nesir Huseyin
Keyword(s):  
Integral Equation ◽  
Control System ◽  
Admissible Control ◽  
Continuous Functions ◽  
Closed Ball ◽  
Compact Set ◽  
Control Vector ◽  
Space Of Continuous Functions ◽  
Control Functions ◽  
Type Integral

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.

scholarly journals APPROXIMATION OF THE SECTIONS OF THE SET OF TRAJECTORIES OF THE CONTROL SYSTEM DESCRIBED BY A NONLINEAR VOLTERRA INTEGRAL EQUATION

2015 ◽  
Vol 20 (4) ◽  
pp. 502-515 ◽  
Author(s):  
Anar Huseyin ◽  
Nesir Huseyin ◽  
Khalik G. Guseinov
Keyword(s):  
Integral Equation ◽  
Control System ◽  
Finite Number ◽  
Volterra Integral Equation ◽  
Admissible Control ◽  
Closed Ball ◽  
Nonlinear Volterra Integral Equation ◽  
Control Functions

Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra integral equation is studied. The admissible control functions are chosen from the closed ball of the space L p , p > 1, with radius µ and centered at the origin. The set of admissible control functions is replaced by the set of control functions, which includes a finite number of control functions and generates a finite number of trajectories. It is proved that the sections of the set of trajectories can be approximated by the sections of the set of trajectories, generated by a finite number of control functions.

scholarly journals On the approximation of the set of trajectories of control system described by a Volterra integral equation

2014 ◽  
Vol 19 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Anar Huseyin
Keyword(s):  
Integral Equation ◽  
Control System ◽  
Volterra Integral Equation ◽  
Admissible Control ◽  
Lipschitz Constant ◽  
Closed Ball ◽  
Continuous Control ◽  
Lipschitz Continuous ◽  
Control Functions ◽  
The Space Lp

In this paper, the set of trajectories of the control system described by a nonlinear Volterra integral equation is studied. It is assumed that the set of admissible control functions is the closed ball of the space Lp, p > 1, with radius µ and centered at the origin. It is shown that the sections of the set of trajectories can be approximated by the sections of trajectories, generated by the mixed constrained and Lipschitz continuous control functions, the Lipschitz constant of which is bounded.

scholarly journals Ulam–Hyers stabilities of a differential equation and a weakly singular Volterra integral equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ozgur Ege ◽  
Souad Ayadi ◽  
Choonkil Park
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Fixed Point ◽  
Volterra Integral Equation ◽  
Continuous Functions ◽  
Banach Fixed Point Theorem ◽  
Space Of Continuous Functions ◽  
Weakly Singular ◽  
Banach Contraction ◽  
The Banach Contraction Principle

AbstractIn this work we study the Ulam–Hyers stability of a differential equation. Its proof is based on the Banach fixed point theorem in some space of continuous functions equipped with the norm $\|\cdot \|_{\infty }$ ∥ ⋅ ∥ ∞ . Moreover, we get some results on the Ulam–Hyers stability of a weakly singular Volterra integral equation using the Banach contraction principle in the space of continuous functions $C([a,b])$ C ( [ a , b ] ) .

scholarly journals Solving volterra integral equation via nonlinear programming

2016 ◽  
Vol 5 (4) ◽  
pp. 192
Author(s):  
Jamal Othman
Keyword(s):  
Integral Equation ◽  
Nonlinear Programming ◽  
Volterra Integral Equation ◽  
Least Square ◽  
Nonlinear Programming Problem ◽  
Programming Technique ◽  
New Approach ◽  
Nonlinear Volterra Integral Equation ◽  
Limited Memory ◽  
Quasi Newton

In this paper we propose an approach to find approximate solution to the nonlinear Volterra integral equation of the second type through a nonlinear programming technique by firstly converting the integral equation into a least square cost function as an objective function for an unconstrained nonlinear programming problem which solved by a nonlinear programming technique (The preconditioned limited- memory quasi-Newton conjugates, gradient method) and as far as we read this is a new approach in the ways of solving the nonlinear Volterra integral equation. We use Maple 11 software as a tool for performing the suggested steps in solving the examples.

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