On the Robustness Property of the Control System Described by an Urysohn Type Integral Equation

2021 ◽  
Vol 27 (3) ◽  
pp. 263-270
Author(s):  
N. Huseyin ◽  
A. Huseyin ◽  
Kh. G. Guseinov
Keyword(s):  
Integral Equation ◽  
Control System ◽  
Robustness Property ◽  
Type Integral

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 Set of Trajectories of the Nonlinear Control System with Limited Control Resources

2018 ◽  
Vol 23 (1) ◽  
pp. 152-166
Author(s):  
Nesir Huseyin ◽  
Anar Huseyin ◽  
Khalik Guseinov
Keyword(s):  
Integral Equation ◽  
Control System ◽  
Nonlinear Control ◽  
Finite Number ◽  
Hausdorff Distance ◽  
Admissible Control ◽  
Nonlinear Control System ◽  
Integral Constraint ◽  
Control Functions ◽  
Type Integral

In this paper the control system described by a Urysohn type integral equation is studied. It is assumed that the control functions have integral constraint. Approximation of the set of trajectories generated by all admissible control functions is considered. Step by step way, the set of admissible control functions is replaced by a set consisting of a finite number of control functions which generates a finite number of trajectories. An evaluation of the Hausdorff distance between the set of trajectories of the system and the set, consisting of a finite number of trajectories is obtained.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

2021 ◽  
Vol 10 (6) ◽  
pp. 2687-2710
Author(s):  
F. Akutsah ◽  
A. A. Mebawondu ◽  
O. K. Narain
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Existence Of Solution ◽  
Volterra Type ◽  
Complete Metric Spaces ◽  
Type Integral ◽  
New Type

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

scholarly journals Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Ming Li ◽  
Wei Zhao
Keyword(s):  
Integral Equation ◽  
Fractional Order ◽  
Operational Calculus ◽  
Point Of View ◽  
Alternative Method ◽  
Type Integral

This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.

Abel-Type Integral Equation

2020 ◽  
pp. 325-334
Author(s):  
A.G. Ramm ◽  
A.I. Katsevich
Keyword(s):  
Integral Equation ◽  
Type Integral

scholarly journals On the Wolff-type Integral System with Negative Exponents

2021 ◽  
Author(s):  
Rong Zhang ◽  
Ling Li
Keyword(s):  
Integral Equation ◽  
Positive Solutions ◽  
Equation System ◽  
Type Condition ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Integral System ◽  
Type Integral ◽  
Bounded Functions ◽  
Type Potential

In this paper, we are concerned with the positive continuous entire solutions of the Wolff-type integral system \begin{equation*} \left\{ \begin{array}{ll} &u(x) =C_{1}(x)W_{\beta,\gamma} (v^{-q})(x), \\[3mm] &v(x) =C_{2}(x)W_{\beta,\gamma} (u^{-p})(x), \end{array} \right. \end{equation*} where $n\geq1$, $\min\{p,q\}>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma\neq n$. In addition, $C_{i}(x) \ (i=1,2)$ are some double bounded functions. If $\beta\gamma\in (0,n)$, the Serrin-type condition is critical for existence of the positive solutions for some double bounded functions $C_{i}(x)$ $(i=1,2)$. Such an integral equation system is related to the study of the $\gamma$-Laplace system and $k$-Hessian system with negative exponents. Estimated by the integral of the Wolff type potential, we obtain the asymptotic rates and the integrability of positive solutions, and studied whether the radial solutions exist.

scholarly journals Non-parametric identification of thermal control system elements

2021 ◽  
pp. 7-20
Author(s):  
Vasilisa Boeva ◽  
◽  
Yuri Voskoboinikov ◽  
Rustam Mansurov ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Control System ◽  
Impulse Response ◽  
Stable Solution ◽  
Parametric Identification ◽  
Thermal Control ◽  
Black Box ◽  
Response Evaluation ◽  
Thermal Control System ◽  
Non Parametric

The thermal control system “Heater-Fan-Room” is represented by three different-type interconnected simpler subsystems. In this paper, a “black-box” whose structure is not specified is used as a mathematical model of the system and subsystems due to complexity of physical processes proceeding in these subsystems. For stationary linear systems, the connection between an input and an output of the “black-box” is defined by the Volterra integral equation of the first kind with an undetermined difference kernel also known as impulse response in the automatic control theory. In such a case, it is necessary to evaluate an unknown impulse response to use the “black-box” model and formulate all subsystems and the system as a whole. This condition complicates significantly the solution search of non-parametric identification problems in the system because an output of one subsystem is an input of another subsystem, so active identification schemes are unappropriated. Formally, an impulse response evaluation is a solution of the integral equation of the first kind for its kernel by registered noise-contaminated discrete input and output values. This problem is ill-posed because of the possible solution instability (impulse response evaluation in this case) relative to measurement noises in initial data. To find a unique stable solution regularizing algorithms are used, but the specificity of the impulse response identification experiment in the “Heater-Fan-Room” system do not allow applying computational methods of these algorithms (a system of linear equations or discrete Fourier transformation). In this paper, the authors propose two specific identification algorithms for complex technical systems. In these algorithms, impulse responses are evaluated using first derivatives of identified system signals that are stably calculated by smoothing cubic splines with an original smoothing parameter algorithm. The results of the complex “Heater-Fan-Room” system modeling and identification prove the efficiency of the algorithms proposed. Acknowledgments: The reported study was funded by RFBR, project number 20-38-90041.

scholarly journals New Explicit Bounds on Gamidov Type Integral Inequalities for Functions in Two Variables and Their Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Kelong Cheng ◽  
Chunxiang Guo
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Integral Inequalities ◽  
Fredholm Integral Equations ◽  
Uniqueness Of Solutions ◽  
Fredholm Type ◽  
Type Integral ◽  
Explicit Bounds ◽  
Linear And Nonlinear

Some linear and nonlinear Gamidov type integral inequalities in two variables are established, which can give the explicit bounds on the solutions to a class of Volterra-Fredholm integral equations. Some examples of application are presented to show boundedness and uniqueness of solutions of a Volterra-Fredholm type integral equation.

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