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Schauder’s fixed-point theorem in approximate controllability problems

Babiarz A., Klamka J., Niezabitowski M.

International Journal of Applied Mathematics and Computer Science

2016

Vol. 26, no. 2

263--275

The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.

On an initial value problem for singular integro-differential equations

Šmarda Z.

Demonstratio Mathematica

2002

Vol. 35, nr 4

803-811

The purpose of this paper is to study the existence and asymptotic behaviour of solutions of a nonlinear singular integro-differential equation.

Schauder's fixed-point theorem in nonlinear controllability problems

Klamka J.

Control and Cybernetics

2000

Vol. 29, no 1

153-165

This paper refers to application of the Schauder's fixed point theorem together with linear controllability results in getting the sufficient controllability conditions for various kinds of controllability and for different types of nonlinear control systems. The following nonlinear control systems are considered : finite-dimensional systems, systems with delays in control or in the state variables, and infinite-dimensional systems. The paper presents the review of results existing in the literature which show how Schauder's fixed-point theorem can be practically used to solve several controllability problems for different types of nonlinear control systems.