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1

Schauder’s fixed-point theorem in approximate controllability problems

Babiarz A., Klamka J., Niezabitowski M.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 2

263--275

EN

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.

2

Approximate controllability of semilinear impulsive strongly damped wave equation

Larez H., Leiva H., Rebaza J., Rios A.

Journal of Applied Analysis

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2015

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Vol. 21, nr 1

45--57

EN

Rothe’s fixed-point theorem is applied to prove the interior approximate controllability of a semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in the space Z1/2 = D((-Δ)1/2) × L2(Ω), where Ω is a bounded domain in Rn (n ≥ 1). Under some conditions we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state z0 to a neighborhood of the final state z1 at time τ > 0.

3

Controllability and observability of distributed parameter systems : a survey

Klamka J.

Studia z Automatyki i Informatyki

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2002

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T. 27

7-19

EN

In this paper, various concepts of controllability and observability of linear infinite-dimensional continuous-time control systems with constant coefficients have been studied. The paper contains a survey of various fundamental definitions, theorems and corollaries concerning approximate controllability, approximate boundary controllability and observability. Moreover, the different relations among them are also listed. As an illustration of the general theory, constrained approximate controllability, constrained approximate boundary controllability and observability of linear distributed parameter dynamical systems described by partial differential state equations and linear output equation are considered.

PL

W pracy rozważono różne koncepcje sterowalności i obserwowalności liniowych, nieskończenie wymiarowych, ciągłych układów sterowania o stałych współczynnikach. Praca zawiera przegląd różnych podstawowych definicji, twierdzeń i wniosków dotyczących sterowalności aproksymacyjnej, sterowalności aproksymacyjnej brzegowej i obserwowalności. Ponadto przedstawiono różne związki między nimi. Jako ilustrację ogólnej teorii rozważono sterowalność aproksymacyjną z ograniczeniami, sterowalność aproksymacyjną brzegową z ograniczeniami oraz obserwowalność liniowych układów o parametrach rozłożonych, opisanych cząstkowymi równaniami różniczkowymi stanu i liniowym równaniem wyjścia.