Controllability of second order infinite dimensional dynamical systems with delays

Jerzy Respondek

doi:10.2478/v10065-008-0013-5

Abstract

The article is devoted to analysing the approximate absolute and approximate relative controllability of a given type second order infinite dimensional system. The considered dynamical system is governed by the evolution equation with three damping terms and three terms without derivatives. Following this aim, spectral theory for linear unbounded operators is involved. At first the representation of considered infinite dimensional dynamical system by the infinite series of finite dimensional systems is given. Next, two theorems on necessary and sufficient conditions of approximate absolute and approximate relative controllability of the considered system are formulated and proved. Finally, proven theorems are applied to the analysis of the elastic beam.

Highlights

The article is devoted to analysing the approximate absolute and approximate relative controllability of a given type second order infinite dimensional system
The obtained theorems of the approximate controllability without constraints, with the cone type constraints, and with delays in control hold true for the second order of the verified infinite dimensional dynamical system
A possible way of further investigations can be the generalisation of the presented results into the case of arbitrary eigenvalues multiplicities of the state operator

Summary

The transformation of the state equation

Using the spectral resolution of the state operator A and its properties (4)-(9). we can transform the infinite dimensional dynamical system, given by the abstract differential equation (1), into equivalent form of the infinite series of the finite dimensional second order linear dynamical systems with constant coefficients of the following form [5]:. We can transform the infinite dimensional dynamical system, given by the abstract differential equation (1), into equivalent form of the infinite series of the finite dimensional second order linear dynamical systems with constant coefficients of the following form [5]:. The coefficients are explicitly given by the inner product between element in the state space X and the appropriate eigenfunctions φij of the operator A: xij (t) =< x(t),φij > X i = 1, 2,3,... Basing on the infinite series of the equations (10) we can transform given system (1) to the more convenient form in the control theory, namely the form of infinite series of the set of first order finite-dimensional ordinary finite dimensional differential equations with constant coefficients (16) as follows [5]:.

The Jordan decomposition of the state matrix

Basic notions

Approximate controllability analysis

The definition of the state differential operator

The state equation

Summary of the mechanical example