Publication detail

# A PROBLEM OF FUNCTIONAL MINIMIZING FOR SINGLE DELAYED DIFFERENTIAL SYSTEM

DEMCHENKO, H. DIBLÍK, J.

Original Title

A PROBLEM OF FUNCTIONAL MINIMIZING FOR SINGLE DELAYED DIFFERENTIAL SYSTEM

English Title

A PROBLEM OF FUNCTIONAL MINIMIZING FOR SINGLE DELAYED DIFFERENTIAL SYSTEM

Type

conference paper

Language

en

Original Abstract

In the contribution, a linear differential system with a single delay $$\frac{dx(t)}{dt} =A_{0} x(t)+A_{1} x(t-\tau)+bu(t), t\ge t_{0}$$ where $A_{0}$, $A_{1}$ are $n\times n$ constant matrices, $x\in R^{n}$, $b\in R^{n}$, $\tau>0$, $t_0\in{\mathbb{R}}$, $u\in R$, is considered. A problem of minimizing (by a suitable control function $u(t)$) a functional \begin{multline*} I=\int _{t_0}^{\infty }(x^{T}(t)C_{11}x(t)+x^{T}(t)C_{12} x(t-\tau) \\ +x^{T}(t-\tau)C_{21}x(t)+x^{T}(t-\tau)C_{22}x(t-\tau)+du^{2}(t))\mathrm{d}t, \end{multline*} where $C_{11}$, $C_{12}$, $C_{21}$, $C_{22}$ are $n\times n$ constant matrices, $d>0$, and the integrand is a positive-definite quadratic form, is considered. To solve the problem, Malkin's approach and Lyapunov's second method are utilized

English abstract

In the contribution, a linear differential system with a single delay $$\frac{dx(t)}{dt} =A_{0} x(t)+A_{1} x(t-\tau)+bu(t), t\ge t_{0}$$ where $A_{0}$, $A_{1}$ are $n\times n$ constant matrices, $x\in R^{n}$, $b\in R^{n}$, $\tau>0$, $t_0\in{\mathbb{R}}$, $u\in R$, is considered. A problem of minimizing (by a suitable control function $u(t)$) a functional \begin{multline*} I=\int _{t_0}^{\infty }(x^{T}(t)C_{11}x(t)+x^{T}(t)C_{12} x(t-\tau) \\ +x^{T}(t-\tau)C_{21}x(t)+x^{T}(t-\tau)C_{22}x(t-\tau)+du^{2}(t))\mathrm{d}t, \end{multline*} where $C_{11}$, $C_{12}$, $C_{21}$, $C_{22}$ are $n\times n$ constant matrices, $d>0$, and the integrand is a positive-definite quadratic form, is considered. To solve the problem, Malkin's approach and Lyapunov's second method are utilized

Keywords

delayed differential system, Lyapunov-Krasovskii functional, integral quality criterion, optimal control.

Released

15.06.2017

Publisher

University of Defence

Location

Brno

ISBN

978-80-7582-026-6

Book

Mathematics, Information Technologies and Applied Sciences 2017

Edition number

1

Pages from

55

Pages to

62

Pages count

8

URL

Documents

BibTex


@inproceedings{BUT142612,
author="Hanna {Demchenko} and Josef {Diblík}",
title="A PROBLEM OF FUNCTIONAL MINIMIZING FOR SINGLE DELAYED DIFFERENTIAL SYSTEM",
annote="In the contribution, a linear differential system with a single delay

$$\frac{dx(t)}{dt} =A_{0} x(t)+A_{1} x(t-\tau)+bu(t), t\ge t_{0}$$

where $A_{0}$, $A_{1}$ are $n\times n$ constant matrices, $x\in R^{n}$, $b\in R^{n}$, $\tau>0$, $t_0\in{\mathbb{R}}$,
$u\in R$, is considered.

A problem of minimizing (by a suitable control function $u(t)$) a functional

\begin{multline*}

I=\int _{t_0}^{\infty }(x^{T}(t)C_{11}x(t)+x^{T}(t)C_{12} x(t-\tau)
\\

+x^{T}(t-\tau)C_{21}x(t)+x^{T}(t-\tau)C_{22}x(t-\tau)+du^{2}(t))\mathrm{d}t,

\end{multline*}

where $C_{11}$, $C_{12}$, $C_{21}$, $C_{22}$ are $n\times n$ constant matrices, $d>0$,
and the integrand is a positive-definite quadratic form,
is
considered. To solve the problem, Malkin's approach and
Lyapunov's second method are utilized",
}