Publication detail

# Optimality conditions for a linear differential system with a single delay

DEMCHENKO, H. DIBLÍK, J.

Original Title

Optimality conditions for a linear differential system with a single delay

English Title

Optimality conditions for a linear differential system with a single delay

Type

conference paper

Language

en

Original Abstract

In the contribution, linear differential system with a single delay $$\frac{dx(t)}{dt}= A_0x(t) + A_1x(t -\tau) + bu(t), t \geq t_0$$ where A_0, A_1 are $n \times n$ constant matrices, $x \in R^n$, $b \in R^n$, $\tau > 0$, $t_0 \in R$, $u \in R$, is considered. A problem of minimizing (by a suitable control function u(t)) a functional $$I =\int _t_0 ^ \infty (x^T(t)C_{11}x(t) + x^T (t)C_{12}x(t-\tau) + x^T (t-\tau)C_{21}x(t) + ^T (t-\tau)C_{22}x(t-\tau) + du^2(t))dt,$$ 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, linear differential system with a single delay $$\frac{dx(t)}{dt}= A_0x(t) + A_1x(t -\tau) + bu(t), t \geq t_0$$ where A_0, A_1 are $n \times n$ constant matrices, $x \in R^n$, $b \in R^n$, $\tau > 0$, $t_0 \in R$, $u \in R$, is considered. A problem of minimizing (by a suitable control function u(t)) a functional $$I =\int _t_0 ^ \infty (x^T(t)C_{11}x(t) + x^T (t)C_{12}x(t-\tau) + x^T (t-\tau)C_{21}x(t) + ^T (t-\tau)C_{22}x(t-\tau) + du^2(t))dt,$$ 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

Univerzita obrany v Brně

Location

Brno

ISBN

978-80-7231-417-1

Book

Matematika, informační technologie a aplikované vědy (MITAV 2017)

Edition number

1

Pages from

1

Pages to

7

Pages count

7

URL

Documents

BibTex


@inproceedings{BUT142610,
author="Hanna {Demchenko} and Josef {Diblík}",
title="Optimality conditions for a linear differential system with a single delay",
annote="In the contribution, linear differential system with a single delay
$$\frac{dx(t)}{dt}= A_0x(t) + A_1x(t -\tau) + bu(t), t \geq t_0$$
where A_0, A_1 are $n \times n$ constant matrices, $x \in R^n$, $b \in R^n$, $\tau > 0$, $t_0 \in R$,
$u \in R$, is considered. A problem of minimizing (by a suitable control function u(t)) a functional
$$I =\int _t_0 ^ \infty (x^T(t)C_{11}x(t) + x^T (t)C_{12}x(t-\tau) + x^T (t-\tau)C_{21}x(t) + ^T (t-\tau)C_{22}x(t-\tau) + du^2(t))dt,$$
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.",
}