Invariant stabilization of a certain class of functional differential equations
type: Item
instance of: scholarly article
title: Invariant stabilization of a certain class of functional differential equations
zbMATH Open document ID: 1264.34152
DOI: 10.3103/S1063454110020068
author: Irina Efremovna Zuber
author: Arkadiĭ Khaimovich Gelig
published in: Vestnik St. Petersburg University. Mathematics
publication date: 2012-11-21T00:00:00Z
review text: The system of functional differential equations \[ \dot x=A(\cdot)x+b(\cdot)u,\;\;u=s^{\top}(\cdot)x \] is considered with coefficients which are functionals of arbitrary nature, of which only the range of their variation is known. The first problem is to find \(b(\cdot)\) and \(s(\cdot)\) for which the system is globally asymptotically stable. Six classes of systems are considered for which this problem can be solved by using a quadratic Lyapunov function with constant coefficient-matrix. Also, the system \[ \dot x=A(\cdot)x+b_1(\cdot)u_1+b_2(\cdot)u_2+g(\cdot) \] is considered with the same restrictions on the right-hand side. The second problem is to determine controls \(u_1(\cdot)\) and \(u_2(\cdot)\) such that, for any initial point and for any \(g(\cdot)\), the output of the system \(\sigma=(c,x)\) would satisfy the invariance condition \(\dot \sigma+\varepsilon\sigma=0\) and boundedness condition \(\overline{\lim}_{t\to\infty}\|x(t)\|\leqslant\kappa\overline{\lim}_{t\to\infty}\|g(\cdot)\|\). In the paper, \(u_1\) and \(u_2\) are obtained provided the matrix \(A(\cdot)\) lies in either of the classes considered.
reviewed by: Nataliya O. Sedova
Mathematics Subject Classification ID: 34K35
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 6109210
zbMATH Keywords: stabilization
zbMATH Keywords: functional differential equation
zbMATH Keywords: Lyapunov function
MaRDI profile type: Publication
cites work: Universal regulators in invariance and tracking problems
cites work: Design of stabilizing controllers that ensure that the output variable of a control system is independent of the external action.
cites work: Approximate solution to the problem of the invariance of a control system
cites work: The problem of absolute invariance of a linear discrete-time control system
cites work: Synthesis of an adaptive regulator in the stabilization of an uncertain discrete linear system
cites work: Global stabilization of nonlinear systems by quadratic Lyapunov functions
cites work: Stability of indefinite systems
full work available at URL: s1063454110020068
OpenAlex ID: W1969128456
description: scientific article; zbMATH DE number 6109210