On the existence of equations of evolution
type: Item
instance of: scholarly article
title: On the existence of equations of evolution
zbMATH Open document ID: 0568.93033
DOI: 10.1155/S0161171284000430
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/45946
review text: The author considers nonlinear controlled systems (time-independent and non-autonomous) whose equation of evolution is not of the usually taken form \(u'=f(u,x)\) (u is the output, x the control and the prime denotes differentiation with respect to time) but is of the more general form (*) \(u'=f(u,x,x',...,x^{(n)})\). The need for an equation of this type occurs, for example, in the mechanics of inelastic continua. However, the existence of such equation cannot be assumed a priori for an arbitrary system; the value of n depends on the way in which the system smooth the control x. The author proves that, if the system is governed by a principle of determinism (i.e. the value of u at time \(t+\tau\) is determined by its value at time t and by history of the control in \([t,t+\tau])\), then it is described by an evolution equation of the type (*). The value n is such that the state u is continuously differentiable with respect to time whenever the control x is of class \(C^ n\).
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 70G10
Mathematics Subject Classification ID: 74D99
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 3908272
zbMATH Keywords: nonlinear controlled systems
zbMATH Keywords: equation of evolution
zbMATH Keywords: mechanics of inelastic continua
zbMATH Keywords: principle of determinism
author: Jacob Lubliner
reviewed by: Augusto Muracchini
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3908272