Abstract

Symmetry Recognition of Mobile Systems

G.Yakovenko (Moscow Institute of Physics and Technolodgy, Dolgoprudny, Moscow Region, Russia )

e-mail: yakovenko_g@mtu-net.ru

Let control system (u - control)

dx/dt=f(t,x,u) (1)

be quite controlled. The set N(x0 -> x1) is set of such pairs {T, u(t)}, that for the decision x(t), appropriated to control u(t) is carried out: x(0)=x0, x(T)=x1. The system (1) is called a mobile, if for any initial condition x*0 there will be such final condition x*1, that appropriate sets N(x*0 -> x*1) and N(x0 -> x1) will coincide: (other terminology: portability, multiplicity of controllability) The system

dx/dt=xu (2)

is a mobile system: the set N(x0 -> x1) contains pairs {T, u(t)}, and these pairs translate an initial point x*0 in the same point x*1. The system

dx/dt=x+u (3)

is not a mobile system: the set N(x0 -> x1) contains pairs {T, u(t)}, and these pairs translate an initial point x*0 in points x*1, which arrangement depends on duration T of process. It is proved that only quite controlled L-systems have mobility: systems that admit maximal - n-parametrical - group of symmetries. In particular, the system (2) is the L-system (it supposes one parametrical group x^=ax). The system (3) is not the L-system (the unique transformation of symmetry is identical: x^=x). Supported by Russian Foundation for Basic Reseach (grant 02-01-00697) and Programms Council of support of Leading Scientific Schools (grant 00-15-96137).

Section : 12