Stabilization of non-linear dynamic systems with chaos
V.V. Zelenskyi
National Technocal University of Ukraine "KPI"
The structure instability caused by small disturbances is the compulsory feature for many chaotic processes. The present paper is dealt with the objects described by non-linear operator equations of the form:
Here 
is non-linear operator, X - Banach space,
- its dual, 
.
Let we have the following informationabout the object: the system in the dual
space behaves itself chaotically inside the ball with radius 
. Besides it we know that
if 
.
The problems of stabilizability and stability of such systems, i.e. existing
of such solutions of the operator equation (1) which are inside the ball with
radius 
such that 
are
solved.
Definition 1. The system (1) will be called stabilizable if the set
is not empty.
Definition 2. The system (1) will be called stable if 
there exists such 
, that 
satysfying the condition the set 
is not empty and 
.
Theorem. Let non-linear monotone operator is radially
continuous and satisfies the condition
if 
Then there exists such regulator 
that the set
is not empty and 
.
It means that the system (1) is stabilizable and stable.
The map is obtained.
