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Zelenskyi V.V.

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:

equation2195

Here tex2html_wrap_inline4566 is non-linear operator, X - Banach space, tex2html_wrap_inline4570 - its dual, tex2html_wrap_inline4572 .

Let we have the following informationabout the object: the system in the dual space tex2html_wrap_inline4570 behaves itself chaotically inside the ball with radius tex2html_wrap_inline4576 . Besides it we know that
tex2html_wrap_inline4578 if tex2html_wrap_inline4580 .
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 tex2html_wrap_inline4582 such that tex2html_wrap_inline4584 are solved.

Definition 1. The system (1) will be called stabilizable if the set tex2html_wrap_inline4586

is not empty.

Definition 2. The system (1) will be called stable if tex2html_wrap_inline4588 there exists such tex2html_wrap_inline4590 , that tex2html_wrap_inline4592 satysfying the condition tex2html_wrap_inline4584 the set tex2html_wrap_inline4596 is not empty and tex2html_wrap_inline4598 .

Theorem. Let non-linear monotone operator tex2html_wrap_inline4566 is radially continuous and satisfies the condition tex2html_wrap_inline4602   if   tex2html_wrap_inline4604

Then there exists such regulator tex2html_wrap_inline4606 that the set tex2html_wrap_inline4608

is not empty and tex2html_wrap_inline4610 .

It means that the system (1) is stabilizable and stable.

The map tex2html_wrap_inline4606 is obtained.

displaymath4564



Book of abstracts
ICND-96