Solutions stability

of quasi-linear mixed difference-differential equations

G.A. Kamenskii, A.A. Zaitsev

Moscow Aviation Institute

Volokolamskoe Shosse, 4, Moscow, 125871, Russia

 

There is considered the mixedљ quasilinear equation

љ(1)

where

љљ

with initial and boundary conditions

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (2)

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (3)

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (4)

Here

There is assumed that (1) is already the system describing the disturbed motion, i.e. љis a solution of (1) with љand we investigate the stability of this solution.

Solution љis called - stable, if for any љthere is such a љthat for љany functions љthere exists a solution љof problem (1)-(4) with љљand if , then љfor .

There is investigated separately the disturbations of boundary and initial conditions and introduced the following more weak definition of stability.

Solution of problem (1)-(4) is called љ- stable, if for any there is such a љthat for any љthere exists a solution of problem (1)-(4) with љand , if . If additionally љas , then the solution љis asymptotically stable. And, if , then the solution љis exponential - stable.

Suppose that equation (1) can be written in the form

(7)

Equation (7) is transformed to the equation

, љљљљљљљљљљљљљљљљљљљљљљљљљљљљ(9)

where

The initial point љwe put equal to zero and investigate the љ- stability of trivial solution. Then (9) has the form

љљљљљљљљљљљљљљљљљљ (10)

Theorem

Suppose that for equation (7) there are fulfilled following conditions

1)      The matrix љis continuous on љand the spectrum

2)      љuniformly with respect to

3)      љuniformly with respect to , љand .

Then the trivial solution of system (7) an exponential љ- stable.

There is proved an analogous theorem about the љ- stability by the first approximation and a theorem concerning the sufficient conditions of non-stability.