ON THE STABILIZATION OF THE ENERGY OF A HARMONIC OSCILLATOR DISTURBED BY RANDOM PROCESSES OF THE " WHITE AND SHOT NOISES " TYPES GRIGORI

In this paper the behavior of the instantaneous energy of a harmonic oscillator is investigated in the case when at a certain angle to the vector of the 
phase velocity of the oscillator, random disturbances of the white and 
shot noises types are acting.


Introduction
By harmonic oscillator without friction we mean an oscillating system for which motion is described by the following linear differential equation of the second order (t) + 2(t) o, (o) o, (o) o, (1) where u 0 is the initial position and /t 0 is the initial velocity of the oscillator (u0 + /t0 > 0); k > 0 is a parameter of the oscillator; u(t)is the position and/t(t) is the velo- city of the oscillator at the moment of time t > 0, and (t) [kl 2u2(t) / /t 2(t)] is the instantaneous energy of the oscillator.
Equation (1) is equivalent to the system of first order differential equations (2) In the present paper we investigate the behavior of instantaneous energy c(t) in the case when, at a certain angle to b (kx2(t),-kxl(t)) where b is the vector of the phase velocity of system (2), fluctuations of the "white noise" type (tb(t) is a "derivative" of a Wiener process w(t)) and fluctuations of the "shot noise" type (/([0, t),R) is a "derivative" of a Poisson measure u([0, t), R)) are acting.In this case system (2) is considered as the following system of stochastic differential equations without aftereffect (see [2]): where a(t.x) (ql(t.x)xlq-q2(t.x)x2.-q2(t.x)xl/ ql(t.x)x2).X (Xl, X2) E / X R, x 1(0) k/to, x2(O -/to, b(t.x) (gl(t.x)xl+ g2(t.x)x2.g2(t.x)xI -k-gl(t.x)x2).C(t,X, U) (71(t, x, U)x q-/2(t,x, u)x2) 72(t,X, u)x I q-")'l(t,x, U)X2) u E R is a non-random vector function, w(t) is a one-dimensional Wiener process, u([0, t),A)) is a Poisson measure with parameter tII(A), such that II(R)< oc.The process w(t) and the measure u([0, t),A), are defined on the probability space (f,F,P).They are jointly independent and Ft-measurable for any t _> 0 and A, where F C F is a nondecreasing family of (r-algebras. Qualitative analysis of the behavior of the harmonic oscillator without friction under the random perturbation along the vector of the phase velocity by stochastic process of the "white noise" type is made in paper [5] and qualitative analysis of the behavior of the harmonic oscillator with friction is made in paper [6].Book  [8] gives a formula for the fundamental matrix for linear equations of type (3) with varying co- efficients.For equations with constant coefficients, conditions are given under which [x(t)[0 with probability 1 as tc as well as conditions under which EIx(t) 12--,O as t--<x.The behavior of the instantaneous energy of the harmonic os- cillator under the random perturbation only of the second component of the vector of the phase velocity was investigated by many authors (see, for example [3,4,7,9]).
In the present paper, we investigate the sufficient conditions under which the in- stantaneous energy does not change: e(t)-e(0) (Corollary 1 of Theorem 1), the suffi- cient conditions under which the instantaneous energy e(t) changes only step-wise (Theorem 2), as well as the sufficient conditions of stability e(t) (Theorems 3-5) are established for equation (3) in terms of functions qi(t,x), gi(t,x), 7i(t,x,u).It is shown that it is possible to control the behavior of e(t) by the choice of function ql(t, x) (determined disturbance).
x(t) 2 x(0) 2[(1 + ")'1 -- '2] This means that under the first impulse disturbance, the considered system moves into the equilibrium state and does not leave it with probability 1.Thus in this case with small disturbances of coefficients 7i(t,u), it is possible to achieve equality (7)  and then obtain (8) by passing to the limit.Theorem 3: If for all t >_ 0 then P{ t_>oSUp (t) u for any 1 > O, 2 > 0 as soon as Ix(O)[ < 6; 5 > O.
Remark 3: If the system is perturbed by "centralized shot noise" ( ([0, t),A)is a "derivative" of a Poisson nature) instead of "shot noise" and other perturbations are fixed then only the orientation of a(t,x) changes in equation (3), that is, a(t,x)-('l(t,x)xl +2(t,x)x2,-2(t,x)x I +'l(t,x)x2) where