ВЕСТНИК ВГУ, Серия физика, математика, 2003, ¹ 1 МАТЕМАТИКА UDC 514.8;517.9;519.248.2 ON A SECOND ORDER DIFFERENTIAL INCLUSION WITH RANDOM PERTURBATION OF VELOCITY* © 2003 Yuri E. Gliklikh, Andrei V. Obukhovskii ¢ Voronezh State University A second order stochastic differential inclusion in Rn is investigated that describes the motion of a mechanical system with set-valued deterministic lower semicontinuous force and random influence on the velocity. <...> The notion of relaxed solution is introduced and an existence theorem for that is proved. <...> For the sake of simplicity let us call the first equation of above system horizontal and the second one vertical. <...> As it was mentioned in [4], this leads to several possibilities for arising random perturbations in the second order equation: when the randomness appears in the vertical equation, in the horizontal one and in both equations. <...> All three cases have natural physical meaning and the corresponding stochastic equations describe three kinds of mechanical systems with randomness. <...> A more interesting and complicated problem is to consider the situation when the right-hand side of the second order equation is set-valued, i.e., the equations is in fact replaced by a second order differential inclusion. <...> Such inclusions with randomness describe mechanical systems with discontinuous forces or with control under the influence of stochastic parameters. <...> It should be pointed out that the types of stochastic differential inclusions, corresponding to three types of stochastic perturbations, mentioned * The research is partially supported by Grant 9900559 from INTAS, by Grant 03-01-00112 from RFBR, by Grant UR.04.01.008 of the Program «Universities of Russia» and by U.S. CRDF RF Ministry of Education Award VZ-010-0. 93 & represented as a first order system on the space of doubled dimension is above, require different methods for their investigation. <...> In [5] we studied the case where the vertical equation was perturbed by a stochastic term expressed via white noise. <...> The physical meaning of this case is that there is a stochastic summand in the force field of mechanical system. <...> Inclusions of this sort are called the stochastic differential inclusions of Langevin type since for single-valued forces they are transformed into the well-known Langevins equations. <...> In [5] the general case of Langevin type inclusions on Riemannian manifolds was considered that allowed us to cover the mechanical systems <...>