On equi-asymptotic stability with respect to part of the variables

A system of equations of perturbed motion in which the right-hand sides are almost periodic functions of time is considered. A sufficient condition for the trivial solution of the system to be equi-asymptotically stable with respect to part of the variables is proved.     

Problems of stability with respect to part of the variables

The Lyapunov-Malkin theorem on stability and (simultaneously) exponential asymptotic stability with respect to part of the variables in the linear approximation in critical cases (in Lyapunov's sense) has served as a point of departure for various previous results. These results are strengthened by relaxing all additional assumptions (other than continuity) regarding the coefficients of the linear part of the non-linear system under consideration. The result is extended to the problem of polystability with respect to part of the variables. In addition, a method for narrowing down the admissible domain of variation of “uncontrollable” variables is worked out as applied to problems of asymptotic stability with respect to part of the variables. Examples are considered.     

Asymptotic stability and instability with respect to part of variables for solutions to impulsive systems

We obtain sufficient conditions for asymptotic stability with respect to part of variables for the zero solution to an impulsive system with the fixed moments of impulse effects.     

The controllability of dynamical systems with respect to part of the variables

A particular formulation of the problem of control with respect to part of the variables, in which the initial and terminal states of the system belong to the same subspace, is considered. Necessary and sufficient conditions are established for linear autonomous systems of this type to be partially controllable, i.e. controllable with respect to part of the variables. Using the method of oriented manifolds [1], several theorems concerning the partial controllability of non-linear autonomous systems are proved. The control of the rotational motion of a rigid body by a single rotor is investigated.     

Stability with respect to part of the variables in systems with impulse effect

Effective sufficient conditions are found for stability with respect to part of the variables in systems of ordinary differential equations with impulse effect. The approach presented is based on the specially introduced piecewise continuous Ljapunov functions.     

On polynomials hypoelliptic with respect to a group of variables

For a class of polynomials a necessary and sufficient condition is found for those polynomials to be hypoelliptic with respect to a group of variables.     

Random variables as pathwise integrals with respect to fractional Brownian motion

We give both necessary and sufficient conditions for a random variable to be represented as a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand. We also show that any random variable is a value of such integral in an improper sense and that such integral can have any prescribed distribution. We discuss some applications of these results, in particular, to fractional Black–Scholes model of financial market.     

Lyapunov direct method in the analysis of Lagrange instability with respect to part of the variables

We obtain a sufficient condition using two Lyapunov functions for the Lagrange instability with respect to part of the variables. We also obtain another sufficient condition for the Lagrange instability with respect to part of the variables; this condition uses one Lyapunov function and is an analog of the Chetaev theorem on the instability with respect to part of the variables. If the considered part of the variables coincides with the set of all variables, then the sufficient criterion with one Lyapunov function is a corollary of the sufficient criterion with two Lyapunov functions.     

On asymptotic stability and instability with respect to a fading stochastic perturbation

We develop necessary and sufficient conditions for the a.s. asymptotic stability of solutions of a scalar, non-linear stochastic equation with state-independent stochastic perturbations that fade in intensity. These conditions are formulated in terms of the intensity function: roughly speaking, we show that as long as the perturbations fade quicker than some identifiable critical rate, the stability of the underlying deterministic equation is unaffected. These results improve on those of Chan and Williams; for example, we remove the monotonicity requirement on the drift coefficient and relax it on the intensity of the stochastic perturbation. We also employ different analytic techniques.     

On the stability of a balanced system with respect to its asymptotic state

Siberian Mathematical Journal -     

The stability of a non-autonomous functional-differential equation relative to part of the variables

The asymptotic stability and instability of the trivial solution of a functional-differential equation of delay type relative to part of the variables are investigated using limit equations and a Lyapunov function whose derivative is sign-definite. The theorems thus obtained are used to solve the problem of stabilizing mechanical control systems with delayed feedback. As examples, solutions of problems of the uniaxial and triaxial stabilization of the rotational motion of a rigid body with a delay in the control system are presented.     

On the Wiener integral with respect to a sub-fractional Brownian motion on an interval

The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , k≠0, is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions.     

On stochastic calculus with respect to q-Brownian motion

Following the approach and the terminology introduced in Deya and Schott (2013) [6], we construct a product Lévy area above the q-Brownian motion (for $q\in [0,1)$) and use this object to study differential equations driven by the process.We also provide a detailed comparison between the resulting “rough” integral and the stochastic “Itô” integral exhibited by Donati-Martin (2003) [7].     

Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls

We investigate the problem of the existence of a discontinuous feedback that guarantees the stabilization of a nonlinear control system with respect to a part of variables. A solution of the system is defined in the Filippov sense. We establish a necessary condition for stabilization with respect to a part of variables in the class of discontinuous controls, which generalizes the Ryan condition to the case of stabilization with respect to a part of variables. An example of a mechanical system that cannot be stabilized with respect to a part of variables is considered. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1434–1440, October, 2006.