# stability

## Artificial libration points in the task of towing space debris by an ion beam

The work is devoted to the problem of towing space debris from a geostationary orbit to a graveyard orbit by a non-contact method using an ion beam created by the engine of an active spacecraft. For the planar case using the modified Hill task, the points of relative equilibrium (libration points) of the active spacecraft relative to the object of removal are determined and their stability is estimated. It is shown that, depending on the values of radial acceleration, there are up to 6 libration points, but only one point is suitable for towing a space debris object.

## Mathematical modelling of critical speed of the multistage core at longitudinal blow

Mathematical modeling of longitudinal elastic central blow of a multistage core about a rigid barrier is carried out, at not keeping communications. Mathematical modeling is carried out by the exact analytical decision of the wave differential equation by a method of Dalamber with the task of necessary initial and boundary conditions. With application of the formula of Euler analytical expression for calculation of critical pretonic speed at which there comes loss of stability of a step core is received.

## Mathematical modelling of loss of stability of system of step and homogeneous cores at blow about the rigid barrier Tymoshenko's method

Mathematical modeling of longitudinal elastic central blow of system of step and homogeneous cores about a rigid barrier is carried out, at not holding communications by a solution of the wave equation by Dalamber's method. On the basis of the law of conservation of energy by Tymoshenko's method the size of critical compressing loading according to which, the size of critical pretonic speed leading to loss of stability of considered rod system further pays off.

## On Functional Stability of the Solution for the Problem of Convex Body Best Approximating by a Ball with Fixed Radius

A finite-dimensional problem of finding a uniform estimate (approximation in the Hausdorff metric) of a convex body by a fixed-radius ball in an arbitrary norm is considered. It is known that this problem can be reduced to a linear programming problem in the case, when the convex body and the norm ball are polytops. Therefore, we prove the functional stability of the optimal value of the objective function with respect to accuracy of the given convex body and accuracy of the unit ball for the norm used. The stability rating is derived.

## Analyticity Conditions of Characteristic and Disturbing Quasipolynomials of Hybrid Dynamical Systems

Hybrid dynamical systems (HDS) are connected by means of the boundary conditions and the constraint’s conditions systems of ordinary differential equations and partial differential equations with the corresponding initial conditions. Check the stability of HDS can be performed on the basis of the "fast"algorithm for the application which requires analytic characteristic and disturbing quasipolynomials of HDS in the right half-plane and near the imaginary axis.

## The Characteristic of Stability of the Solution in the Problem of Convex Compact Set Asphericity

We consider the problem of stability of the solution in the problem of asphericity of a convex set with respect to the error of defining the compact set. It is shown that the optimal value of the criterion function (an asphericity indicator) is stable. Properties of the setvalued mapping, that puts to a convex compact compact set the centers of its asphericity are also investigated. It is proved that this mapping is semicontinious from above everywhere in the space of convex compact sets.

## Stability of Periodic Billiard Trajectories in Triangle

The problem of stability of periodic billiard trajectories in triangles is considered. The notion of stability means the preservation of a period and qualitative structure of a trajectory (its combinatorial type) for sufficiently small variations of a triangle. The geometric, algebraic and fan unfoldings are introduced for stable trajectories description. The new method of fan coding, using these unfoldings, is proposed. This method permits to simplify the stability analysis.