Irina Yu. Pototskaya

Control with expenditure criteria in rotational motion of the satellite moving along a circular orbit

In the neighborhood of any of it’s 24 equilibrium points, the controlled rotational motion of a satellite whose center of mass moves along a circular orbit is considered. The admissible control is a piecewise constant function that blanks selected frequency components of the solution of linear equations at the moment T. As a functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T ]. The formulas and algorithms to find the switching points of the control that satisfy the necessary conditions of the functional having an extremum are proposed

Optimal damping of fast-linear oscillations of stationary satellite with the flywheel

The problem of damping of the satellite fast-linear oscillations about its center of mass on yaw and roll channels is considered. The satellite is moving along a stationary orbit and equipped with a hard spun flywheel with a kinematic momentum H. The controlled motion of the satellite can be represented by the linear ODE system with constant coefficients. The admissible control is a piecewise constant function that blanks fast frequency components of the solution of linear equations at the moment T. As “the expenditure” functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the expenditure. To solve this problem the method is proposed which leads to explicit formulas.

Expenditure optimization in a problem of controlled motion of mechanical systems

The controlled motion which is represented by the linear ODE system with constant coefficients is considered. The admissible control is a piecewise constant function that blanks selected frequency components of the solution of linear equations at the moment T. As “the expenditure” functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the Expenditure. To solve this problem the method is proposed which leads to explicit formulas. All results of research are formulated as the theorem. These results can be applied not only in mechanical controlled systems, but also in any problem that can be described by the system of ordinary differential equations with control.

Fuel optimal control of plane oscillations of a satellite on elliptical orbit

Active damping of the free plane oscillations of a satellite on elliptical orbit is discussed. Satellite oscillations are described by Whittaker-Hill equation. Fuel optimal control method is offered. Numerical examples are submitted.

Problems of stability with respect to a part of variables

Mathematical models based on nonlinear differential equations for dynamic systems of classical mechanics and biophysics are considered. The research concentrates on these equations integration features, their solutions properties, stability and behavior nearby an equilibrium point. Factors which can change the solution stability such as a form of a problem definition, a choice of the generalized coordinates and equations describing the process, the presence of small periodic or random perturbation are taken into account in the research.

Error estimates for numerical integration of ODEs in the minimax formulation

Many of ordinary differential equations one can reduce to the form of a polynomial differential system. One of the best methods for the numerical solution of such systems is the method of Taylor series. In this work we consider the Cauchy problem for the polynomial ODE system, and then — a theorem about the accuracy of its solutions by this method. In connection with this theorem, a special minimax problem is formulated. Using this problem one can improve significantly the estimates of the theorem. The solutions of the corresponding minimax problems for a number of ODE systems are given.

Piecewise polynomial control in mechanical systems

The controlled motion of a mechanical system is represented by the linear ODE system with constant coefficients. The admissible control is a piecewise polynomial function that blanks selected frequency components of the solution of linear equations at the moment T. As ”the expenditure” functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the Expenditure. To solve this problem the method that consists of analytical and numerical parts is proposed. All results of research are formulated as the theorem.

Taylor series method of numerical integration of the N-body problem

Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations with polynomial (in unknowns) right-hand sides. First, we show how to obtain such polynomial system fifth, fourth and third degree for classical Newtonian N-body problem. After that, we present comparative data (the relative errors of the coordinates and velocities of bodies and CPU times) for numerical integration of these systems on the interval [0, T] using two different Taylor series method algorithms.Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations with polynomial (in unknowns) right-hand sides. First, we show how to obtain such polynomial system fifth, fourth and third degree for classical Newtonian N-body problem. After that, we present comparative data (the relative errors of the coordinates and velocities of bodies and CPU times) for numerical integration of these systems on the interval [0, T] using two different Taylor series method algorithms.

Fuel optimal control of non-linear oscillations of a satellite on elliptical orbit

The active damping of the free large oscillations of a satellite on elliptical orbit is discussed. The satellite oscillations are described by non-linear differential equation of second order. According to Pontryagins principle of maximum the fuel optimal control has piecewise constant form. Solving and fundamental matrix of the differential equation is determined with Erugins expansion in symbolic form. The problem is reduced to the optimization task with linear quality function and two restrictions. For solving the Sequential Linear Programming method is offered. The ability of the satellite oscillations damping with fuel optimal control is demonstrated by numerical examples.