In this paper we use the method of finite-dimensional approximations to derive the second-order necessary optimality conditions for singular Pontryagin local minimizers in optimal control problems with equality endpoint constraints.

The properties of spaces equipped with a topology defined by a distance function are studied. The considered distance function is not necessarily symmetric but satisfies the so-called f-triangle inequality, which is a weakened version of the usual triangle inequality. Sufficient conditions for metrizability of such spaces are proposed. A construction of a quasimetric topologically equivalent to a given f-quasimetric is proposed

In the paper the concept of covering (regularity) for set-valued mappings in partially ordered spaces is introduced. The coincidence points problem for set-valued mappings in partially ordered spaces is considered. Sufficient conditions for the existence of coincidence points of isotone and orderly covering set-valued mappings are obtained. It is shown that the known theorems on coincidence points of covering and Lipschitz mappings in metric spaces can be deduced from the obtained results.

The actuator based on a single-phase thyristor current inverter (CI) for fast plasma position control systems in tokamaks is described. It has been successfully applied in the operating Globus-M/M2 (Ioffe Institute, SPetersburg, Russia) and T-11M (TRINITI, Troitsk, Russia) tokamaks for plasma position control. The mode of operation for the CI, allowing to control the current value in an inductive load, is described. The control algorithm for the CI, forming the average (on a period of self-oscillations) output voltage, proportional to the value of the input control signal at minimal amplitude of the voltage self-oscillations, is designed. The initial phase of the CI operation (current increasing through the inductive energy storage and the CI starting up into the selfoscillating mode of operation) is described. The autonomous protection elements for the CI are presented. The application results from the present tokamaks have demonstrated the reliability of the designed control algorithm and the declared quality of the CI operation reaching. The linear identification performed defines the linear CI's model for linear control system synthesis showing the dynamical properties of the CI in terms of the linear system analysis.

Abstract: The structure of the low-order model is characterized by the fact the eigenvalues of its dynamics matrix coincide with the weakly stable eigenvalues of the system, which are calculated using the spectrum of the system dynamics matrix and the method of the gramians spectral decompositions. The optimal synthesis of the simplified model parameters is based on the approximate solution of the matrix polynomial equation for the matrices of the numerator of the system matrix transfer function and its simplified model. The solution of the synthesis problem leads to the solution of the linear matrix algebraic equation formed by the adaptability matrices. The energy functionals of the system and its simplified model can be determined by means of spectral decompositions of the squared H2 -norm of the matrix transfer function of the system and its simplified model. When the system approaches to the stability boundary, the stability margins of the system and its simplified model determined by the dominant weakly stable eigenvalues coincide asymptotically. This makes it possible not only to guarantee the stability of the simplified model, but also to determine the stability margins of a high-order system with help of the stability margins of a low-order model. The method makes it possible to estimate the oscillational instability caused by the resonant interaction of intraregional and inter-area oscillations and also to evaluate the impact of the input and output system matrices on the system stability using input and output matrices of its simplified model.