V. A. Trapeznikov Institute of Control Sciences

В работе выводится принцип максимума Понтрягина для общей задачи оптимального управления, где основным инструментом является абстрактная лемма об обратной функции, доказательство которой существенно опирается на теорему Шаудера о неподвижной точке. Такой подход позволяет сделать доказательство принципа максимума Понтрягина достаточно коротким и весьма прозрачным.

The paper considers a game of three players on the plane attacker, target and defender (an ADT-game). The attacker’s dynamics is described by simple motions, while target and defender are moving in a strait line. At the same time in the game model the attacker receives information about the environment through an acoustic channel in which the target and the defender broadcasts some signals. The field of view of this channel is limited by a cone. The attacker’s goal is to intercept the target and the goal of the target-defender coalition is to delay the interception as much as possible by jamming the attacker’s observation channel. During the chase the attacker uses the law of proportional navigation and various algorithms for target’s bearing estimation which are considered known to the target-defender coalition. The behavior of the system is investigated when using a well-known and a promising intelligent estimation algorithm. For each case an optimization problem is set to optimize the defender’s release angle in order to increase the interception time. The dynamics of all players and various trajectories of the defender in conditions of noiseless and noisy observation channels are simulation, the obtained results are compared.

Stochastic first-order methods are standard for training large-scale machine learning models. Random behavior may cause a particular run of an algorithm to result in a highly suboptimal objective value, whereas theoretical guarantees are usually proved for the expectation of the objective value. Thus, it is essential to theoretically guarantee that algorithms provide small objective residuals with high probability. Existing methods for non-smooth stochastic convex optimization have complexity bounds with the dependence on the confidence level that is either negative-power or logarithmic but under an additional assumption of sub-Gaussian (light-tailed) noise distribution that may not hold in practice. In our paper, we resolve this issue and derive the first high-probability convergence results with logarithmic dependence on the confidence level for non-smooth convex stochastic optimization problems with non-sub-Gaussian (heavy-tailed) noise. To derive our results, we propose novel stepsize rules for two stochastic methods with gradient clipping. Moreover, our analysis works for generalized smooth objectives with Hölder-continuous gradients, and for both methods, we provide an extension for strongly convex problems. Finally, our results imply that the first (accelerated) method we consider also has optimal iteration and oracle complexity in all the regimes, and the second one is optimal in the non-smooth setting.

The paper is devoted to the study of information characteristics which play an important role in solving problems of detection of weak signals and their classification. We formulate and prove lemmas on estimating upper and lower boundaries in statistical and spectral complexity diagrams for different signal-noise mixtures. The obtained theoretical results are verified by numerical experiments, which have confirmed the efficiency of the theoretical estimates. We establish some important laws of the behavior of noise-like and weak signals and the possibilities of their detection in white and blue noise conditions, formulate and prove the corresponding lemmas. Also, we obtain estimates on conditions for the possibility of classifying weak signals by means of information characteristics.