SUMMARYThe paper contains a survey of results devoted to one of the numerical methods of optimal control-the method of successive approximations. This method is based on Pontryagin's maximum principle and is known in the English literature as the min-H method. Various modifications of the method and some theoretical results on its convergence are presented. Examples of applications of the method for the calculation of optimal trajectories are given. The method of small parameters which is close to the method of successive approximations, is also described.

Spectral analysis of 100-year time series of Arctic surface temperature (Arctic dT), mean temperature of 200-m water column along the Kola meridian and global surface temperature anomaly (Global dT) was performed. It is shown that climatic indices of the Arctic region undergo long-term 50Á70-year fluctuations similar to fluctuations of Global dT and Arctic dT for the last 1500-year reconstructed period and the recent 140 years of instrumental measurements. Long-term changes of Atlantic spring-spawning herring and Northeast Arctic cod commercial stocks also show 50Á70-year fluctuations that are synchronous with the fluctuations of climatic indices. A simple stochastic model is suggested that makes it possible to predict the probable trends of basic climatic indices and populations of major commercial fish species for up to 20Á30 years into the future. The results obtained also elucidate the old discussion: which factor is more influential on the longterm fluctuations of major commercial stocks, climate or commercial fisheries? We think that what we have learned is that climate is of priority, but which factor(s) cause(s) what, for which species, seems as yet unanswered.

Analysis of the long-term dynamics of World Fuel Consumption (WFC) and the Global Temperature anomaly (dT) for the last 140 years shows that unlike the monotonously and exponentially increasing WFC, the dynamics of global dT against the background of a linear, age-long trend, undergo quasi-cyclic fluctuations with about 60 a year period. No true linear correlation has taken place between the dT and WFC dynamics in the last century.Spectral analysis of reconstructed temperature for the last 1420 years and instrumentally measured for the last 140 years global dT shows that dominant period for its variations for the last 1000 years lies in the 50-60 years interval.Modeling of roughly 60-years cyclic dT changes suggest that the observed rise of dT will flatten in the next 5-10 years, and that we might expect a lowering of dT by nearly 1-0.15°C to the end of the 2020s.

The prognostic properties of four low-frequency seismic noise statistics are discussed: multi- fractal singularity spectrum support width, wavelet-based smoothness index of seismic noise waveforms, minimum normalized entropy of squared orthogonal wavelet coefficients and index of linear predictability. The proposed methods are illustrated by data analysis from broad-band seismic network F-net in Japan for more than 15 years of observation: since the beginning of 1997 up to 15 of May 2012. The previous analysis of multi-fractal properties of low-frequency seismic noise allowed a hypothesis about approaching Japan Islands to a future seismic catastrophe to be formulated at the middle of 2008. The base for such a hypothesis was statistically significant decreasing of multi-fractal singularity spectrum support width mean value. The peculiarities of correlation coefficient estimate within 1 year time window between median values of singularity spectra support width and generalized Hurst exponent allowed to make a decision that starting from July of 2010 Japan come to the state of waiting strong earthquake. This prediction of Tohoku mega-earthquake, initially with estimate of lower magnitude as 8.3 only (at the middle of 2008) and further on with estimate of the time beginning of waiting earthquake (from the middle of 2010) was published in advance in a number of scientific articles and abstracts on international conferences. It is shown that other 3 statistics (except singularity spectrum support width) could extract seismically danger domains as well. The analysis of seismic noise data after Tohoku mega-earthquake indicates increasing of probability of the 2nd strong earthquake within the region where the north part of Philippine sea plate is approaching island Honshu (Nankai Trough)

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Data of continuous records of low-frequency (periods from 2 to 1,000 min) seismic noise on a global network of 229 broadband stations located around the world for 23 years, 1997–2019, are analyzed. The daily values of the entropy of the distribution of the squares of the orthogonal wavelet coefficients are considered as an informative characteristic of noise. An auxiliary network of 50 reference points is introduced, the positions of which are determined from the clustering of station positions. For each reference point, a time series is calculated, consisting of 8,400 samples with a time step of 1 day, the values of which are determined as the medians of the entropy values at the five nearest stations that are operable during the given day. The introduction of a system of reference points makes it possible to estimate temporal and spatial changes in the correlation of noise entropy values around the world. Estimation in an annual sliding time window revealed a time interval from mid-2002 to mid-2003, when there was an abrupt change in the properties of global noise and an intensive increase in both average entropy correlations and spatial correlation scales began. This trend continues until the end of 2019, and it is interpreted as a feature of seismic noise which is connected with an increase in the intensity of the strongest earthquakes, which began with the Sumatran mega-earthquake of December 26, 2004 (M = 9.3). The values of the correlation function between the logarithm of the released seismic energy and the bursts of coherence between length of day and the entropy of seismic noise in the annual time window indicate the delay in the release of seismic energy relative to the coherence maxima. This lag is interpreted as a manifestation of the triggering effect of the irregular rotation of the Earth on the increase in global seismic hazard.

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