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This paper concerns the robust ‘practical’ stabilization for a class of linear controlled stochastic differential equations subject to both multiplicative and additive stochastic noises. Sufficient conditions of the stabilization are provided in two senses. In the first sense, it is proven that almost all trajectories of the stochastic model converge in a ‘mean-square sense’ to a bounded zone located in an ellipsoidal set, while the second one ensures the convergence to a zero zone in probability one. The considered control law is a linear state feedback. The stabilization problem is converted into the corresponding attractive averaged ellipsoid ‘minimization’ under some constraints of bilinear matrix inequalities (BMIs) type. Some variables permit to represent the BMIs problem in terms of linear matrix inequalities (LMIs) problem, which are resolved in a straight manner, using the conventional LMI-MATLAB toolbox. Finally, the numerical solutions of a benchmark example and a practical example are presented to show the efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

We present analysis of lateral variations of density in the upper mantle in the area of contact of the precambrian East European Craton (EEC) and the Palaeozoic Platform (PP) in Poland, obtained by analysis of the gravimetric geoid undulations. A precise 3-D density model of the crust in the study area down to a depth of 50 km, discussed in the first part of this paper ( Majdański et al. , in press) did not explain all features of the observed geoid. This suggests that these features can be due to density inhomogeneities in the upper mantle. To estimate them, we performed inversion of a residual between the observed geoid and undulations caused by the 3-D density distribution in the crust. Basing on the assumption of local isostatic compensation and Pratt–Hayford isostasy model, the density distribution in the upper mantle was parametrized as a 40-km-thick layer located above the assumed compensation depth of 140 km and subdivided into irregular blocks. The boundaries of the blocks were defined according to boundaries of major tectonic units in the study area and position and shape of the most pronounced anomalies in the residual geoid. A series of sensitivity tests calculated for such density heterogeneities in the upper mantle showed that they can produce geoid undulations of the order of several metres. The density values in each unit were taken as model parameters for the inversion procedure, and inverse problem was solved using global optimization with constraints. The density variations in the upper mantle in the final model correlate well with the surface heat flow. This suggests that these variations can be due to diversity in mantle temperature. The Trans-European Suture Zone (TESZ), which is a major suture separating the EEC from the PP, is not observed as a distinct unit in the mantle. Instead, our study suggests continuation of the lithosphere of the EEC beneath the PP and confirms subdivision of the TESZ into terranes with distinctly different evolution. A simple calculation shows that condition of local isostatic equilibrium is almost achieved at the assumed compensation depth, with the exception of several small-scale units. However, different loads and mass deficiencies in the lithosphere are compensated at different depth levels. This proves that the lithosphere is alternation of mechanically strong and weak layers, both in the crust and upper mantle, and that different loads can be compensated at different depth. However, the total compensation of all lithospheric loads occurs in the upper mantle. [ABSTRACT FROM AUTHOR]

An up to date determination of a high-resolution geoid requires the use of best available databases concerning digital terrain model (DTM), bathymetry, global geopotential model and gravity field. The occasion to revisit methods to validate and merge different data sets has been created by a new project for the determination of a new European Geoid. Since the computation of the latest European geoid and quasi-geoid model (EGG97), significant new or improved data sets have become available, such as new global geopotential models from CHAMP and GRACE missions, new national and global DTMs and new or upgraded gravity data sets. In the context of the new European Gravity and Geoid Project (EGGP), within the IAG Commission 2, some data validation tests have been performed in the Italian zone. In the area 19°× 17° wide, covering Italy, three kinds of tests have been performed: comparison among different DTMs in order to choose the best one to be used; comparisons in terms of geoid computation in some coastal areas, to evaluate bathymetry effects, and the validation of the EIGEN-CG01C and EIGEN-CG03C new global models up to degree and order 360. These preliminary tests lead to the choice of SRTM DTM (integrated in no-data holes), with an added bathymetry derived by the Italian 1:25 000 official cartography near the coasts and the NOAA bathymetry in high seas. The validation of the new global models and the comparison with EGM96 model show that, in terms of geoid computation, the EGM96 yields better results. Moreover, the validation of new available land gravity data and the cross-validation of two sets of gravity data on sea have been completed. [ABSTRACT FROM AUTHOR]