We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial.

We propose a method for the construction and investigation of invariant sets of differential systems described by cone inequalities with the use of the operator of differentiation along the trajectories of the system. Well-known conditions for the positivity of linear and nonlinear differential systems with respect to typical classes of cones are generalized. A method for comparison and ordering is developed for a family of dynamical systems.

New methods of robust stability analysis for equilibrium states and optimization of linear dynamic systems are developed. Sufficient stability conditions of the zero state are formulated for a linear control systems with uncertain coefficient matrices and measurable output feedback. In addition, a general quadratic Lyapunov function and ellipsoidal set of stabilizing matrices for the feedback amplification coefficients are given. Application of the results is reduced to solving the systems of linear matrix inequalities.

The article examines and analyzes the state and economic development of the tourist market of Ukraine. The tourism sector is one of the strategic sectors of the economy, through which further socio-economic development of the state takes place. The tourism industry is an important factor in improving the quality of life in Ukraine, creating additional jobs, replenishing the state's foreign exchange reserves and increasing its authority in the international arena. Economic processes in the tourism industry that change over time can be studied on the basis of linear and nonlinear Dynamic models. The article applies an economic and mathematical model of competition of the Lotki - Volterra type, which describes the dynamics of changes in market shares by types of tourism (inbound, outbound, domestic). The projected market share by type of tourism is calculated within the framework of the proposed model. Unlike classical econometric models of competition, the Lotka-Volterra economic model allows us to fully assess the dynamics of economic processes, achieve a state of equilibrium of the studied competitive systems, and theoretically predict and control the main parameters of the model. A mathematical dependence of the dynamics of changes in the income of tour operators and travel agents on the number of tourists is also constructed. In the course of the research, methods of statistical analysis, methods of economic and mathematical modeling were used to determine the share of the tourist market and its dynamics. As a result of the study, it was found that the rate of income change is most influenced by the number of incoming tourists. A decrease in the flow of domestic tourists has a negative impact on income growth. According to the constructed Lotkа-Volterra competitive model, there is a decrease in the share of incoming and domestic tourists. The development of inbound tourism, the share of which in Ukraine has remained the smallest since 2005, will significantly increase the income of tour operators and travel agencies. These dynamic models allow you to analyze processes in the tourism industry, perform forecast assessments, develop a strategy, make management decisions and carry out planning at various levels.

Purpose. The aim of the article is the analysis of inequality of the population of Ukraine by sources of income. Methodology of research. A set of general scientific and specific methods of economic research was used in the course of the research, in particular, methods of theoretical generalizations: analysis, abstract and logical method (at substantiation of modern tendencies of inequality of the population of Ukraine on incomes); economic and statistical methods (in assessing the dynamics of inequality of the population of Ukraine in terms of monetary income); methods of mathematical statistics (when calculating the Gini index and the decomposition of income); methods of graphic display of the received results of research. Findings. The Gini index was calculated using the spline interpolation method to construct the Lorentz function. The Gini index is compared with known calculation methods, the description of the extended method of decomposition of the Gini index is carried out, and also this method is applied on statistical data of distribution of households of Ukraine on the level of monetary incomes. The influence of changes in sources of income on the general level of differentiation is analysed. According to the results of the study, the main factors influencing the level of income distribution and the possibility of reducing the level of stratification of the population of Ukraine were identified. Originality. It is proposed to use the spline interpolation method to construct the Lorentz function, which involves the calculation of the Gini index. The method of the Gini index decomposition identifies the sources of income that have the greatest impact on the growth of income inequality in Ukraine in the period 2014-2020. Practical value. The obtained results of the research are useful in the study of determining the reasons for the influence of certain factors on the level of income distribution and the possibility of reducing the level of stratification of the population of Ukraine. Key words: population inequality, Lorentz curve, Gini index, Gini index decomposition, spline interpolation.