Mikhail Z. Zgurovsky

On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions

In the first chapter, we considered the existence and properties of global attractors for autonomous multivalued dynamical systems. When the equation is nonautonomous, new and challenging difficulties appear. In this case, if uniqueness of the Cauchy problem holds, then the usual semigroup of operators becomes a two-parameter semigroup or process [38, 39], as we have to take into account the initial and the final time of the solutions.

Structure of Uniform Global Attractor for General Non-Autonomous Reaction-Diffusion System

In this paper we study structural properties of the uniform global attractor for non-autonomous reaction-diffusion system in which uniqueness of Cauchy problem is not guarantied. In the case of translation compact time-depended coefficients we prove that the uniform global attractor consists of bounded complete trajectories of corresponding multi-valued processes. Under additional sign conditions on non-linear term we also prove (and essentially use previous result) that the uniform global attractor is, in fact, bounded set in \(L^{\infty }(\varOmega )\cap H_0^1(\varOmega )\).

Multivalued Dynamics of Solutions for Autonomous Operator Differential Equations in Strongest Topologies

We consider nonlinear autonomous operator differential equations with pseudomonotone interaction functions satisfying \((S)\)-property. The dynamics of all weak solutions on the positive time semi-axis is studied. We prove the existence of a trajectory and a global attractor in a strongest topologies and study their structure. As a possible application, we consider the class of high-order nonlinear parabolic equations.

Uniform Trajectory Attractors for Nonautonomous Dissipative Dynamical Systems

For all global weak solutions of the general classes of nonautonomous evolution equations and inclusions that satisfy standard sign and polynomial growth conditions, the multivalued dynamics as time \(t\rightarrow +\infty \) is studied. The existence of a compact uniform trajectory attractor is justified. The obtained results allow to investigate long-time behavior of distributions of state functions for various mathematical models in geophysics, mechanics, biology, medicine etc.

Optimality Conditions for Partially Observable Markov Decision Processes

This note describes sufficient conditions for the existence of optimal policies for Partially Observable Markov Decision Processes (POMDPs). The objective criterion is either minimization of total discounted costs or minimization of total nonnegative costs. It is well-known that a POMDP can be reduced to a Completely Observable Markov Decision Process (COMDP) with the state space being the sets of believe probabilities for the POMDP. Thus, a policy is optimal in POMDP if and only if it corresponds to an optimal policy in the COMDP. Here we provide sufficient conditions for the existence of optimal policies for COMDP and therefore for POMDP.

Uniform Global Attractors for Nonautonomous Evolution Inclusions

In this note, we prove the existence and provide basic structure properties of compact (in the natural phase space) uniform global attractor for all global weak solutions of the general classes of nonautonomous evolution equations and inclusions that satisfy standard sign and polynomial growth conditions. The obtained results allow to reduce the problem of the complete qualitative investigation of various nonlinear systems into the “small” (compact) part of the natural phase space.

Adaptive Control of Impulse Processes in Complex Systems Cognitive Maps with Multirate Coordinates Sampling

Cognitive map (CM) is a popular method of complex systems description. The system of first-order difference equations in variables increment, based on weighting coefficients of CM, is used to describe impulse process of the system. If different vertices coordinates of CM are measured with different frequencies multirate sampling impulse process model should be developed. The current paper proposes such a model and adds external control vectors with multirate sampling to allow to affect impulse process dynamics. To stabilize this multirate system’s coordinates at predefined levels two optimality criteria are proposed and correspondent control laws are derived. Controls are also multirate, i.e. frequently measured coordinates are affected by controls frequently and infrequent coordinates are affected with longer sampling period. For the case when weighting coefficients of CM are unknown or varying special algorithm of their estimation is developed. The results are verified by simulation performed for CM of a bank.

Application of Fuzzy Logic Systems and Fuzzy Neural Networks in Forecasting Problems in Macroeconomy and Finance

This chapter is devoted to numerous applications of fuzzy neural networks in economy and financial sphere. In the Sect. 4.2 the problem of forecasting macroeconomic indicators of Ukraine with application of FNN is considered. The goal of this investigation was to estimate the efficiency of different fuzzy inference algorithms. Fuzzy algorithms of Mamdani, Tsukamoto and Sugeno were compared in forecasting Consumer Price Index (CPI) and GDP of Ukraine. As result of this investigation the best forecasting algorithm is detected.

Fuzzy Neural Networks in Classification Problems

The purpose of this chapter is consideration and analysis of fuzzy neural networks in classification problems, which have a wide use in industry, economy, sociology, medicine etc. In the Sect. 5.2 a basic fuzzy neural network for classification—NEFClass is considered, the learning algorithms of rule base and MF of fuzzy sets are presented and investigated. Advantages and lacks of the system NEFClass are analyzed and its modification FNN NEFClass M, free of lacks of the system NEFClass is described in the Sect. 5.3. The results of numerous comparative experimental researches of the basic and modified system NEFClass are described in Sect. 5.4. The important in a practical sense task of recognition of objects on electro-optical images (EOI) is considered and its solution with application of FNN NEFClass is presented in the Sect. 5.5. The comparative analysis of different learning algorithms of FNN NEFClass at the task of recognition of EOI objects in the presence of noise is carried out. Problem of hand-written mathematical expressions recognition is considered in the Sect. 5.6 and its solution with application of FNN NEFClass is presented.

Inductive Modeling Method (GMDH) in Problems of Intellectual Data Analysis and Forecasting

This chapter is devoted to the investigation and application of fuzzy inductive modeling method known as Group Method of Data Handling (GMDH) in problems of intellectual data analysis (Data Mining), in particularly its application in the forecasting problem in macroeconomy and financial sphere. The problem consists in forecasting models construction and finding unknown functional dependence between given set of macroeconomic indices and forecasted variable using experimental data. The advantage of inductive modeling method GMDH is a possibility of constructing adequate model directly in the process of algorithm run. The specificity of fuzzy GMDH is getting the interval estimates for forecasting variables. In this chapter the review of main results concerning GMDH and fuzzy GMDH is presented, analysis of application of various membership functions (MF) and perspectives of fuzzy GMDH application for forecasting in macroeconomy and financial sphere are estimated. The Sect. 6.2 contains the problem formulation. In the Sect. 6.3 main principles and ideas of GMDH are considered. In the Sect. 6.4 the generalization of GMDH in case of uncertainty—new method fuzzy GMDH suggested by authors is described which enables to construct fuzzy models almost automatically. The Sect. 6.5 contains the algorithm of fuzzy GMDH. In the Sect. 6.6 the fuzzy GMDH with Gaussian and bell-wise membership functions MF are considered and their similarity with triangular MF is shown. In the Sect. 6.7. fuzzy GMDH with different partial descriptions in particular orthogonal polynomials of Chebyshev and Fourier are considered. In the Sect. 6.8 the problem of adaptation of fuzzy models obtained by GMDH is considered and the corresponding adaptation algorithms are described. The Sect. 6.9 contains the results of numerous experiments of GMDH and fuzzy GMDH application for forecasting share prices and Dow Jones index at New York stock exchange (NYSE). The extension and generalization of fuzzy GMDH in case of fuzzy inputs is considered and its properties are analyzed in the Sect. 6.11.