(Project Head: Professor V. N. Afanasyev)
Work Objective: Optimization of stochastic systems, research in functional analysis and algebra; synthesis of optimal and guaranteeing controls in nonlinear systems.
The formal description of mathematical models of the estimation and control of safety is suggested on the base of a semi-Markov stochastic process with catastrophes, conditions of existence and an explicit expression for the functional of the type of a long-run average income for the semi-Markov process are received. Using methods of the moment theory, the form of optimum strategy of insurance and reinsurance in the model of individual risk is obtained; optimality equations are found. Fourier's method is used to construct optimal control of oscillations; it turns out that the trajectories have an infinite number of switchings.
Studies on the general theory of operators with the detailed description of properties of the linear operators possessing property of a coercitivity are done; results on a structure of the trees corresponding to algebraic polynomials of Chebyshev and Zolotaryov type are obtained. Limit ergodic theorems for functional of the fracture form defined on trajectories of the Markov process are proved; an estimation of the norm of Fourier coefficients that guarantees linearity of a function on a circle is found.
A method for the synthesis of guaranteeing the control of objects with analytical nonlinearities is proposed. This uses the transformation of the initial object to an adequate system having linear structure with parameters depending on the system state. The transformed object is used for description of a model having "the worst parameters for the solution of optimal control problem". The model is used in the synthesis of the needed guaranteeing control. A method of defining the penalty matrixes in a quality functional is presented. The received results are published in journals of the Russian Academy of Sciences.