Tarasenko O. The asymptotic solution of linear singularly perturbed of optimal control problems with the degenerations

The dissertation research is devoted to the construction of asymptotic solution of linear singularly perturbed optimal control problem. We considered the cases when matrix at derivatives is non-singular, and when it degenerates with convergence of the small parameter to zero and in case of full degeneration. We investigated the case of the degeneration matrix of kriteriya quality. The possibility of applying L.S. Pontryagin's principle of maximum to linear of optimal control problem with full degeneration matrix at derivatives is investigated in the case of reducing system, which describes the process to the Central canonical form. The sufficient terms of controllability and observation for linear non-stationary processes that describe systems of differential equations with full degeneration matrix at derivatives are found. Using the theory of asymptotic integration of singularly perturbed systems of differential equations with degenerations, L.S. Pontryagin's principle of maximum and O.A. Boychuk's methods of analisys of Noether's boundary-value problems, the method of construction of asymptotic solutions of the given optimal control problem is suggested. This method is based on the idea of reducing of given pairpoint boundary-value problem. The algorithm of construction of the asymptotics in cases of simple and multiple spectrum of the boundary bundle of matrixes is worked out. The existence and uniqueness conditions for the solution of this optimal control problem in each case have been found. The recurrent formulas for finding the coefficients of the corresponding series in explicit form are developed, the associated asymptotic estimates are obtained.