Asymptotics of the Solution of Discrete-Time Linear-QuadraticOptimal Control Problems with Small Step and Weak Control in the CriticalCase

An algorithm is proposed for constructing an asymptotic approximation to the solution of a discrete-time weakly controllable linear-quadratic optimal control problem with small step size in the critical case. The asymptotic expansion consists of a sum of a regular series and two boundary-layer series containing boundary functions in neighborhoods of two fixed endpoints. The construction of the asymptotic expansion is based on the decomposition of the state space into orthogonal sums of subspaces and the use of the corresponding orthogonal projectors. Explicit relations for determining the terms of the asymptotic expansion of any order are provided. An example illustrating the proposed method is presented.