Projector approach for constructing the zero order asymptotic solution for the singularly perturbed linear-quadratic control problem in a critical case

The paper deals with linear-quadratic optimal control problems with a weak control and the fixed left point in a critical case, where a matrix standing in front of a state variable in the state equation is singular for any argument value if the small parameter is equal to zero.Using the direct scheme method, consisting in immediate substituting a postulated asymptotic solution into the problem condition and obtaining problems for finding asymptotics terms, the zero order asymptotic solution is constructed under some conditions.In contrast to the paper: N. T. Hoai, J. Optim. Theory Appl., vol. 175, 324–340 (2017), where the considered problem was studied, the projector approach is applied here. This approach allows us to make the algorithm of constructing asymptotic solution clearer and to correct some inaccuracies in the paper mentioned.