Zero-order asymptotic solution of a class of singularly perturbed linear-quadratic problems with weak controls in a critical case

We consider a linear-quadratic optimal control problem where the second power of a small parameter stands in front of the derivative and a control in a state equation and in front of a quadratic form with respect to control in a performance index; moreover, in the state equation, the nonhomogeneity has the first order of the power of a small parameter, and a matrix in front of the state variable is singular if the small parameter is equal to zero. Using immediate substituting a postulated asymptotic expansion of a solution, containing a regular series and four boundary layer functions series, into the problem condition, we obtain problems for finding asymptotic terms of the zero order for the optimal control and the first order for the optimal trajectory. An illustrative example is given.