Asymptotic solution of a linear-quadratic problem with discontinuous coefficients and cheap control

This paper is devoted to a linear-quadratic problem with cheap control and discontinuous coefficients. Using the direct scheme, consisting of an immediate substitution of a postulated asymptotic expansion of a solution into the problem condition and finding a family of control problems to define the terms of the asymptotic expansion, the asymptotic solution of the considered problem is obtained. The proximity of the exact solution to the approximate asymptotic one is estimated for the control, trajectory, and functional. It is shown that the minimized functional does not increase when a higher order asymptotic approximation to the optimal control is used. An illustrative example is presented.