Decomposition of linear optimal singularly perturbed systems with retardation

An optimal control problem for singular perturbed systems with aftereffect and with a quadratic performance criterion is considered. A change of variables is constructed, which splits the boundary value problem of the maximum principle into a boundary value problem for the slow variables without delay and two initial problems for the fast variables. This splitting allows us to reduce the problem of the design (with a large degree of accuracy) to the solution of an ordinary Riccati differential equation and a system of algebraic equations. One obtains the asymptotic expansion, with respect to the powers of a small parameter of the change of variables, of the “split problems”, and the reinforcement coefficient.