The use of the models with aftereffect in the problem of synthesis of optimal digital control systems

The linear-quadratic optimal control problem for digital systems with small discretization period is considered. The discrete-continuous description for the systems in the form of differential equations with aftereffect is used. The initial boundary value problem (maximum principle) is decomposed in Coshi problem for the first order equations in partial derivatives and in the boundary value problem for the system of usual differential equations. These results allow to reduce the problem of optimal control synthesis to usual differential equation (Riccati type) and to construct the asymptotics of optimal solution.