Estimation of solutions of neutral type systems with two incommensurate delays

This paper presents the algorithm for estimating solutions of differential-difference systems of neutral type with two incommensurate delays in the neutral part. It is worth mentioning an important assumption about the commutativity of matrices in the left-hand side of the system. The idea of the approach is to represent the system’s solutions in terms of initial functions and the fundamental matrix and then to construct an exponential estimate for this representation. At the first step, the system's initial conditions are set. Next, the system is rewritten in an integral form and the delay operator is introduced. After recursive application of this operator to the right-hand side of obtained system, the system’s solutions are expressed via binomial coefficients, initial functions and the fundamental matrix. At the final step these expressions are used to make an exponential estimate of the solution. It is proved that the estimate of the fundamental matrix of the system also has an exponential form. In practice, the proposed method allows optimizing the control choice for neutral-type delay systems in sense of one of the crucial characteristics of the controlled systems, i.e. the overshoot value.