Valery Y. Glizer, Stabilizability of a singularly perturbed functional-differential system with two fast time scales, Vol. 2025 (2025), Article ID 1, pp. 1-18

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DOI: 10.23952/cot.2025.1

Received October 8, 2023; Accepted February 3, 2024; Published online October 17, 2024

Abstract. A singularly perturbed linear functional-differential controlled system is considered. The coefficients of the system are time-invariant. The functional nature of the system is because of the presence of the state delays in the general form of Stieltjes integrals. The singular perturbations are due to the presence of two small positive multipliers for some of the derivatives in the system. These multipliers (the parameters of singular perturbations) are of different orders of the smallness. The delay in the slow state variable is non-small (of order of 1). The delays in the fast state variables are small of orders of the corresponding parameters of singular perturbations. The original system is decomposed asymptotically into three simpler parameters-free subsystems. It is established in the paper that the $L^{2}$ -stabilizability of these subsystems yields the $L^{2}$ -stabilizability of the original singularly perturbed system for all sufficiently small values of the parameters of singular perturbations. The theoretical results are illustrated by example.

How to Cite this Article:
V.Y. Glizer, Stabilizability of a singularly perturbed functional-differential system with two fast time scales, Commun. Optim. Theory 2025 (2025) 1.