Explicit Commutativity for Lamé Linear Time-Varying Differential Systems

This article studies the commutativity and sensitivity of the Lamé linear time-varying systems (LTVSs), and investigates the effects of disturbances on such systems. The commutative pair for the Lamé LTVS (Formula presented.) of order (Formula presented.) is found, that is, a new Lamé LTVS (Formula presented.) of order (Formula presented.) is derived using the explicit commutative theories for zero initial conditions (ICs). For the case of nonzero ICs, the commutativity between the connected input–output of Lamé systems (Formula presented.) and (Formula presented.) is studied. New and simple explicit commutative theories and conditions for second-order LTVSs are derived, simplifying the use of commutativity for practical and industrial scenarios. These findings enable us to analyse the commutativity, sensitivity, robustness and stability of Lamé systems, and to determine the effects of disturbances. The explicit results presented in this article are supported by simulations and verified by examples and constitute a significant contribution to science and engineering applications.