A Matrix-Operator Formulation of Conformable Sylvester and Lyapunov Differential Matrix Equations

This paper presents a systematic matrix-operator formulation for linear Sylvester and Lyapunov matrix differential equations governed by the conformable derivative. For differentiable functions, the change of variable ξ=tr/r converts the conformable problem into a classical matrix ordinary differential equation. Within this local time–rescaled setting, vectorization, Kronecker products, and Kronecker sums yield transparent variation-of-constants representations for homogeneous and nonhomogeneous problems, with the Lyapunov equations obtained as direct corollaries of the Sylvester formulation. The emphasis is on a careful structural derivation, corrected operator identities, explicit regularity and stability conditions, and a reproducible computational interpretation rather than on a new nonlocal solution mechanism. We compare the matrix-oriented representation with direct Kronecker vectorization and RK4 time stepping, discuss their computational costs and appropriate use cases, and add a parameterized RLC-circuit controllability–Gramian case study. Examples 3–5 are independently verified by classical fourth-order Runge–Kutta integration in the rescaled time variable.