Lagrange Adjoint System for Linear Fractional Systems with Distributed Delays

In the present work we introduce an appropriate definition of the notion of a Lagrange (formal) adjoint system for homogeneous systems with incommensurate-order derivatives in Caputo’s sense and distributed delays. Under natural conditions, the existence and uniqueness of the adjoint fundamental matrix for these systems are proven, and an explicit relation between the resolvent of the formal adjoint system and the adjoint fundamental matrix is established. A generalization of the well-known Halanay relation is also introduced and proven. Furthermore, an explicit relation between the fundamental matrix and its adjoint fundamental matrix is established.