State-space/descriptor models and asymptotic behavior of continuous-time positive control systems

This paper deals with new aspects of the theory of nonnegative matrices and their applications to some problems arising in the studies of continuous-time control systems. The criterion for asymptotic stability of linear state-space control systems with positive state matrix A and respective initial conditions is established and proven. It states that the system is asymptotically stable if and only if the state variables are linearly dependent. Moreover, the case of dynamical systems in descriptor forms with positive leading and state matrices is studied. The asymptotic stability of descriptor systems characterized by positive leading and state matrices is expressed in terms of the geometric relations between matrices of a proper part of the corresponding regular matrix pencil inverse and the initial conditions. Several illustrating examples are also included.