Absolute stability of a class of nonlinear systems with nonpositive linear parts

A dynamical system is called positive if its trajectory starting from any nonnegative initial state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive theory is given in the monographs and papers [1, 2, 6, 11, 12]. Variety of models having positive behavior can be found in engineering, economics, social sciences, biology and medicine. The stability of linear and nonlinear standard and positive fractional systems has been addressed in [3–6, 8, 15, 16, 20–23]. The stabilization of positive descriptor fractional systems has been investigated in [10, 11, 20, 21]. The superstable linear systems have been addressed in [17, 18]. Positive linear systems with different fractional orders have been introduced in [14, 13] and their stability has been analyzed in [3, 20]. The absolute stability of a class of positive nonlinear systems has been investigated in [7]. In this paper the positivity and absolute stability of a class of nonlinear continuous-time and discrete-time systems with nonpositive linear parts will be addressed. The paper is organized as follows. In section 2 some preliminaries concerning positivity and stability of linear systems are recalled. The positivity and absolute stability of positive continuous-time nonlinear systems with nonpositive linear