Stability, D-Stability, Strong D-Stability of Positive Linear Time-invariant Systems with Applications

Abstract

This paper presents new results on the analysis of positive linear time-invariant systems with non-negative state variables and output data for non-negative initial conditions and inputs. The positive linear time-invariant systems are characterized by the state-space equations which offer a formal mathematical structure of form $$\frac{dx(t)}{dt} = Ax(t)$$ where the system matrix $A\in \mathbb{R}^{n,n}$ is Metzler, with non-negative off-diagonal components. These systems exhibit essential characteristics such as monotonicity, stability, and non-negativity, making them fundamental in applications such as biological systems, mathematical economics, chemical reaction networks, and transportation models. We present the theoretical foundations utilizing a mathematical framework from algebraic systems, matrix theory, and stability analysis to investigate stability, $\mathfrak{D}$-stability, and strong $\mathfrak{D}$-stability of positive linear time-invariant systems in the presence of Metzler and Hurwitz matrices. The numerical testing supports the spectrum analysis and $\epsilon$-pseudospectrum of Metzler matrices.