EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6047 ISSN 1307-5543 – ejpam.com Published by New York Business Global Stability, D-stability, Strong D-Stability of Positive Linear Time-invariant Systems with Applications Mutti-Ur Rehman1,∗, Jehad Alzabut2,3,∗, Amanullah Phulpoto4, Mohamed Tounsi5, Rajaa Al Naimi6 1 Center of Research and Innovation, Asia International University, Yangiobod MFY, G‘ijduvon Street, House 74, 200100 Bukhara, Uzbekistan 2 Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia 3 Department of Industrial Engineering, OSTIM Technical University, 06374 Ankara, Türkiye 4 Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan 5 Computer Science Department, Prince Sultan University, 11586 Riyadh, Saudi Arabia 6 Department of Mathematics, Faculty of Art and Science, University of Petra, 11196, Amman, Jordan 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 dx(t) dt = Ax(t) where the system matrix A ∈ Rn,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, D-stability, and strong D-stability of positive linear time-invariant systems in the presence of Metzler and Hurwitz matrices. The numerical testing supports the spectrum analysis and ϵ- pseudospectrum of Metzler matrices. 2020 Mathematics Subject Classifications: 15A18, 15A16, 15A23 Key Words and Phrases: Singular values, structured singular values, strong D-stable matrix, positive linear systems, Metzler matrices ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6047 Email addresses: muttiur.abbasi@oxu.uz (M. U. Rehman), jalzabut@psu.edu.sa (J. Alzabut), amanullah@iba-suk.edu.pk (A. Phulpoto), mtounsi@psu.edu.sa (M. Tounsi), rajaa.alnaimi@uop.edu.jo (R. Al Naimi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 2 of 33 1. Introduction The stability theory concerning positive linear time-invariant (LTI) systems is well- documented and widely accessible in the literature. However, when extending this theory to nonlinear systems, systems with uncertainties, or positive LTI systems, the framework remains less developed and not well established. The stability characteristics of positive LTI systems have been generalized to include positive descriptor systems and were further analyzed in [1]. Additionally, in [2], stability theory for positive LTI systems was expanded to switched positive linear systems. Moreover, the concept of D-stability for positive switched linear systems was further refined, leading to the derivation and presentation of new findings in [2]. The time-invariant linear system characterized by dx(t) dt = Ax(t), is classified as positive if and only if A ∈ Rn,n is a Metzler matrix, meaning that all its off-diagonal elements are non-negative. As demonstrated in [3], such a system attains global asymptotic stability if and only if the system dx(t) dt = DAx(t), is asymptotically stable, where D represents a positive diagonal matrix. In [4], it was demonstrated that a delayed positive linear system of the form dx(t) dt = Ax(t) + Bx(t− τ), τ ≥ 0, where A is a Metzler matrix and B is a non-negative matrix, exhibits global asymptotic stability. Furthermore, [5] explored theoretical developments in cooperative systems that maintain homogeneity of arbitrary degree under certain dilation mappings. Considering the linear time-invariant system: dx(t) dt = Ax(t), where A is an n × n real Metzler matrix and x(t) ∈ Rn,1, the system is deemed stable if the real parts of all eigenvalues of A are strictly negative. Such a system satisfies the Hurwitz condition, ensuring that all eigenvalues reside within the open left half-plane of the complex domain. For a system of this type, one can deduce that stability implies the existence of a positive matrix P such that: ATP + PA < 0. Additionally, the stability condition ensures that there exists a positive diagonal matrix D. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 3 of 33 The notion of D-stability for real-valued matrices was given by Arrow and McManus [6] in their classical paper. It was then introduced by Enthoven and Arrow [7] for the analysis and interpretations of stability of competitive markets. The concept of D-stability holds significant importance across various fields, for instance, in the study of models in eco- nomics, see [8–14]. An n-dimensional real matrix A is considered D-stable if there exists a positive diagonal matrix D such that either DA or AD remains positive stable. Charac- terizing D-stability is highly challenging, and in certain cases, it leads to computationally intractable (NP-hard) problems; see [15, 16]. The problems concerning D-stability for the linear systems have received a great atten- tion over the last few decades. Numerous significant findings and mathematical approaches were established in the literature; refer to [17–25] and the associated references. The se- lection of an appropriate region(s) in the complex plane, for instance, D(α, r) is a disk, centered at α + i0, having radius r (see [26]) plays a vital role for the characterization of D-stability. The analysis on D-stability for singular systems is indeed very complicated and a challenging problem as compared to normal state space systems. The establishment of numerous significant results for D-stability, regularity, and causality is documented in [27–30] and the cited sources. For a given stable matrix A ∈ Rn,n, it is well established that the matrix A+M remains stable, provided that ∥M∥ < γ for some γ > 0. see [31, 32]. This further ensures that the stability is a property that is very much robust to some small admissible perturbations appearing across the system. For given D, a positive diagonal matrix, assume that A ∈ Rn,n, is D-stable matrix, then one can easily observe that matrix products DA, and D(A+M) are also stable. The important question that arises at this stage is then to ask if it is possible to search for a suitable γ > 0, so that A + M is a D-stable matrix for |M | < γ. But, Unfortunately, in general, the response is a NO. However, in certain special cases, it is indeed feasi- ble to identify a class of D-stable matrices where small, permissible perturbations result exclusively in D-stable matrices. The given A ∈ Rn,n is strongly D-stable if there exist a γ > 0 such that A + M is a D-stable for each M ∈ Rn,n, with |M | < γ. Furthermore, one can easily notice that each strongly D-stable matrix must be a D-stable matrix. The mathematical problem of optimization of spectral abscissa over the families of Metzler matrices is helpful to study and analyze the stabilization of the Metzler matrices. The objective function is neither convex not concave, not Lipschitz and this cause the problem to be very hard. Further, from this one might obtained many local extrema which in turn are very hard to localize, see [33]. The necessary and sufficient condition for a given matrix to be D-stable by using Kalman-Yacubovich-Popov lemma were given in [34]. It was shown that obtained condition is mathematically equivalent to requirement that pair of linear-time-invariant systems have common Lyapunov function in the lower dimensions. Furthermore, the simple conditions for Hurwitz stability of a given Metzler matrix were derived. In recent years, the study of fractional differential equations (FDEs) has gained signif- M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 4 of 33 icant attention due to their ability to model memory and hereditary properties in various physical and engineering systems. Analytical methods such as the Laplace transform method, Adomian decomposition method (ADM), and the homotopy analysis method (HAM) have been widely applied to obtain exact or approximate solutions to FDEs. These approaches are especially useful for linear or weakly nonlinear problems, offering insight into the qualitative behavior of solutions. On the numerical side, methods like the finite difference method (FDM), finite element method (FEM), and spectral methods have proven effective in handling more complex or strongly nonlinear problems, particu- larly when closed-form solutions are not attainable. More recent advancements include the development of Grünwald–Letnikov and Caputo-based numerical schemes, which are particularly suited for time-fractional models, as well as predictor–corrector algorithms and adaptive mesh techniques that improve accuracy and efficiency. For a comprehensive overview, the works of [35–40] provide foundational insights into both the theoretical and practical aspects of these methods. The spectra and pseudo-spectra of structured matrices to different linear and non- linear mathematical problems helps to analyze their behavior. The spectrum and pseudo- spectrum of D-stable matrices for an economic model was the subject of some recent novel mathematical studies, see [41]. In [42] a characterization of D-stable matrices from transportation problems were analyzed. A in-depth mathematical analysis on stability, D- stability, and pseudo-spectrum of structured matrices arising from economic models was studied in [43]. The most recent mathematical findings on interaction between µ-values and Schur stability were studied and analyzed in [44]. Our main objective in this article is to extend the results on stability, D-stability, and strong D-stability theory. Mainly, we target problems of linear time-invariant systems which are positive and have following mathematical formulations: Problem-I: To extend and construct some new insights on stability, D-stability and strong D-stability analysis of LTI system dx(t) dt = Ax(t); x(t) ∈ Rn,1, A ∈ Rn,n, where A is Metzler and Hurwitz. Problem-II: To extend and construct some new findings on stability, D-stability and strong D-stability analysis of LTI system dx(t) dt = A(t)x(t); x(t) ∈ Rn,1, A(t) ∈ {A1, A2}, with A1, A2 being asymptotically stable matrices. Problem-III: To extend and construct some new findings on stability, D-stability and strong D-stability analysis of LTI system dx(t) dt = A(t)x(t); x(t) ∈ Rn,1, A(t) ∈ {D1A1, D2A2}, with A1, A2 being irreducible matrices, and D1, D2 > 0. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 5 of 33 Overview of article: Section 2 is about notations and the preliminary findings on stabil- ity, D-stability, and strong D-stability. In section 3, we recall some basic and fundamental outcomes on Metzler matrices and their spectrum. In section 4, we present some new re- sults on the extension of concepts to stability, D-stability, and strong D-stability of Metzler and Hurwitz matrices. For this purpose, we apply different approaches drawn from linear algebra, matrix analysis, and system theory. The numerical illustration is presented in section 5. The applications to positive linear time-invariant systems having Metzler and Hurwitz as their coefficient matrices are presented in section 6, and then finally in section 7, we present the conclusion to our paper. 2. Preliminaries The problem on the analysis of stability and its properties for linear time-invariant system with positive constraints has been thoroughly investigated and examined in the literature in much greater detail. The positive system stability is an important focus of many disciplines, for instance, applied mathematics, engineering, and computational sciences. We study and analyze the positive system of linear time-invariant of the form dx(t) dt = Ax(t), A ∈ Rn,n, x(t) ∈ Rn,1, with A being Metzler, and Hurwitz. Definition 1. [45] A matrix A ∈ Rn,n is classified as Metzler if all of its off-diagonal entries aij , ∀ i ̸= j are non-negative. Definition 2. The matrix A ∈ Rn,n is Hurwitz if λi(A) ∀i is such that Re(λi(A)) < 1 ∀ i. The D-stability analysis aims to study and analyze the stability of an equilibrium of dynamic models appearing in competitive market. A given n-dimensional real-valued matrix A is a D-stable matrix if for any diagonal matrix D = diag(dii), dii > 0, ∀ i, the matrix DA or AD is a stable matrix. Remark 1. The classification of D-stability is generally difficult, except in the case of matrices with size 3, or less, see [15, 46]. The concept of strong D-stability for n-dimensional real-valued matrix A, introduced in [47], seeks to investigate and characterize the properties of D-stability. Definition 3. The matrix A ∈ Rn,n is stable if real-part of all of its eigenvalues are strictly positive (sometime in literature it maybe consider as strictly negative). Definition 4. [6] The matrix A ∈ Rn,n is D-stable if real-part of all eigenvalues of DA or AD are strictly positive (sometime in literature it maybe consider as strictly negative) for a positive diagonal matrix D. Definition 5. [47] The matrix A ∈ Rn,n is strongly D-stable if there exists γ > 0 so that A + M is a D-stable matrix, for every M ∈ Rn,n with |M | < γ. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 6 of 33 Remark 2. The D-stability of a matrix needs to be a robust property. This can be easily understand with an example (see [47]), for instance, the matrix A(0) = ( 0 1 −1 −1 ) , is a D-stable matrix, but the perturbed matrix, that is, A(γ) = ( γ 1 −1 −1 ) , is not a D-stable matrix for γ > 0. The following Lemma is taken from [48]. Lemma 1. If A is a D-stable matrix, then A is non-singular and each of following matrices are D-stable. 1. AT 2. A−1 3. P TAP for a permutation matrix P 4. DAE for positive diagonal matrices D, E. The following Theorem is taken from [48]. Theorem 1. Let A ∈ Cn,n be a stable matrix. Then following statements are equivalent. 1. A is a D-stable matrix 2. det(A± iD) ̸= 0 for each positive diagonal matrix D. Let B denotes the set of block-diagonal matrices and is defined as B = {diag(δ1Ir1 , δ2Ir2 , · · · , δSIrS ; ∆1,∆2, · · · ,∆F ) : δi ∈ K,∆j ∈ Kmj ,mj ,K = R or C}. The structured singular value, denoted as µB(A), of a real matrix A ∈ Rn,n with respect to B is defined such that µB(A) = 0 if and only if there exists no ∆ ∈ B for which the matrix In − A∆ has at least one eigenvalue equal to zero, where In represents the n × n identity matrix. Otherwise, µB(A) takes a nonzero (positive) value, that is µB(A) := 1 min{||∆||2 : det(In −A∆) = 0} , where min is taken over all ∆ ∈ B. Remark 3. The necessary condition for µ-value is that if det(In − A∆) ̸= 0 for any ∆ ∈ B. Then, 0 ≤ µB(A) < 1. Remark 4. The matrix A ∈ Rn,n is D-stable iff it is stable, and det(A+iD) ̸= 0, i = √ −1 for some specific positive diagonal matrix D, see [13]. Remark 5. The sufficient condition for µ-value is that if 0 ≤ µB(A) < 1, then det(In − A∆) ̸= 0, ∀∆ ∈ B. The analysis and investigation on the behaviour of A ∈ Rn,n subject to an admissible perturbation ∆p, for ϵ > 0 such that ||∆p|| ≤ ϵ, is to study the pseudo-spectrum [49]. For given A ∈ Rn,n and ϵ > 0, the ϵ-pseudospectrum is given by Λϵ(A) := {z ∈ C : ||(A− zIn)−1||−1 < ϵ}, where || · || is matrix-norm. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 7 of 33 Remark 6. If || · || is being Euclidean norm, then Λϵ(A) := {z ∈ C : ||(A− zIn)−1||−1 < ϵ} = {z ∈ C : σmin(A− zIn) < ϵ}, where σmin(·) denotes smallest singular-value of a matrix. 2.1. Sufficient conditions for D-stability: We review C.R. Johnson’s [15] 13 sufficient conditions for the D-stability for a given n-dimensional real-valued matrix A. C1 : All the eigenvalues λi ( DA + ATD ) > 0, ∀i, D is a diagonal matrix which is positive. C2 : The A-matrix means that all of the principal minors are positive and all of the off- diagonal entries are non-positive. C3 : There exists a positive diagonal matrix D such that AD = B = (bij) which satisfies the condition that Re(bii) > n∑ j=1 |bij |; i = 1 : n, j ̸= i. C4 : Given A ∈ Rn,n is a triangular matrix and the real part of all the off-diagonal entries aii is strictly positive. C5 : Given A ∈ Rn,n is a sign stable matrix. C6 : Every principal minor of A ∈ Rn,n is positive, and A is a tri-diagonal matrix. C7 : Given A ∈ Rn,n is an oscillatory matrix, that is, A is totally non-negative matrix. C8 : For each x ∈ Rn,1, x ̸= 0, a positive diagonal matrix D exists such that the real part of xTDAx is positive. C9 : For given A ∈ Rn,n, and for every positive definite matrix P , the Hadamard product of P and A is a stable matrix. C10 : The given A ∈ Rn,n is a strictly sign symmetric matrix and its every minor is positive. C11 : Given A ∈ Rn,n such that A ∈ R2,2 ∩ P+ 0 . C12 : Given A ∈ Rn,n such that A ∈ R3,3 ∩ P+ 0 , and A = x a b α y c β α z  . C13 : Given A ∈ Rn,n such that A ∈ Rn,n ∩ P+ 0 satisfies GKK condition with n ≤ 4. 2.2. Sufficient condition for strong D-stability [50]: For a given A ∈ Rn,n, the sufficient conditions for the strong D-stability are: C1 : For a positive diagonal matrix D, all the eigenvalues λi ( DA + ATD ) < 0, ∀i. C2 : Given A ∈ Rn,n is an A-matrix, that is, all the off-diagonal entries are non-positive and all the principal minors are positive. C3 : There exists a positive diagonal matrix D such that AD = B = (bij) which satisfies the condition that Re(bii) < − n∑ 1≤j≤n |bij |; 1 ≤ i ≤ n, j ̸= i. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 8 of 33 C4 : Given A ∈ Rn,n is a sign triangular matrix, and aii < 0, i = 1 : n.. C5 : Given A ∈ Rn,n is a sign stable matrix without having a any of non-zero entry. C6 : For given A ∈ Rn,n is a jocabi matrix, and each of jth-order principal minor is of sign (−1)j . C7 : Given A ∈ Rn,n is an oscillatory matrix, that is, A is totally non-negative matrix. C8 : For each x ∈ Rn,1, x ̸= 0, there exists a positive diagonal matrix D such that real part of xTDAx is strictly positive. C9 : For given A ∈ Rn,n, the Hadamard product (H ◦ (A + G)) is Schur stable matrix for each positive definite symmetric matrix H, and a perturbation matrix G such that ||G||2 < α, α ∈ R. C10 : For given A ∈ Rn,n each jth-order principal minor is of sign (−1)j . C11 : Given A ∈ R2,2 is strongly D-stable iff its jth-order principal minors are of sign (−1)j . C12 : Given A ∈ R3,3 with all of its jth-order principal minors are with sign (−1)j , and a11a22a33 < a12a23a31 + a21a32a13 2 . C13 : Given Given A ∈ Rn,n is strongly D-stable matrix if for n ≤ 4, and A satisfies GKK condition. 3. Metzler Matrices and their spectrum In this section, we provide an overview of Metzler matrices and their spectral properties, specifically the set of eigenvalues. Metzler matrices are fundamentally linked to the study and analysis of continuous-time linear systems. The spectrum of a Metzler matrix can be determined by shifting the spectrum of a non-negative matrix by αIn, where In represents the n-dimensional identity matrix, as noted in [51]. It is well established that the spectrum of a positive matrix is positive and corresponds exactly to its spectral radius ρ, see [52]. Moreover, as demonstrated in [53], the spectrum of a non-negative matrix is confined to specific regions within the complex plane. These regions, referred to as Kerpelevich regions, define the spectral behavior of such matrices. In [45], the Metzler matrix spectrum has been proven to reside within the Kerpelevich region, shaped like a cone in the complex plane. Additionally, it has been demonstrated that for 3 × 3 matrices, both necessary and sufficient conditions for the spectrum of Met- zler matrices are established. Metzler matrices share a strong connection with positive matrices, particularly in the context of analyzing linear time-invariant systems. In [45], following dynamical system was considered for a demonstration. dx(t) dt = Ax(t) x(0) = x0 x ∈ Rn,1, A ∈ Rn,n. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 9 of 33 Lemma 2. [45] The dynamical system (given as above) is positive iff A is a non-negative matrix, and for all t ≥ 0, the matrix eAt is non-negative. Lemma 3. [45] For A ∈ Rn,n, the following statements are equivalent: (i) eAt ≥ 0, ∀ t ≥ 0. (ii) The matrix A is Metzler matrix. Remark 7. The Metzler matrtix A ∈ Rn,n can be expressed as the sum of a non-negative matrix N , and αIn, where arbitrary number α ∈ R satisfies α ≥ diag(N). For Sr n := {λ ∈ C such that ∃ N ∈ Rn,n, λ ∈ σ(N), N ≥ 0, ρ(N) = r}, one may easily find that the set Sr n can be obtained by the following Theorem 2. Theorem 2. [45] The set Sr n is characterized as: (i) The set Sr n is in a unit disk of complex plane, and it’s spectrum is symmetric about real-axis. (ii) The set Sr n may intersect the unit circle in a finite number of vertices. 4. New Results In this section, we provide recent findings on the stability, D-stability and strong D- stability analysis on positive dynamical systems whose coefficient matrices are Metzler matrices. Mainly, We analyze positive linear time-invariant systems, where the develop- ment of these findings incorporates various principles from linear algebra, matrix theory, and system theory. We use results on the interconnection between µ-theory and D-stability theory to formulate and bring forth new findings on stability, D-stability and strong D- stability. 4.1. The stability of positive linear time-invariant systems: We present some recent findings on stability analysis of positive linear time-invariant systems having the appearance of Metzler, and Hurwitz matrices. Theorem 3 gives the conditions under which linear time-invariant system with n-dimensional real-valued Met- zler, and Hurwitz matrices, is stable. Theorem 3. Let A1, A2 ∈ Rn,n be Metzler matrices and Hurwitz. The linear system: dx(t) dt = A(t)x(t); A(t) ∈ {D1A1, D2A2} : D1, D2 > 0, x(t) ∈ Rn,1, is stable if Re ( λ1(D1(A1 + γD−1 1 D2A2)) ) > ∣∣Re ( λk(D1(A1 + γD−1 1 D2A2) )∣∣ , where λ1 is largest eigenvalue, and λk denotes all the remaining eigenvalues other than λ1, and γ ≥ 0. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 10 of 33 Proof. For the largest eigenvalue λ1, we have that, Re ( λ1(D1(A1 + γD−1D2A2)) ) > 0 because λ1 ∈ R, and∑ i=1 λi ( D1(A1 + γD−1D2A2) ) = Tr ( D1(A1 + γD−1 1 D2A2) ) , where trace of the matrix is denoted by Tr(·). The above expression is strictly positive, and from this it follows that λ1 ( D1(A1 + γD−1 1 D2A2) ) > 0. Next, we aim to show that λ1 ̸= λk, and Re ( λ1(D1(A1 + γD−1 1 D2A2)) ) > ∣∣Re ( λk(D1(A1 + γD−1 1 D2A2) )∣∣ . Consider for λ1 > λk, v⃗1 be normalized eigenvector then,∑ i ai,j v⃗1 = λkv⃗1, and let |v⃗1| = x1, then 0 < Re ( λ1(D1(A1 + γD−1 1 D2A2)) ) = ∑ ij aij v⃗1v⃗k = ∣∣∣∣∣∣ ∑ ij v⃗1v⃗k ∣∣∣∣∣∣ ≤ ∑ ij aijx1x2. Suppose that x2 be an eigenvector corresponding to λ1 ( D1(A1 + γD−1 1 D2A2) ) , then we have ∑ ij aijx2 = λ1x1, x2 ̸= 0⃗. The condition that x2 ̸= 0⃗ is a non-degeneracy condition, and it then allows to have Re ( λ1(D1(A1 + γD−1 1 D2A2) ) > ∑ ij aij |v1||vk| ≥ ∣∣∣∣∣∣ ∑ ij aijv1vk ∣∣∣∣∣∣ = ∣∣λk ( D1(A1 + γD−1 1 D2A2) )∣∣ = ∣∣Re ( λk(D1(A1 + γD−1 1 D2A2)) )∣∣ > 0. Theorem 4 shows that the positive linear time-invariant system with coefficient ma- trices A1, A2 ∈ Rn,n being Metzler and Hurwitz, is stable system for x(t) ∈ Rn,1, γ > 0, the quantity xt(t) ( D1(A1 + γD−1 1 D2A2) ) x(t) is strictly positive if and only if Re ( λi(D1(A1 + γD−1 1 D2A2)) ) is strictly positive. Theorem 4. Let A1, A2 ∈ Rn,n be Metzler matrices and Hurwitz. The linear system dx(t) dt = A(t)x(t), where A(t) ∈ {D1A1, D2A2}, D1, D2 > 0, and x(t) ∈ Rn,1 is stable then xT (t) ( D1(A1 + γD−1 1 D2A2) ) x(t) > 0 ⇐⇒ Re ( λi(D1(A1 + γD−1 1 D2A2)) ) > 0. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 11 of 33 Proof. To ensure stability, it is sufficient to demonstrate that for z = x+iy : x, y ∈ Rn,1, the quadratic form zT ( D1(A1 + γD−1 1 D2A2) ) z > 0, z ∈ Cn,1. Further yT ( D1(A1 + γD−1 1 D2A2)x )T = xt ( D1(A1 + γD−1 1 D2A2) )T y. Also, zT ( D1(A1 + γD−1 1 D2A2) ) z = xT ( D1(A1 + γD−1 1 D2A2) ) x+yT ( D1(A1 + γD−1 1 D2A2) ) y+i ( xT (D1(A1 + γD−1 1 D2A2) ) y −yT ( (D1(A1 + γD−1 1 D2A2))x ) = xT ( D1(A1 + γD−1 1 D2A2 ) x+yT ( D1(A1 + γD−1 1 D2A2) ) y > 0. This is true if x, y ̸= 0⃗, and hence this implies that Re ( λi(D1(A1 + γD−1 1 D2A2) ) > 0,∀ i. Theorem 5. Let A1, A2 ∈ Rn,n be Metzler matrices and Hurwitz. The linear system dx(t) dt = A(t)x(t), where A(t) ∈ {D1A1, D2A2}, D1, D2 > 0, and x(t) ∈ Rn,1, is stable then xT ( D1(A1 + γD−1 1 D2A2) ) x > 0 ⇐⇒ Re ( λi(D1(A1 + γD−1 1 D2A2)) ) > 0, ∀ i = 1 : n. Proof. The quantity xT ( D1(A1 + γD−1 1 D2A2) ) x is real, and positive for the given ma- trix (D1(A1+γD−1 1 D2A2)). Then for x ∈ Rn,1, the quadratic form xT ( D1(A1 + γD−1 1 D2A2) ) x, x ∈ R2,1. Also, Re ( λi(D1(A1 + γD−1 1 D2A2)) ) > 0,∀i = 1 : 2 because for v1 ∈ Rn,1, Re ( λi(D1(A1 + γD−1 1 D2A2)) ) = vT1 (λiv1) = vT1 ( D1(A1 + γD−1 1 D2A2) ) v1, where v1 is unit eigenvector corresponding to λi ∀i = 1 : n. If we consider the matrix( D1(A1 + γD−1 1 D2A2) ) such that it has only one positive eigenvalue, then( D1(A1 + γD−1 1 D2A2) ) = UΛUT , UTU = UUT = In and Λ = diag(λ1, λ2, · · · , λn). This yield xT ( D1(A1 + γD−1 1 D2A2) ) x = xTUΛUTx = (U tx)TΛ(UTx) = ∑ i Re(λi)|vT1 x|2 ≥ 0, for Re ( λi(D1(A1 + γD−1 1 D2A2) ) > 0, and some vT1 x ̸= 0, with x ̸= 0. Theorem 6 shows that the positive linear time-invariant system with coefficient matri- ces A1, A2 ∈ Rn,n being Metzler and Hurwitz, is a stable system for ST = S, γ > 0, the perturbed matrix D1 ( A1 + γD−1 1 D2A2) )T S ( D1(A1 + γD−1 1 D2A2) ) − S, has strictly positive real part for all of it’s eigenvalues. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 12 of 33 Theorem 6. Let A1, A2 ∈ Rn,n be Metzler matrices and Hurwitz. The linear system dx(t) dt = A(t)x(t), where A(t) ∈ {D1A1, D2A2}, D1, D2 > 0, and x(t) ∈ Rn,1, is stable if Re [ λi ( D1(A1 + γD−1 1 D2A2) )T S ( D1(A1 + γD−1 1 D2A2) ) − S ] > 0 ∀i = 1 : n with St = S and S > 0 is positive definite matrix. Proof. Consider α ∈ {1, 2, · · · , n}, and let x(t) ∈ Rn,1, t ∈ R+. Also, assume that x[α] ̸= 0. For the proof of our result, we take a non-zero vector x(t), so that x[α]T ( D1(A1 + γD−1 1 D2A2 )T S ( D1(A1 + γD−1 1 D2A2) − S) ) x[α] = xT (t) ( (D1(A1 + γD−1 1 D2A2 ) S ( D1(A1 + γD−1 1 D2A2) − S)x(t) ) > 0. Further, we have xT (t)Re ( λi(D1(A1 + γD−1 1 D2A2) ) x(t) > 0, ∀i = 1 : n. In turn, this yields Re(λi ( D1(A1 + γD−1 1 D2A2) ) > 0, ∀i = 1 : n and ||x(t)|| = 1. Thus, finally we have that given linear system is stable. Theorem 7. Let A1, A2 ∈ Rn,n be Metzler matrices and Hurwitz. The linear system dx(t) dt = A(t)x(t), where A(t) ∈ {D1A1, D2A2}, D1, D2 > 0, and x(t) ∈ Rn,1 is stable if Re [ λi ( (D1(A1 + γD−1 1 D2A2) )T S ( D1(A1 + γD−1 1 D2A2)) − S )] > 0 ∀i = 1 : n. Let (λi(t), xi(t)) be an eigenpair, and assume that ρ = ∣∣∣Re ( λi(D1(A1 + γD−1 1 D2A2 )T S ( D1(A1 + γD−1 1 D2A2) ) − S ∣∣∣ , then |x(t)| > 0, t ∈ R+ and ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S ) |x(t)| = ρ ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) ) |x(t)|. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 13 of 33 Proof. Consider that x̃(t) = ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) ) |x(t)| > 0. This further implies that x̃(t) = Re [ λi ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) )] |x(t)|. Also, from this we have that, x̃(t) = ρ ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) ) |x(t)|. Let x̂(t) = x(t) − ρ ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) ) |x(t)| ≥ 0. Also, ρ ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) ) ≥ 0, and |x(t)| > 0, x̂(t) = 0. For x̃(t) ̸= 0, we have( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) ) x(t) > ρ ( (D1(A1 + γD−1 1 D2A2)) TS(D1(A1 + γD−1 1 D2A2)) − S) ) x(t), which is only possible if we don’t consider x(t), and this is not possible, and hence x̃(t) ̸= 0. 4.2. The D-stability of positive linear time-invariant systems: We derive some new results on D-stability analysis of positive time-invariant linear systems having the presence of Metzler, and Hurwitz matrices. The characterization of D-stability [54] for a given real-valued n-dimensional matrix A in terms of the real structured singular values is given by the following Theorem 8. Theorem 8. Let A ∈ Rn,n be the given matrix. Then A is a D-stable matrix if and only if it is stable and none of the eigenvalues of A± iD is exactly equal to zero, and 0 ≤ µB ( (iI + A)−1(iI −A) ) < 1. Theorem 9 gives the conditions under which linear time-invariant system with n- dimensional real-valued Metzler, and Hurwitz matrices, is D-stable. Theorem 9. Let A1, A2 ∈ Rn,n be Metzler matrices and Hurwitz.The time-invariant linear system dx(t) dt = A(t)x(t), where A(t) ∈ {A1, A2}, and x(t) ∈ Rn,1, is D-stable if Re (λi(A1 + DA2)) ̸= 0, ∀i = 1 : n with D = diag(dii) : dii > 0. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 14 of 33 Proof. Let A(t) ∈ {A1, A2} is D-stable, which means that Re (λi(A1 + DA2)) > 0, ∀i = 1 : n. Further (A1+DA2)+iD = D ( D−1(A1 + A2D) + iIn ) so that λi ( D(D−1(A1 + A2D) + iIn) ) ̸= 0, ∀i = 1 : n, with In being n-dimensional identity matrix. Assume that A(t) ∈ {A1, A2} is not D-stable matrix, which implies that Re(λi(D(A1 +DA2))) ̸̸> 0, ∀i = 1 : n. For each positive diagonal matrix D̃, the matrix D̃D(A1+A2D) is not stable, but D̃(A1+A2D) is a stable matrix for 0 < t ≤ 1 and for some γ > 0, we have that 1 γ (tD+(1−t)In)D̃(A1+DA2) with D = γ(tD + (1 − t)In)−1 has an eigenvalue i = √ −1. In Theorem 10, it has been proven that the positive linear time-invariant system is D- stable if the product of all eigenvalues of the perturbed matrix (A1 +DA2)D −1 +D(A1 + DA2) −1 is strictly positive. Theorem 10. Let A1, A2 ∈ Rn,n be Metzler, and Hurwitz matrices. The positive linear time-invariant system dx(t) dt = A(t)x(t), where A(t) ∈ {A1, A2} and x(t) ∈ Rn,1, is D-stable if ∏n i=1 λi ( (A1 + DA2)D −1 + D(A1 + DA2) −1 ) > 0, ∀ i = 1 : n. Proof. We start with the partitioned matrix of the form( A1 + DA2 −D D A1 + DA2 ) . The Schur complement of the partitioned matrix is given as (A1 + DA2) + D(A1 + DA2) −1D = D(A1 + DA2)D −1D + D(A1 + DA2) −1D = ((A1 + DA2) −1 + D(A1 + DA2) −1D). The eigenvalues λi ∀ i = 1 : n are with n∏ i=1 λi((A1 + DA2) + D(A1 + DA2) −1) = n∏ i=1 λi((A1 + DA2)D −1 + D(A1 + DA2) −1)D = n∏ i=1 λi((A1 + DA2) n∏ i=1 λi((A1 + DA2) −1D−1 + D(A1 + DA2) −1) n∏ i=1 λi(D). We know that n∏ i=1 λi ( (A1 + DA2) + D(A1 + DA2) −1D ) > 0 ⇔ n∏ i=1 λi ( (A1 + DA2)D −1 + D(A1 + DA2) −1 ) > 0, M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 15 of 33 because λi(A1 + DA2) > 0, λi(D) > 0, ∀i = 1 : n. Also, λi((A1 + DA2) −1) > 0, which implies λi(D(A1 + DA2) −1) > 0, ∀i = 1 : n. The results established in Theorem 11 show that positive linear time-invariant system is D-stable for n-dimensional real-valued Metzler, and Hurwitz matrices A1, A2 if λi((A1+ DA2) + iD) ̸= 0, ∀i = 1 : n. Theorem 11. The dynamical system dx(t) dt = A(t)x(t), A(t) ∈ {A1, A2} where A1, A2 ∈ Rn,n are Metzlern and Hurwitz, is D-stable if λi((A1 + DA2) + iD) ̸= 0,∀i = 1 : n,with D = diag(dii), where dii > 0. Proof. Suppose that A(t) ∈ {A1, A2} is a D-stable matrix. This implies that Re(λi(D(A1 + DA2))) > 0,∀i = 1 : n. The eigenvalue i = √ −1 is not an eigenvalue of the matrix product D(A1 + DA2). Fur- thermore, we have that (A1 + DA2) + iD = D(D−1(A1 + DA2) + iIn), such that λi(D(D−1(A1 + DA2) + iIn)) ̸= 0,∀i = 1 : n where In is n× n identity matrix. This also hold for a positive diagonal matrix D̃ so that the matrix D̃D(A1 + DA2) is not stable, but D̃(A1 + DA2) is stable, means that Re(λi(D̃(A1 + DA2)) > 0,∀i = 1 : n. This follow from fact that if 0 < t ≤ 1, and for γ > 0, the matrix 1 γ (tD+ (1− t)In)D̃(A1 + DA2)), with D = γ(tD + (1 − t)In)−1 has an eigenvalue i = √ −1. Theorem 12 provides insights into the D-stability of positive linear time-invariant systems. It has been established that a positive linear time-invariant system, where A1, A2 ∈ Rn,n are Metzler and Hurwitz matrices, is D-stable if the µ-value of (A1 +DA2) −2 greater than or equal to 0 and less than 1, for some positive diagonal matrix D. Theorem 12. The dynamical system dx(t) dt = A(t)x(t), where A(t) ∈ {A1, A2} and x(t) ∈ Rn,1, where A1, A2 ∈ Rn,n are D-stable Metzler and Hurwitz matrices iff 0 ≤ µB((A1 + DA2) −2) < 1, with D = diag(dii); dii > 0. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 16 of 33 Proof. From [55], it is evident that the matrix (A1 + DA2) is D-stable iff (A1 + DA2) is stable and det ( A1 + DA2 −D D A1 + DA2 ) ̸= 0. As we know that for a stable matrix (A1 + DA2), we have that det ( A1 + DA2 −D D A1 + DA2 ) ̸= 0. This implies that 0 ≤ µB((A1+DA2) −2) < 1. Also from [56], we have det ( A1 + DA2 −D D A1 + DA2 ) ̸= 0, in turn this implies det ( (A1 + DA2) 2 −D(A1 + DA2) −1D(A1 + DA2) ) ̸= 0, which further yields det ( (A1 + DA2) 2 −D(A1 + DA2) −1D(A1 + DA2) ) ̸= 0. This implies that det ( I2 − (A1 + DA2) −2D̃ ) ̸= 0, where D̃ = D = diag(d11, d22); dii > 0. Thus finally, det(In − (A1 + DA2) −2D̃) ̸= 0 =⇒ 0 ≤ µB((A1 + DA2) −2) < 1. Theorem 13 establishes results regarding the D-stability of positive linear time-invariant systems. It has been demonstrated that a positive linear time-invariant system with A1, A2 ∈ Rn,n, where both are Metzler and Hurwitz matrices, is D-stable if the real parts of all eigenvalues of the matrix D(A1 +DA2) + (A1 +DA2) TD are strictly positive. Additionally, for a positive diagonal matrix D, the system remains D-stable if the µ-value of the matrix M , derived from A1, A2, and D, satisfies 0 ≤ µ(M) < 1. Theorem 13. The dynamical system dx(t) dt = A(t)x(t), where A(t) ∈ {A1, A2}, x(t) ∈ Rn,1, where A1, A2 ∈ Rn,n are Metzler, and Hurwitz matrices, is D-stable if and only if Re ( λi(D(A1 + DA2) + (A1 + DA2) TD) ) > 0, ∀i = 1 : n with D = diag(dii), dii > 0, and 0 ≤ µB(M) < 1. The matrix M is defined as M = ( iIn + D(A1 + DA2) + (A1 + DA2) TD )−1 (iIn −D(A1 + DA2) − (A1 + DA2)D) . Proof. To prove that (A1 + DA2) is D-stable iff 0 ≤ µB(M) < 1, we assume that (A1 +DA2) is D-stable matrix, means that for all D = diag(dii), λi((A1 +DA2) + iD) ̸= 0, ∀ i = 1 : n. Let ∆ ∈ B has block-diagonal structure, that is, ∆ = (iIn −D)(iIn + D)−1. Then, D = (iIn + ∆)−1(iIn − ∆),∆ ∈ B. Since λi((A1 + DA2) + iD) ̸= 0,∀ i = 1 : n for some D, a positive diagonal matrix. This further implies that λi((A1 + DA2) + i(iIn + ∆)−1(iIn − ∆)) ̸= 0 ∀∆ ∈ B,∀ i = 1 : n. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 17 of 33 We also observe that, rank ( (A1 + DA2) + i(iIn + ∆)−1(iIn − ∆) ) ≈ rank ((iIn + (A1 + DA2) − (iIn − (A1 + DA2)∆)) . This allows us to arrive at following expression, that is, (iIn+(A1+DA2))−(iIn−(A1+DA2)∆)) = (In−(iIn+(A1+DA2) −1)(iIn−(A1+DA2)∆)),∀∆ ∈ B. Thus λi((In − (iIn + (A1 + DA2) −1(iIn − (A1 + DA2))∆) ̸= 0,∀∆ ∈ B,∀i = 1 : n which yield that 0 ≤ µB(M) < 1. Conversely, we assume that 0 ≤ µB(M) < 1 and we show that (A1 + DA2) is D-stable matrix. For 0 ≤ µB(M) < 1, a sufficient condition is that λi((A1 + DA2) + iD) ̸= 0,∀ i = 1 : n, and for some D = diag(dii), a positive diagonal matrix and this shows that (A1 + DA2) is a D-stable matrix. 4.3. The strong D-stability of positive linear time-invariant systems: We present some recent findings on strong D-stability analysis of positive linear time- invariant systems having the presence of Metzler, and Hurwitz matrices. The characteri- zation of strong D-stability [57] for a given real-valued n-dimensional matrix A in terms of the real structured singular values is given by the following Theorem 14. Theorem 14. Let A ∈ Rn,n be the given matrix. Then A is a strongly D-stable matrix if and only if it is stable and 0 ≤ µB ( (iI + A)−1(iI −A) ) < 1. Theorem 15 gives the conditions under which linear time-invariant system with n- dimensional real-valued Metzler, and Hurwitz matrices, is strongly D-stable. Theorem 15. Let M ∈ Rn,n and let A1, A2, ..., Ar ∈ Rn,n. The linear system dx(t) dt = MA(t)x(t) ;A(t) ∈ {A1, A2, ..., Ar}; x(t) ∈ Rn,1 is strongly D-stable if there exist γi > 0, the matrix log(M)+ ( (log(M) ⊗ (A1 + γ2A2 + .... + γrAr) T∆ + ∆((log(M) ⊗ (A1 + γ2A2 + ... + γrAr) ) is a D-stable matrix with ∆ ∈ B and ⊗ denotes entry-wise product of matrices. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 18 of 33 Proof. Consider that ∆ = ∆(t) ∈ B be an admissible perturbation and B is the set of block diagonal matrices with real or complex uncertainties. Let λ(t) = |λ(t)|(cos θ+i sin θ) for 0 < θ ≤ 2π be the largest eigenvalue. Assume that x̃(t), ỹ(t) be the right and left eigenvectors and let z = ( (log(M) ⊗ (A1 + γ2A2 + · · · + γrAr)) T ∆ + ∆ (log(M) ⊗ (A1 + γ2A2 + · · · + γrAr)) ) y(t). We make use of an eigenvalue perturbation result by [32] on the largest and simple eigen- value λ(t) to have that d dt |λ(t)| ∣∣∣∣2 t=0 = 2ϵ |λ(t)| α Re(zT∆(t)x), with α = (cos θ + i sin θ)yTx > 0, ϵ > 0. In turn, this implies that (log(M) ⊗ (A1 + γ2A2 + · · · + γrAr)) T ∆ + ∆ (log(M) ⊗ (A1 + γ2A2 + · · · + γrAr)) > 0, which further yields that log(M)+ ( (log(M) ⊗ (A1 + γ2A2 + · · · + γrAr)) T ∆ + ∆ (log(M) ⊗ (A1 + γ2A2 + · · · + γrAr)) ) is D-stable. Theorem 16. Let M ∈ Rn,n, and let M = e{A1+γ2A2+...+γrAr} where A1, A2, ..., Ar are n-dimensional Hermition matrices, then M is strongly D-stable matrix if M is stable and for some α > 0 the matrix (M + G) is D-stable where G = ( M ⊗ (A1 + γ2A2 + ... + γrAr) T )∆ + ∆(M ⊗ (A1 + γ2A2 + ... + γrAr) ) ,∆ ∈ B, ∥G∥ < α. Proof. Assume that ∆ = ∆(t) ∈ B, where B is set of block diagonal matrices. Let λ(t) = |λ(t)|(cos θ+i sin θ), 0 < θ ≤ 2π be the largest eigenvalue. In addition, assume that x(t), y(t) are left- and right-eigenvectors. Consider z = GT y. The eigenvalues perturbation findings by [32] on λ(t) yield d dt |λ(t)|2 ∣∣∣∣ t=0 = 2ϵ |λ(t)| α Re(zT ∆̇(t)x) with α = (cos θ + i sin θ)ytx; α ≥ 0. As we know that, Re(zT ∆̇(t)x) > 0, and in turn this implies that( M ⊗ (A1 + γ2A2 + ... + γrAr) T )∆ + ∆(M ⊗ (A1 + γ2A2 + ... + γrAr) ) ,∆ ∈ B is PD-matrix. Finally for D = diag(dii); dii > 0, the matrix D ( e{A1+γ2A2+···+γrAr} + G ) and from this it follows that M + G = M + ( M ⊗ (A1 + γ2A2 + ... + γrAr) T )∆ + ∆(M ⊗ (A1 + γ2A2 + ... + γrAr) ) is D-stable matrix. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 19 of 33 5. Numerical Testing The numerical experimentation illustrate the computation, and to visualize the spec- trum, the singular values, the structured singular values, and the pseudo-spectra for Met- zler matrices appearing across the positive dynamical systems. The graphical representa- tions of the ϵ-pseudospectrum are the level sets which are corresponding to resolvent norm ||(zIn −A)−1||. Example 1. Consider dx(t) dt = Ax(t) is a positive dynamical system with A Metzler matrix give by A =  −10−9 109 0 0 0 0 −10−9 0 0 109 0 109 −10−9 0 0 109 109 0 −10−9 109 0 0 0 0 −10−9  . The spectral properties like the computation of spectrum, singular, structured singular values, and pseudo-spectrum of Metzler matrix A are presented in Figure 1. In Figure 2, we plot the eigenmode corresponding to the eigenvalues. The top plot in the figure (left) shows an envelope which is produced by plotting the absolute value of an eigenmode and minus the absolute value. The real part is shown with a cyan line. The plot at the bottom level show absolute value of eigenmode being ploted at a log scale. Further, it shows that how quickly an eigenmode is decaying with time. The condition number computed for an eigenvalue is shown in the top plot. The large condition number means that eigenvalue is sensitive to perturbations. We plot the value of inverse of the resolvent norm (right). We show real part of pseudomode in magenta.The right singular vector corresponding to the smallest singular value of the matrix zIn − A is depicted in pseudomode. Example 2. Consider a positive Frobenius matrix [58] F =  0 1 0 0 0 1 0.4714 0.1953 0.3333  , and a characteristic polynomial for F given as P (β, F ) = β3 − 0.3333β2 − 0.1953β − 0.4714, with β = λ + η, η = 1.1 > ρ(F ) = 1. The matrix A = F − ηI3, is a Metzler matrix with A =  −1.1 1 0 0 −1.1 1 0.4714 0.1953 0.7667  . The spectral properties like the computation of spectrum, singular, structured singular values and pseudo-spectrum of Metzler matrix A are presented in Figure 3. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 20 of 33 Figure 1: Spectral properties of Metzler matrix A in Example-1. In Figure 4, we plot the eigenmode corresponding to the eigenvalues. The top plot (left) in the figure shows an envelope which is produced by plotting the absolute value of an eigenmode and minus the absolute value. The real part is shown with a cyan line. The plot at the bottom level show absolute value of eigenmode being ploted at a log scale. Further, it shows that how quickly an eigenmode is decaying with time. The condition number computed for an eigenvalue is shown in the top plot. The large condition number means that eigenvalue is sensitive to perturbations. We plot the value of inverse of the resolvent norm (right). We show real part of pseudomode in magenta. The right singular vector corresponding to the smallest singular value of the matrix zIn − A is depicted in pseudomode. Example 3. We consider Metzler matrices with sizes 50, 100, and 500, respectively. The spectral properties like the computation of spectrum, singular, structured singular values and pseudo-spectrum of Metzler matrix A are presented in Figures 5,7, and 9. In Figures 6,8, and 10, we plot (left) the eigenmode corresponding to the eigenvalues. The top plot in the figure shows an envelope which is produced by plotting the absolute value of an eigenmode and minus the absolute value. The real part is shown with a cyan line. The plot at the bottom level show absolute value of eigenmode being ploted at a log scale. Further, it shows that how quickly an eigenmode is decaying with time. The condition number computed for an eigenvalue is shown in the top plot. The large condition number means that eigenvalue is sensitive to perturbations. We plot the value of inverse of the resolvent norm (right). We show real part of pseudomode in magenta. The right singular vector corresponding to the smallest singular value of the matrix zIn −A is depicted in pseudomode. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 21 of 33 Figure 2: Eigenmode (left) and inverse of resolvent norm (right) of Metzler matrix A in Example-1 Example 4. We consider Hurwitz matrices with sizes 50, 100, and 500, respectively. The spectral properties like the computation of spectrum, singular, structured singular values and pseudo-spectrum of Hurwitz matrix A are presented in Figures 11,13, and 15. In Figures 12,14, and 16, we plot (left) the eigenmode corresponding to the eigenvalues. The top plot in the figure shows an envelope which is produced by plotting the absolute value of an eigenmode and minus the absolute value. The real part is shown with a cyan line. The plot at the bottom level show absolute value of eigenmode being ploted at a log scale. Further, it shows that how quickly an eigenmode is decaying with time. The condition number computed for an eigenvalue is shown in the top plot. The large condition number means that eigenvalue is sensitive to perturbations. We plot the value of inverse of the resolvent norm (right). We show real part of pseudomode in magenta. The right singular vector corresponding to the smallest singular value of the matrix zIn − A is depicted in pseudomode. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 22 of 33 Figure 3: Spectral properties of Metzler matrix A in Example-2. Figure 4: Eigenmode (left) and inverse of resolvent norm (right) of Metzler matrix A in Example-2 M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 23 of 33 Figure 5: Spectral properties of Metzler matrix (size = 50) in Example-3. Figure 6: Eigenmode (left) and inverse of resolvent norm (right) of Metzler matrix (size = 50) in Example-3 M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 24 of 33 Figure 7: Spectral properties of Metzler matrix (size = 100) in Example-3. Figure 8: Eigenmode (left) and inverse of resolvent norm (right) of Metzler matrix (size = 100) in Example-3 M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 25 of 33 Figure 9: Spectral properties of Metzler matrix (size = 500) in Example-3. Figure 10: Eigenmode (left) and inverse of resolvent norm (right) of Metzler matrix (size = 500) in Example-3 M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 26 of 33 Figure 11: Spectral properties of Hurwitz matrix (size = 50) in Example-4. Figure 12: Eigenmode (left) and inverse of resolvent norm (right) of Hurwitz matrix (size = 50) in Example-4 M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 27 of 33 Figure 13: Spectral properties of Hurwitz matrix (size = 100) in Example-4. Figure 14: Eigenmode (left) and inverse of resolvent norm (right) of Hurwitz matrix (size = 100) in Example-4 M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 28 of 33 Figure 15: Spectral properties of Hurwitz matrix (size = 500) in Example-4. Figure 16: Eigenmode (left) and inverse of resolvent norm (right) of Hurwitz matrix (size = 500) in Example-4 M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 29 of 33 6. Applications 1. Mathematical Modeling: The Metzler matrices play an important role for mathematical modeling of various problems, see [59–61]. For an input data vector x(t) ∈ Rn,1, the most common problem is to analyze the given linear system having the form dx(t) dt = Ax(t), with A being a Metzler matrix. The analysis on positive linear switching systems also play an important and vital role to the mathematical modeling. An extensive amount of research work has already been done in this directions, see [18, 62, 63]. 2. Dynamics of Love: A mathematical model was proposed by S. Rinaldi [64] to describe the dynamics of love of Laura de Noves and Petrarch, which leads to a periodic dynamics. For the implementation of model, 23 dated poems out of 366 poems Petrarch wrote in Canzoniere during 21 years were analyzed, see [65]. Rinaldi’s model describes love dynamic in three variables, L denotes love of Laura for Petrarch, P denotes love of Petrarch for Laura, and Z denotes the poetic inspiration. The mathematical model is: d dt(L(t)) = −α1L(t) + RL(P (t)) + Ap d dt(P (t)) = −α2P (t) + RP (L(t)) + β2AL(Z(t)) d dtZ(t) = −α3Z(t) + β4P (t), where all parameters involved are defined and described in [65]. This love model can be expressed as dx(t) dt = A0x(t) + b = ϕ(t), where A0 = −α1 0 0 0 −α2 −β3 0 β4 −α3  ; b = β1β2 0  ; ϕ(t) = [RL(P (t), 0, 0)]t. For a matrix Q̃, and using matrix A0, one can have that A = Q̃A0Q̃, where the matrix A is same as Metzler matrix. Thus, finally, the analysis on A can lead to interesting results for the love dynamic model. 7. Conclusion In this paper, we have studied and analyzed mathematical problems related to positive linear time-invariant systems whose coefficient matrices are Metzler and Hurwitz matrices. We have established and presented some new theoretical results on stability, D-stability, and strong D-stability to the following three mathematical problems: Problem-I: To extend and construct some recent findings on stability, D-stability and strong D-stability analysis of LTI system dx(t) dt = Ax(t), where x(t) ∈ Rn,1, A ∈ Rn,n, with A a Metzler and Hurwitz matrix. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 30 of 33 Problem-II: To extend and construct some new findings on stability, D-stability and strong D-stability analysis of LTI system dx(t) dt = A(t)x(t), where x(t) ∈ Rn,1, A(t) ∈ {A1, A2}, with A1, A2 being asymptotically stable matrices. Problem-III: To extend and construct some new findings on stability, D-stability and strong D-stability analysis of LTI system dx(t) dt = A(t)x(t), where x(t) ∈ Rn,1, A(t) ∈ {D1A1, D2A2}, with A1, A2 being irreducible matrices, and D1, D2 > 0. Conflicts of Interest The authors confirm that they have no conflicts of interest concerning the publication of this paper. Acknowledgements J. Alzabut and M. Tounsi express their sincere thanks to Prince Sultan University for its endless support. M. Rehman expresses his gratitude to Asia International University for its assistance. References [1] Abraham Berman and Robert J Plemmons. Nonnegative matrices in the mathematical sciences. SIAM, 1994. [2] Oliver Mason, Vahid S Bokharaie, and Robert Shorten. Stability and d-stability for switched positive systems. In Positive Systems: Proceedings of the third Multi- disciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2009) Valencia, Spain, September 2-4, 2009, pages 101–109. Springer, 2009. [3] Lorenzo Farina and Sergio Rinaldi. Positive linear systems: theory and applications. John Wiley & Sons, 2011. [4] Wassim M Haddad and VijaySekhar Chellaboina. Stability theory for nonnegative and compartmental dynamical systems with time delay. In Proceedings of the 2004 American Control Conference, volume 2, pages 1422–1427. IEEE, 2004. [5] Vahid Samadi Bokharaie, Oliver Mason, and Mark Verwoerd. D-stability and delay- independent stability of homogeneous cooperative systems. IEEE Transactions on automatic control, 55(12):2882–2885, 2010. [6] Kenneth J Arrow and Maurice McManus. A note on dynamic stability. Econometrica: Journal of the Econometric Society, pages 448–454, 1958. [7] Alain C Enthoven and Kenneth J Arrow. A theorem on expectations and the stability of equilibrium. Econometrica: Journal of the Econometric Society, pages 288–293, 1956. [8] Giorgio Giorgi. Stable and related matrices in economic theory. Control and Cyber- netics, 32(2):397–410, 2003. [9] Frank Hahn. Stability. Handbook of mathematical economics, 2:745–793, 1982. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 31 of 33 [10] Murray C Kemp and Yoshio Kimura. Introduction to mathematical economics. Springer, 1978. [11] Peter K Newman. Some notes on stability conditions. The review of economic studies, 27(1):1–9, 1959. [12] James P Quirk and Rubin Saposnik. Introduction to general equilibrium theory and welfare economics. McGraw-Hill, 1968. [13] John E Woods. Mathematical economics. (No Title), 1978. [14] Eugenius Kaszkurewicz and Amit Bhaya. Matrix diagonal stability in systems and computation. Springer Science & Business Media, 2012. [15] Charles R Johnson. Sufficient conditions for d-stability. Journal of Economic Theory, 9(1):53–62, 1974. [16] Cheng-Ching Yu and Michael KH Fan. Decentralized integral controllability and d-stability. Chemical Engineering Science, 45(11):3299–3309, 1990. [17] Mahmoud Chilali, Pascal Gahinet, and Pierre Apkarian. Robust pole placement in lmi regions. IEEE transactions on Automatic Control, 44(12):2257–2270, 2002. [18] Valter JS Leite and Pedro LD Peres. An improved lmi condition for robust d-stability of uncertain polytopic systems. IEEE Transactions on Automatic Control, 48(3):500– 504, 2003. [19] Dimitri Peaucelle, Denis Arzelier, Olivier Bachelier, and Jacques Bernussou. A new robust d-stability condition for real convex polytopic uncertainty. Systems & control letters, 40(1):21–30, 2000. [20] D Arzelier, D Henrion, and D Peaucelle. Robust d stabilization of a polytope of matrices. International Journal of Control, 75(10):744–752, 2002. [21] Olga I Kosmidou. Robust control with pole shifting via performance index modifica- tion. Applied mathematics and computation, 182(1):596–606, 2006. [22] Dong Hwan Lee, Jin Bae Park, and Young Hoon Joo. A less conservative lmi con- dition for d-stability of polynomial matrix polytopes—a projection approach. IEEE Transactions on Automatic Control, 56(4):868–873, 2010. [23] Mutti-Ur Rehman et al. Spectrum and pseudspectrum of d-stable matrices of economy models. J. Math. Computer Sci, 38:298–312, 2025. [24] Pierre Rostan and Alexandra Rostan. The versatility of spectrum analysis for fore- casting financial time series. Journal of Forecasting, 37(3):327–339, 2018. [25] Alaa khadim Mohammed and Salam Jasim Majeed. Bifurcation analysis of an eco- epidemiological model involving prey refuge, fear impact and hunting cooperation. Journal of Education for Pure Science-University of Thi-Qar, 14(2), 2024. [26] Katsuhisa Furuta and S Kim. Pole assignment in a specified disk. IEEE Transactions on Automatic control, 32(5):423–427, 1987. [27] Chun-Hsiung Fang and Li Lee. Robustness of regional pole placement for uncer- tain continuous-time implicit systems. IEEE transactions on automatic control, 39(11):2303–2307, 1994. [28] Chun-Hsiung Fang, Li Lee, and Fan-Ren Chang. Robust control analysis and design for discrete-time singular systems. Automatica, 30(11):1741–1750, 1994. [29] Shengyuan Xu, James Lam, and Chengwu Yang. Robust h control for uncertain M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 32 of 33 discrete singular systems with pole placement in a disk. Systems & Control Letters, 43(2):85–93, 2001. [30] Tung-Kuan Liu, Shinn-Horng Chen, Jyh-Horng Chou, and Cheng-Yi Chen. Regional eigenvalue-clustering robustness of linear uncertain multivariable output feedback pid control systems. Journal of the Franklin Institute, 346(3):253–266, 2009. [31] Roger W Brockett. Finite dimensional linear systems. SIAM, 2015. [32] Tosio Kato. Perturbation theory for linear operators, volume 132. Springer Science & Business Media, 2013. [33] Aleksandar Cvetković. Stabilizing the metzler matrices with applications to dynam- ical systems. Calcolo, 57(1):1, 2020. [34] Kumpati S Narendra and Robert Shorten. Hurwitz stability of metzler matrices. IEEE Transactions on Automatic Control, 55(6):1484–1487, 2010. [35] Igor Podlubny. Fractional differential equations: an introduction to fractional deriva- tives, fractional differential equations, to methods of their solution and some of their applications, volume 198. elsevier, 1998. [36] Diethelm Kai. The analysis of fractional differential equations. an application-oriented exposition using differential operators of caputo type. Lecture Notes in Mathematics, 2010. [37] Anatoliui Aleksandrovich Kilbas, Hari M Srivastava, and Juan J Trujillo. Theory and applications of fractional differential equations, volume 204. elsevier, 2006. [38] Muhammad Shoaib Arif, Kamaleldin Abodayeh, and Yasir Nawaz. Numerical schemes for fractional energy balance model of climate change with diffusion effects. Emerging Science Journal, 7(3):808–820, 2023. [39] Shafiullah, Kamal Shah, Muhammad Sarwar, and Thabet Abdeljawad. On theoretical and numerical analysis of fractal–fractional non-linear hybrid differential equations. Nonlinear Engineering, 13(1):20220372, 2024. [40] Mutti-Ur Rehman et al. Spectrum and pseudspectrum of d-stable matrices of economy models. J. Math. Computer Sci, 38:298–312, 2025. [41] Mutti-Ur Rehman et al. Spectrum and pseudspectrum of d-stable matrices of economy models. J. Math. Computer Sci, 38:298–312, 2025. [42] Mutti-Ur Rehman, Behkzod Aminov, Mohammed N Alshehri, Mustafa M Mo- hammed, Arafa O Mustafa, Nhla A Abdalrahman, Mona Magzoub, Hala S Mahgoub, Sakeena EM Hamed, Runda AA Bashir, et al. Spectral properties of structured matri- ces in transportation problems. European Journal of Pure and Applied Mathematics, 18(1):5637–5637, 2025. [43] Mutti-Ur Rehman, Sakeena EM Hamed, Nidal E Taha, Arafa O Mustafa, Khurshid- bek Dilmurodov, Hala S Mahgoub, Mona Magzoub, Runda AA Bashir, Mustafa M Mohammed, and Awad A Bakery. Analysis of stability, d-stability, and pseudospec- tra in economic modeling. European Journal of Pure and Applied Mathematics, 18(1):5657–5657, 2025. [44] Mutti-Ur Rehman, Jehad Alzabut, Muhamad Tayyab, and Fouzia Amir. Intercon- nection between schur stability and structured singular values. Contemporary Math- ematics, pages 63–72, 2025. M.U. Rehman et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6047 33 of 33 [45] Micha l Domka and Wojciech Mitkowski. On spectrum of metzler matrices. Przeglad Elektrotechniczny, 98(12), 2022. [46] Dragoslav D Siljak. Large-scale dynamic systems: stability and structure. (No Title), 1978. [47] Eyad H Abed. Strong d-stability. Systems & control letters, 7(3):207–212, 1986. [48] Olga Y Kushel. How to check d-stability: a simple determinantal test. arXiv preprint arXiv:2210.05711, 2022. [49] Lloyd N Trefethen and Mark Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. 2020. [50] Wasfi Kafri. Robust d-stability. Applied Mathematics Letters, 15(3):7–10, 2001. [51] Wojciech Mitkowski. Dynamical properties of metzler systems. Bulletin of the Polish Academy of Sciences: Technical Sciences, 56(4), 2008. [52] Oskar Perron. Zur theorie der matrices. Mathematische Annalen, 64(2):248–263, 1907. [53] Fridrikh Izrailevich Karpelevich. On the characteristic roots of matrices with non- negative elements. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 15(4):361–383, 1951. [54] Jie Chen, Michael KH Fan, and Cheng-Ching Yu. On d-stability and structured singular values. Systems & control letters, 24(1):19–24, 1995. [55] Russell Allan Johnson, Alberto Tesi, et al. On the d-stability problem for real matri- ces. Bollettino Dell’Unione Matematica Italiana. B, 2:299–314, 1999. [56] John R Silvester. Determinants of block matrices. The Mathematical Gazette, 84(501):460–467, 2000. [57] Jietae Lee and Thomas F Edgar. Real structured singular value conditions for the strong d-stability. Systems & control letters, 44(4):273–277, 2001. [58] Wojciech Mitkowski. Dynamical properties of metzler systems. Bulletin of the Polish Academy of Sciences: Technical Sciences, 56(4), 2008. [59] Lorenzo Farina and Sergio Rinaldi. Positive linear systems: theory and applications. John Wiley & Sons, 2011. [60] David G Luenberger. Introduction to dynamic systems: theory, models, and applica- tions. (No Title), 1979. [61] Corentin Briat. Sign properties of metzler matrices with applications. Linear Algebra and its Applications, 515:53–86, 2017. [62] Daniel Liberzon. Switching in systems and control, volume 190. Springer, 2003. [63] Leonid Gurvits, Robert Shorten, and Oliver Mason. On the stability of switched positive linear systems. IEEE Transactions on Automatic Control, 52(6):1099–1103, 2007. [64] Sergio Rinaldi. Laura and petrarch: An intriguing case of cyclical love dynamics. SIAM Journal on Applied Mathematics, 58(4):1205–1221, 1998. [65] Frederic J Jones. The Structure of Petrarch’s Canzoniere: A Chronological, Psycho- logical, and Stylistic Analysis. Boydell & Brewer, 1995.