EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6072 ISSN 1307-5543 – ejpam.com Published by New York Business Global Common Riccati Stability and Time-Delay Systems Ali Algefary1,∗, Khulud Abdullah Alqufari2 1 Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia. 2 Department of Statistics and Operation Research, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia Abstract. This paper investigates the stability properties of matrix families through the lens of Riccati, Lyapunov, and Schur stability. We focus on establishing connections between these stability concepts, particularly in the context of continuous and discrete-time systems, as well as time-delay systems. The results provide criteria for common Riccati stability, exploring its implications on Lyapunov and Schur stability across matrix families. Furthermore, we examine the effects of scaling transformations and similarity transformations on common Riccati stability, demonstrating its robustness under scalar multiplication and similarity changes. The findings contribute to a deeper understanding of matrix stability in control systems, offering insights into the structural preservation of stability properties under various transformations. This work has potential applications in robust control and the stability analysis of complex systems subject to time delays and structural modifications. 2020 Mathematics Subject Classifications: 15A45, 15B48, 93D05, 34K25, 34K05 Key Words and Phrases: Common Riccati Stability, differential equations, Lyapunov functions, positive definite matrices, Common Lyapunov stability 1. Introduction The work in this paper considers only matrices in the real space Rn×n. We use the notation X ≻ 0 (X ≺ 0, respectively) to indicate that a matrix X ∈ Rn×n is positive definite (negative definite, respectively). Similarly, we denote a positive semidefinite ( negative semidefinite, respectively) matrix by X ⪰ 0 (X ⪯ 0, respectively). Unless stated otherwise, a positive or negative definite or semidefinite matrix is assumed to be symmetric. For a matrix X ∈ Rn×n, we write XT to denote the transpose of X, we also write X−1 to refer to the inverse of X, and X−T for its inverse transpose, defined as X−T = (XT )−1 = (X−1)T . Additionally, we use I to denote the identity matrix with its size implied by the context. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6072 Email addresses: a.algefary@qu.edu.sa (A. Algefary), 432206823@qu.edu.sa (K. A. Alqufari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 2 of 14 Consider a matrix X ∈ Rn×n. In this paper, we use σ(X) to represent the spectrum of X and ρ(X) to indicate its spectral radius. The spectral abscissa of X, represented by α(X), is defined as the largest real part among the eigenvalues of X. Formally, we write: α(X) = max{Re(λ) | λ ∈ σ(X)}. For further details, refer to [1, 2]. Let us start by reviewing some key definitions related to types of matrix stability relevant to this study. Definition 1. [3] Let A be a matrix in Rn×n. The matrix A is called Hurwitz (or Hurwitz stable) if all its eigenvalues are located in the open left half of the complex plane, meaning α(A) < 0. A well-known characterization of Hurwitz stability for a matrix A ∈ Rn×n involves the Lyapunov equation ATP + PA+Q = 0, where P,Q ∈ Rn×n with P ≻ 0 and Q ≻ 0; see [3]. This result is fundamental in assessing the stability of continuous linear systems described by ẋ(t) = Ax(t), (1) where A ∈ Rn×n and x(t) ∈ Rn. Specifically, if there exists a P ≻ 0 satisfying the Lyapunov equation, then the linear system in (1) is associated with a Lyapunov function V (x) = xTPx, confirming the system equilibrium is asymptotic stability. This criterion for Hurwitz stability is presented in the following lemma, known as Lyapunov’s Theorem. Lemma 1. [3, 4] A matrix A ∈ Rn×n is Hurwitz stable if there exist positive definite matrices P,Q ∈ Rn×n such that ATP + PA = −Q. Alternatively, Lemma 1 can be restated using the Lyapunov inequality: A is Hurwitz stable if and only if there exists a positive definite matrix P ∈ Rn×n such that ATP + PA ≺ 0. (2) Within this context, P is referred to as a Lyapunov solution for matrix A or the Lyapunov inequality. The concept of Hurwitz stability for a single matrix A ∈ Rn×n has been extended in the literature (see [5–8]) to consider families of real matrices through the notion of common Lyapunov stability. This extension is particularly significant in control theory, especially in the analysis of switched systems and robustness, where ensuring stability across multiple system configurations is crucial [9, 10]. A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 3 of 14 Specifically, a family of real matrices A = {A1, . . . , Am} is said to have common Lyapunov stability if there exists a positive definite matrix P such that for each i = 1, . . . ,m, AT i P + PAi ≺ 0. (3) The existence of such a matrix P implies simultaneous Hurwitz stability for the ma- trices in A. In other words, the linear systems associated with the matrices in the family A share a common Lyapunov function given by V (x) = xTPx, (4) which serves as a unified measure to demonstrate the stability of each system in the family. The concepts of Hurwitz stability and common Lyapunov stability have been further extended to diagonal Lyapunov stability [4, 11, 12] and common diagonal Lyapunov sta- bility [4, 13, 14], respectively. This extension focuses on identifying a diagonal positive definite matrix P = diag(p1, p2, . . . , pn) such that, for a single matrix A, the Lyapunov inequality (2) is satisfied. This formulation simplifies the task by narrowing it down to finding the positive diagonal elements pi, taking advantage of the diagonal structure for easier computations. In a similar way, for a family of matrices A = {A1, . . . , Am}, common diagonal Lya- punov stability entails finding a single diagonal positive definite matrix P that meets the inequalities (3). In this case, P is referred to as a common Lyapunov solution for the fam- ily A. Equivalently, the family A has common Lyapunov stability if there exist positive definite matrices P,Q1, . . . , Qm ∈ Rn×n such that AT i P + PAi +Qi = 0, for i = 1, . . . ,m. This approach proves especially valuable in large-scale systems or in cases involving decoupled or weakly coupled subsystems, where computational simplicity and efficiency are key. By restricting P to be diagonal, we lower the problem’s complexity while maintaining a unified framework for proving the stability of each system in the family. Building upon the continuous-time case, we now consider discrete-time difference sys- tems, which are fundamental in digital control and signal processing [15]. Specifically, we examine the system: x(k + 1) = Ax(k). The zero equilibrium of this system is asymptotically stable if there exists a positive definite matrix P ∈ Rn×n such that ATPA− P ≺ 0. A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 4 of 14 This condition ensures that the Lyapunov function V (x) = xTPx decreases over time, guaranteeing stability. Similar to the continuous-time scenario, the stability of the discrete- time system can also be characterized by the eigenvalues of A. Specifically, the system is asymptotically stable if all eigenvalues of A lie strictly inside the open unit disk of the complex plane. When this condition is met, A is referred to as Schur stable matrix. Definition 2. [4] A matrix B ∈ Rn×n is called Schur stable if its spectral radius satisfies ρ(B) < 1. Lemma 2. [4] A matrix A ∈ Rn×n is Schur stable if and only if there exist positive definite matrices P,Q ∈ Rn×n such that: ATPA− P = −Q. This equation is known as the Stein equation and plays a significant role in discrete- time stability analysis, analogous to the Lyapunov equation in continuous-time systems. Extending this notion to the common case, consider a family of matricesA = {A1, A2, . . . , Am}. The family A is said to possess common Schur stability if there exists a single positive definite matrix P such that for all i = 1, . . . ,m, the following inequality holds AT i PAi − P ≺ 0. This P is known as the common Schur solution for the family A. The definition of common Schur stability can stated using an equation instead of inequality. That is A has a common Schur stability if there exist positive definite matrices P,Q1, . . . , Qm ∈ Rn×n such that AT i PAi − P = −Qi, i = 1, . . . ,m. The existence of such a common matrix P implies that each system in the family, described by x(k + 1) = Aix(k), is asymptotically stable. Moreover, they all share the common Lyapunov function as in (4) which serves as a unified tool to demonstrate the stability of all systems within the family A. Moving forward, consider the linear differential systems with time delays, which are frequent in many engineering applications where delays are inevitable. Specifically, we examine the system ẋ(t) = Ax(t) +Bx(t− τ), (5) where A,B ∈ Rn×n are constant matrices, x(t) ∈ Rn is the state vector, and τ ≥ 0 represents an arbitrary time delay. Understanding the stability of such time-delay systems is crucial because delays can significantly affect system performance and may lead to instability if not properly accounted for [16]. As demonstrated in [17], the system (5) admits a Lyapunov-Krasovskii functional of the form A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 5 of 14 V (x) = xTPx+ ∫ t t−τ xT (s)Qx(s) ds, provided that there exist positive definite matrices P,Q,R ∈ Rn×n satisfying the Riccati equation ATP + PA+Q+ PBQ−1BTP +R = 0. The existence of this Lyapunov-Krasovskii functional is significant because it guaran- tees that the equilibrium point of system (5) is asymptotically stable for all delays τ ≥ 0 see [18]. The requirement that positive definite matrices P,Q,R satisfy the Riccati equation defines the concept of Riccati stability for a pair of matrices (A,B), which was introduced in [19]. The connections between Riccati stability and classical notions of stability, such as Hurwitz and Schur stability, have been explored in [20]. Moreover, several results establish links between Riccati stability and the stability analysis of time-delay systems, underscoring its importance in this area of study. Similar to the concept of common Lyapunov stability, we introduce the notion of com- mon Riccati stability. This concept involves finding positive definite matrices P,Q,Ri ∈ Rn×n, i = 1, . . . ,m, that satisfy a Riccati equation simultaneously for a family of matrix pairs U = {(Ai, Bi)}mi=1. Definition 3. Let Ai, Bi ∈ Rn×n for i = 1, . . . ,m. We say that the family of pairs U = {(Ai, Bi)}mi=1 has common Riccati stability if there exist positive definite matrices P,Q,Ri ∈ Rn×n such that the following Riccati equations AT i P + PAi +Q+ PBiQ −1BT i P +Ri = 0, i = 1, . . . ,m, hold. When such matrices P,Q,Ri, i = 1, . . . ,m, exist, we refer to the pair (P,Q) as a common Riccati solution for the family U . Essentially, common Riccati stability represents a simultaneous solution to the Riccati equations associated with each pair (Ai, Bi) in U . The existence of a common Riccati solution for the family U implies that V (x) = xTPx+ ∫ t t−τ xT (s)Qx(s) ds serves as a common Lyapunov-Krasovskii functional for all time-delay systems associated with the pairs in U . The main contributions of this paper are organized into two key parts. In the first part, we explore the connections between common Riccati stability for a family of matrix pairs U = {(Ai, Bi)}mi=1 and the concepts of common Lyapunov stability and common Schur stability. By investigating these relationships, we aim to deepen the understanding of how these different stability notions are interrelated within the context of control theory and system dynamics. A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 6 of 14 In the second part, we explore several scaling properties associated with common Riccati stability. Analyzing these properties is significant because scaling can affect the stability of systems, and understanding this impact is crucial for the design and analysis of robust control systems. By identifying how scaling transformations influence common Riccati stability, we provide insights that can lead to more efficient computational methods and enhance the applicability of stability criteria to a broader class of systems. 2. Common Lyapunov, Schur, and Riccati stability In this section, we explore the foundational relationships among common Lyapunov stability, common Schur stability, and common Riccati stability for families of matrices and matrix pairs. These stability concepts play crucial roles in analyzing and ensuring the robustness of complex systems, particularly in control theory. By developing these interconnections, we provide a unified framework that simplifies stability analysis across various system structures and enhances our understanding of stability in both continuous and discrete-time systems. Theorem 1. For i = 1, . . . ,m, suppose that Ai, Bi ∈ Rn×n. If the family U = {(Ai, Bi)}mi=1 has a common Riccati stability, then the family A = {Ai}mi=1 has a common Lyapunov stability. Proof. Let P,Q,Ri ∈ Rn×n, i = 1, . . . ,m be positive definite matrices such that (P,Q) is a common Riccati solution for the family U = {(Ai, Bi)}mi=1. Then, for each i = 1, . . . ,m, the matrices Xi = Q+ PBiQ −1BT i P +Ri are positive definite. Consequently, we have AT i P + PAi +Xi = 0 for every i ∈ {1, . . . ,m}. Thus, P serves as a common Lyapunov solution for the family A = {Ai}mi=1. Theorem 2. For i = 1, . . . ,m, suppose that Ai, Bi ∈ Rn×n. If the family U = {(Ai, Bi)}mi=1 has common Riccati stability, then the family {A−1 i Bi}mi=1 has common Schur stability. Proof. Let P,Q,Ri ∈ Rn×n, i = 1, . . . ,m be positive definite matrices such that (P,Q) is a common Riccati solution for the family U = {(Ai, Bi)}mi=1. Then, for i = 1, . . . ,m, the following equations hold AT i P + PAi +Q+ PBiQ −1BT i P +Ri = 0. (6) According to Theorem 1, the family {Ai}mi=1 has common Lyapunov stability, and there- fore, each Ai is nonsingular. Thus, the equations in (6) can be rewritten as AT i P + PAi +Q+ PAi [ A−1 i BiQ −1BT i A −T i ] AT i P +Ri = 0 A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 7 of 14 for each i. Next, observe that for each i ∈ {1, . . . ,m}, these equations are identical to (PAiQ −1 + I)Q(I +Q−1AT i P ) + PAi [ A−1 i BiQ −1BT i A −T i −Q−1 ] AT i P +Ri = 0. (7) Define Zi = (PAiQ −1 + I)Q(I +Q−1AT i P ) +Ri, for i ∈ {1, . . . ,m}. It is evident that each of Zi is a positive definite matrix. By substituting Zi in (7), we obtain PAi [ A−1 i BiQ −1BT i A −T i −Q−1 ] AT i P + Zi = 0. For i = 1, . . . ,m, note that Ai is nonsingular, and since P is positive definite, it is also nonsingular. Therefore, the matrices PAi and (PAi) T for i = 1, . . . ,m are nonsingular. Now, observe that for i = 1, . . . ,m, we have (PAi) −1 = A−1 i P−1 and (PAi) −T = P−1A−T i . Consequently, for each i ∈ {1, . . . ,m}, we can pre-multiply the ith equation in (7) by (PAi) −1 and post-multiply it by (PAi) −T . Thus, the equations in (7) become A−1 i BiQ −1BT i A −T i −Q−1 + (PAi) −1Zi(PAi) −T = 0, (8) for i = 1, . . . ,m. Next, let Yi = (PAi) −1Zi(PAi) −T , for i = 1, . . . ,m. Recall that each Zi is positive definite; consequently, Yi is positive definite for each i ∈ {1, . . . ,m}. Hence, the equations in (8) reduce to the following (A−1 i Bi)Q −1(A−1 i Bi) T −Q−1 + Yi = 0, for i = 1, . . . ,m. By the definition of common Schur stability, this implies that the family {A−1 i Bi}mi=1 has common Schur stability. Having established that common Riccati stability implies both common Lyapunov and Schur stability, we now examine the reverse relationship in Theorems 3 and 4. Specifi- cally, in Theorem 3, we show that if the family { A−1 i Bi }m i=1 possesses common Schur stability, then there exist orthogonal transformations of the matrices Ai that ensure each transformed matrix is Hurwitz. Furthermore, under these transformations, the pairs (∆iAi,∆iBi) maintain a Riccati solution that shares a common matrix. Theorem 3. For i = 1, . . . ,m, suppose that Ai, Bi ∈ Rn×n. If the family {A−1 i Bi}mi=1 has common Schur stability, then there exist orthogonal matrices ∆i, i = 1, . . . ,m, such that for each i, ∆iAi is Hurwitz, and for each i, the pair (∆iAi,∆iBi) has a Riccati solution (Xi, Y ), i.e. all the pairs share the same Y in their Riccati solution. Proof. As {A−1 i Bi}mi=1 has common Schur stability, there are positive definite matrices P,Q1, . . . , Qm such that (A−1 i Bi)P (A−1 i Bi) T +Qi = P, (9) A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 8 of 14 for i = 1, . . . ,m. Additionally, the existence of the family {A−1 i Bi}mi=1 means that for each i, the matrices A−1 i exist. Now, let Y = P−1 ≻ 0 and suppose that the matrices Y A−1 i for all i ∈ {1, . . . ,m} have singular value decomposition Y A−1 i = UiΣiV T i , where UiU T i = I, ViV T i = I, and the diagonal matrices Σi are positive definite. Next, for each i, define ∆i = −UiV T i . Thus, we have Y A−1 i ∆T i = −UiΣiU T i = −Xi. (10) Clearly, each Xi is a positive definite matrix. Now, for each i, observe that Y = −Xi∆iAi = −AT i ∆ T i Xi, i.e., Y +Xi∆iAi = 0 and Y +AT i ∆ T i Xi = 0. Adding these two equations together, we obtain (∆iAi) TXi +Xi(∆iAi) + 2Y = 0, (11) for i = 1, . . . ,m. This implies that for every i ∈ {1, . . . ,m}, ∆iAi is Hurwitz. Now, from (9), it follows that A−1 i ∆T i ∆iBiY −1BT i ∆ T i ∆iA −T i +Qi = Y −1. By pre-multiplying this last equation by Y and post-multiplying it by Y T , we get Y A−1 i ∆T i (∆iBi)Y −1(BT i ∆ T i )∆iA −T i Y T + Y QiY T = Y, i = 1, . . . ,m. This, by (10), is equivalent to Xi(∆iBi)Y −1(∆iBi) TXi + Y QiY T = Y, (12) i = 1, . . . ,m. Combining (11) and (12) gives (∆iAi) TXi +Xi(∆iAi) + Y +Xi(∆iBi)Y −1(∆iBi) TXi + Zi = 0, i = 1, . . . ,m, where Zi = Y QiY T . Thus, the conclusion of the theorem holds. Next, we extend the results on common Lyapunov solutions to demonstrate that, given a family of matrices with a common Lyapunov solution, we can construct a corresponding family of matrix pairs {(Ai, Bi)}mi=1 that shares a common Riccati solution. This result A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 9 of 14 bridges the gap between Lyapunov stability for a family of matrices and Riccati stability for pairs of matrices, highlighting how solutions in one framework can lead to stability in the other. By constructing appropriate matrices Bi, we show that a single Lyapunov matrix can satisfy the Riccati equations across the family, establishing a unified stability criterion. Theorem 4. For i = 1, . . . ,m, suppose that Ai ∈ Rn×n. If the family A = {Ai}mi=1 has a common Lyapunov solution, then there exist matrices Bi, i = 1, . . . ,m, such that for each i, the pair (Ai, Bi) has a Riccati solution (P, Q̂i), i.e. all the pairs share the same P in their Riccati solution. Proof. Let us assume that the family A has a common Lyapunov solution. Then, there exist positive definite matrices P,Q1, . . . , Qm such that AT i P + PAi +Qi = 0, i = 1, . . . ,m. Now, consider arbitrary scalars α > 0 and β > 0 with α2 + β2 < 1. Therefore, it follows that AT i P+PAi+α2Qi+(1−α2−β2)Qi+P ( P−1αβQi ) Q−1 i α−2 ( QiαβP −1 ) P = 0, i = 1, . . . ,m. (13) Define Bi = αβP−1Qi, and Si = (1− α2 − β2)Qi, i = 1, . . . ,m. Then, equation (13) reduces to AT i P + PAi + α2Qi + Si + PBiQ −1 i α−2BT i P = 0, i = 1, . . . ,m. On letting Q̂i = α2Qi, this implies that for each i ∈ {1, . . . ,m}, the pair (Ai, Bi) has (P, Q̂i) as a Riccati solution. This completes the proof. This section establishes the foundational relationships between common Riccati, Lya- punov, and Schur stability. Theorems 2.1 and 2.2 demonstrate that common Riccati stability implies both common Lyapunov and Schur stability, highlighting how the Riccati framework provides a unifying structure for ensuring system robustness across continuous and discrete-time settings. These results indicate that solving the Riccati equations for a family of matrix pairs yields solutions that inherently satisfy Lyapunov-type inequalities, thus extending classical notions of stability to more complex system structures. In the next section, we build upon these results by exploring the preservation of Riccati sta- bility under various transformations, further emphasizing the robustness of the proposed approach. A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 10 of 14 3. Stability preservation in scaled common Riccati families In this section, we explore how common Riccati stability behaves under transforma- tions, focusing on scaling properties and similarity transformations. These adaptation examine how stability characteristics are preserved or modified when the family of matrix pairs undergoes specific alterations, such as scaling by a positive scalar or transformations by a nonsingular matrix. This analysis is crucial in control theory applications where systems may be subject to rescaling or coordinate transformations, yet stability needs to be maintained across these changes. Theorem 5. For i = 1, . . . ,m, suppose that Ai, Bi ∈ Rn×n. If the family U = {(Ai, Bi)}mi=1 has common Riccati stability, then the family V = {(βAi, βBi)}mi=1 has common Riccati stability for all β > 0. Proof. Assume that there are positive definite matrices P,Q,R1, . . . , Rm ∈ Rn×n such that (P,Q) is a common Riccati solution for U . Thus, the following equations hold AT i P + PAi +Q+ PBiQ −1BT i P +Ri = 0, for i = 1, . . . ,m. Thus, for every β > 0, we have β(AT i P + PAi +Q+ PBiQ −1BT i P +Ri) = 0, for i = 1, . . . ,m, which are equivalent to (βAi) TP + P (βAi) + βQ+ P (βBi) Q−1 β (βBi) TP + (βRi) = 0, for i = 1, . . . ,m. This implies that the pair (P, βQ) is a common Riccati solution for the family V. Theorem 6. For i = 1, . . . ,m, suppose that Ai, Bi ∈ Rn×n. If the family U = {(Ai, Bi)}mi=1 has common Riccati stability, then for any nonsingular matrix H ∈ Rn×n, the pair (H−TPH−1, H−TQH−1) is a common Riccati solution for the family V = {(HAiH −1, HBiH −1)}mi=1. Proof. Suppose that (P,Q) is a common Riccati solution for the U . Thus, there are positive definite matrices R1, . . . , Rm ∈ Rn×n satisfying AT i P + PAi +Q+ PBiQ −1BT i P +Ri = 0, for i = 1, . . . ,m. Consequently, H−T (AT i P + PAi +Q+ PBiQ −1BT i P +Ri)H −1 = 0, for all i, and therefore, we have H−TAT i PH−1 +H−TPAiH −1 +H−TQH−1 +H−TPBiQ −1BT i PH−1 +H−TRiH −1 = 0. A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 11 of 14 Therefore, we get that (HAiH −1)TH−TPH−1 +H−TPH−1(HAiH −1) +H−TQH−1+ H−TPH−1(HBiH −1)(H−TQH−1)−1(HBiH −1)TH−TPH−1 +H−TRiH −1 = 0, for all i. This means that the pair (H−TPH−1, H−TQH−1) is a common Riccati solution for the family V. Theorem 7. For i = 1, . . . ,m, suppose that Ai, Bi ∈ Rn×n provided that Bi have full rank for all i. If the family U = {(Ai, Bi)}mi=1 has common Riccati stability, then the family V = {(AT i , B T i )}mi=1 has common Riccati stability Proof. Let (P,Q) be the common Riccati solution for U , i.e., there are positive definte Ri ∈ Rn×n, i = 1, . . . ,m, such that AT i P + PAi +Q+ PBiQ −1BT i P +Ri = 0. Since, P ≻ 0, then P−1 exists. Thus, by pre and and post-multiply these last equations, we obtain P−1AT i +AiP −1 + P−1QP−1 +BiQ −1BT i + P−1RiP −1 = 0. Define P̂ = P−1 ≻ 0, Q̂i = BiQ −1BT i ≻ 0 and R̂i = P−1RiP −1 ≻ 0, i = 1, . . . ,m. Since each Bi has a full rank, then B−1 i , i = 1, . . . ,m, exists. Therefore, Q̂i −1 ’s are defined. Thus, for i = 1, . . . ,m, the pair (P̂ , Q̂i) is a Riccati solution for (AT i , B T i ). To see this, for each i, observe that (AT i ) T P̂ + P̂AT i + Q̂i + P̂BT i Q̂i −1 (BT i ) T P̂ + R̂i = P−1AT i +AiP −1 + P−1QP−1 +BiQ −1BT i + P−1RiP −1. The results presented in this section demonstrate that common Riccati stability is preserved under scaling transformations and similarity transformations. These findings underscore the robustness of Riccati-based stability criteria, which remain unaffected by coordinate transformations or uniform scaling of system parameters. By establishing that the Riccati solution remains valid across such transformations, we show that the proposed framework is not only theoretically sound but also practically applicable to a wide range of control systems, including those subject to rescaling or changes in system representation. A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 12 of 14 4. Examples and Computational Verification In this section, we provide numerical examples to demonstrate the practical applica- bility of the proposed stability criteria. Example 1. Consider the family U = {(Ai, Bi)}2i=1, where A1 = [ −2 1 0 −3 ] , A2 = [ −1 1 0 −4 ] , B1 = [ 0.1 0 0 0.003 ] , and B2 = [ 0.02 −0.019 0 0.23 ] . It can be easily verified that P = [ 2 1 ] and Q = [ 0.1 0.1 ] form a common Riccati solution for U . A simple calculation shows that P is a common Lyapunov solution for the family A = {Ai}2i=1 and Q−1 is a common Schur solution for the family {A−1 i Bi}2i=1, verifying Theorems 1 and 2. Example 2. Consider the family given in Example 1. A simple calculation shows that the matrix P and Q̄ = [ 0.1β 0.1β ] form a common Riccati solution for the family V = {(βAi, βBi)}2i=1 for any β > 0, as asserted by Theorem 5. Example 3. Consider the nonsingular matrix H = [ 1 1 0 1 ] , and the family U from Ex- ample 1. A straightforward calculation shows that the pair P1 = H−TPH−1 = [ 2 −2 −2 3 ] ≻ 0 and Q1 = H−TQH−1 = [ 0.1 −0.1 −0.1 0.2 ] ≻ 0 form a common Riccati solution for the transformed family V = {(HAiH −1, HBiH −1)}2i=1. This result is consistent with Theorem 6, which asserts that common Riccati stability is preserved under similarity transformations. 5. Conclusion In this work, we have explored fundamental stability properties of matrix families, establishing significant links between common Riccati stability, Lyapunov stability, and Schur stability. By demonstrating that common Riccati stability implies both Lyapunov and Schur stability, we have highlighted the utility of Riccati stability as a unifying concept in stability analysis. Additionally, we showed that common Riccati stability is preserved A. Algefary, K. A. Alqufari / Eur. J. Pure Appl. Math, 18 (2) (2025), 6072 13 of 14 under scaling and similarity transformations, reinforcing its robustness in various applica- tions. These results provide a foundation for future work in stability analysis and control theory, particularly in systems with time delays or those requiring stability across multiple configurations. This study’s findings contribute valuable insights for the design of robust control systems that require stability despite parameter variations and transformations. Acknowledgements The authors gratefully acknowledge Qassim University, represented by the Deanship of Graduate Studies and Scientific Research, on the financial support for this research under the number (QU-J-UG-2-2025-56216) during the academic year 1446 AH / 2024 AD. We also sincerely thank the reviewers for their valuable feedback and constructive comments, which have significantly contributed to enhancing the quality of this paper. References [1] R Horn and C Johnson. Matrix Analysis. Cambridge University Press, 1985. [2] Lloyd N Trefethen and David Bau. Numerical linear algebra. SIAM, 2022. [3] Roger A Horn and Charles R Johnson. Topics in matrix analysis. Cambridge univer- sity press, 1991. [4] Eugenius Kaszkurewicz and Amit Bhaya. Matrix diagonal stability in systems and computation. Springer Science & Business Media, 2012. [5] Robert K Brayton and Charles C Conley. 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