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rsre the preserves ∞ o all for . B 4 ≥  2  . eoe its denotes  5 , . ( i 0 = t , 0 τ } . and . , ) ti not is It k n so and ≥ . t 5000 τ< dτ ≥ is ) t 0 , or 0 = n 1 , , IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2

(UES) uniformly exponentially stable if and only there exist for all t0 τ t< ; positive constants K, α˜ such that P3) The solution≤ of≤ (1) satisfies∞ for all t t the inequalities ≥ 0 t t −α˜(t−τ) Φ(t, τ) Ke for t0 τ t< . − R µ[−A(s)]ds R µ[A(s)]ds t t k k≤ ≤ ≤ ∞ x(t0) e 0 x(t) x(t0) e 0 . We will derive results for unspecified vector norm on Rn, . k k ≤k k≤k k For the matrices, as an operator norm is used the inducedk·k II. THEORYABOUTLINEARPERIODICTIME-VARYING norm, A = max Ax . We will use for both vector norm SYSTEMS k k kxk=1 k k and matrix operator norm the same notation but it will always Although precise stability assessment for general LTV sys- be clear from the context that norm is being used. In particular tems is very difficult, the stability of linear periodic time- cases we will consider the three most common vector norm varying (LPTV) systems

- x 1 , x 2 and x ∞ . We denote by µ[A(t)], t 0, the x˙ = A(t)x, A(t + T )= A(t) for some T > 0, (3) logarithmick k k k norm (LN)k k of a continuous matrix function≥ A(t), can be determined using the Floquet theory, which states that A( ) : [0, ) Rn×n defined as · ∞ → for every LPTV system the associated state-transition matrix I + hA(t) 1 can be expressed as µ[A(t)] , lim k n k− , + h→0 h Φ(t, τ)= P −1(t)eR(t−τ)P (τ), 0 τ t< (4) where I denotes the identity on Rn (see Table I). We note ≤ ≤ ∞ n via a stability preserving Lyapunov transformation z = P (t)x, here that the LN µ is not a norm in the usual sense, because where P (t) is continuously differentiable, nonsingular and it can take negative values. periodic matrix function which reduces the stability analysis to an analysis of the LTI system z˙ = Rz, where we define TABLE I the constant matrix by setting RT LOGARITHMICNORMSFORTHEVECTORNORMS AND n n R e = Φ(T, 0). 1 , 2 ∞ , × [1, P. 54], [11, P. 33]. k·k k·k k·k Consequently, the stability of the original LPTV system is equivalent to that of the LTI system z˙ = Rz. The eigenvalues Vector norm Logarithmic norm of R are known as the Floquet characteristic exponents (FCEs) n [3], [4], [7]. Unfortunately, the application of this theory x 1 = xi µ1[A] = max ajj + aij k k i=1 | | 1≤j≤n i=6 j | |! is hampered by the fact that, in general, the FCEs is hard P P to determine, namely, the Lyapunov transformation P (t) is n −1 −Rt 2 1 T defined via P (t)=Φ(t, 0)e . x 2 = xi µ2[A]= 2 λmax A + A k k si=1 A comprehensive Floquet theory including Lyapunov trans- P
formations was developed and their various stability preserving properties were analyzed in [8]. Colaneri [5] addresses a few x ∞ = max xi µ∞[A] = max aii + aij k k 1≤i≤n | | 1≤i≤n j=6 i | |! theoretical aspects of LPTV systems and methodology which P can be useful to characterize and extend other concepts usually In Table I and elsewhere in the paper, the superscript exploited in the time-invariant case only. On the other hand, T relating computational and numerical aspects, in [26], the ’T’ denotes transposition, the number λmax(A + A ) is the maximum eigenvalue of the matrix A + AT . FCEs are directly calculated for the special types of system Remark 1: Note that the value µ[A] may depends on the matrices, when the coefficient matrices are triangular. In [13], used vector norm, see an example in [1, p. 56]. Thus, we based on the solution of linear differential Lyapunov matrix can verify whether the LTI system x˙ = Ax is stable or not equation, necessary and sufficient numerical conditions for by means of the vector norm with negative value of µ[A], asymptotic stability of LTV systems are given. see Lemma 1 (P3) below. For a Hurwitz matrix A we obtain Our aim in the present paper is to provide a conceptually , √ T new approach to study of general LPTV systems allowing to such LN for a vector norm x H x Hx, where the symmetric positive definite matrixk k H satisfies the Lyapunov estimate the norm of state-transition matrix and subsequently T its stability property without knowing the fundamental matrix equation A H + HA = 2In. Then µH [A]= 1/λmax(H), see Lemma 2.3 in [15].− Thus, the stability in− terms of LN solution, purely on the basis of the matrix A(t) entries (Theorem 2). This approach has the advantage of avoiding becomes a topological notion, while the spectrum σA = λ C : λ is an eigenvalue of A is topologically invariant.{ ∈ the need of calculation of Lyapunov transformation. Moreover, Now we summarize the important} properties of the LN useful the developed technique allows us to find the upper and lower for the stability analysis of linear dynamical systems. bounds for the solutions of LPTV systems (3) in Lemma 5 Lemma 1 ( [7], [10], [11], [23], [24]): and for the FCEs in Remark 2. As is shown, the accuracy of the achieved estimates depends on the used vector norm P1) µ[ A] µ[A]; µ[A] µ[B] A B for any given in Rn. Because the spectrum of a matrix is invariant to n− −n matrices≤ A|and B−; |≤k − k the change of norm on Rn, this problem can be formulated P2) Let× X(t), t 0 is a fundamental matrix solution for as an optimization problem of finding vector norm on Rn x˙ = A(t)x. Then≥ minimizing (separately) λ− and λ+. Definitions of these t t − R µ[−A(s)]ds R µ[A(s)]ds and other important constants and concepts are given in the e τ X(t)X−1(τ) eτ (2) following subsection. ≤ ≤

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A. Notation (continued) Remark 2: As a corollary we obtain for FCEs the inclusion

Let − + σR z C : λ z λ . t t ⊂{ ∈ − ≤ ℜ{ }≤ } ˆ Π+(t) , µ[A(s)]ds, Π−(t) , µ[ A(s)]ds, In fact, if there was an eigenvalue λ σR such that − λˆ > λ+ (analogously for λˆ < ∈λ−), then, taking tZ tZ 0 0 ℜ{into} account (4), there would beℜ{ a solution} − xˆ(t) of (3) with + , + − , − (ℜ{λˆ}+ε)(t−t ) λ Π (t0 + T )/T, λ Π (t0 + T )/T xˆ(t) = xˆ(t ) o(e 0 ) as t for arbitrarily k k k 0 k → ∞ ∗ ∗ ∗ small constant ε > 0 which contradicts with (5). Here the δ , min δ : Π (t) λ (t t0)+ δ, t [t0,t0 + T ] , U { ≤ − ∀ ∈ } asymptotics of xˆ(t) is expressed by the ”little-o” Bachmann- ∗ ∗ ∗ δ , max δ : Π (t) λ (t t0)+ δ, t [t0,t0 + T ] , Landau notation. L { ≥ − ∀ ∈ } = +, . All functions and constants are well-defined because∗ µ−[A(t)] is continuous as follows from Lemma 1 (P1) C. Main results and from the assumption of continuity of matrix function ∗ ∗ ∗ ∗ The sufficient conditions for stability of the LPTV systems t A(t). The functions eλ (t−t0)+δL and eλ (t−t0)+δU , → can be expressed in terms of an integral over one period T of = +, will be called a lower and upper barrier function, the LN µ[A(t)] or µ[ A(t)]. ∗ − ∗ ∗ − respectively. The constants δL, δU can be calculated by Theorem 2: If for some vector norm and associated induced applying global extrema–searching procedure for the function norm for matrices is ∗ ∗ Π (t) λ (t t0) on the interval [t0,t0 + T ]. + − − I) Π (t0 + T ) < 0, then the LPTV system (3) is UES; + II) Π (t0 + T )=0, then the LPTV system (3) is US; B. Auxiliary results − III) Π (t0 + T ) < 0, then the LPTV system (3) is unstable In all lemmas below it is assumed that A(t) is periodic with (specifically, the norms of all nonzero solutions converge period T > 0. to infinity as t ). Lemma 2: δ∗ 0, δ∗ 0, =+, . → ∞ U L Proof: I)+II): For all t0 τ t< we have, by (2), Proof: Both≥ inequalities≤ follows∗ immediately− from the ≤ ≤ ∞ fact that Π∗(t + T )= λ∗T, =+, . t t τ 0 R µ[A(s)]ds R µ[A(s)]ds−R µ[A(s)]ds − + ∗ − −1 t t Lemma 3: Π (t) Π (t) for all t t0. X(t)X (τ) eτ = e 0 0 Proof: This− property≤ follows from Lemma≥ 1 (P1). ≤ + + + + + + ∗ ∗ ∗ Π (t)−Π (τ) (λ (t−t0)+δU )−(λ (τ−t0)+δL ) Lemma 4: For all t t0 is Π (t) λ (t t0)+ δU and = e e ∗ ∗ ≥∗ ≤ − ≤ Π (t) λ (t t0)+ δL, =+, . Π+(t +T ) ≥ − ∗ − λ+(t−τ) δ+−δ+ 0 (t−τ) δ+−δ+ Proof: Let tˆ [t , ) is chosen arbitrarily. Then tˆ = e e U L = e T e U L , ∈ 0 ∞ ∈ [t + (k 1)T,t + kT ) for some k 1. As follows from the + + + 0 0 δ −δ Π (t0+T ) − ≥ that is, we set K = e U L and α˜ = in definition of LN, µ[A(t)] is also periodic which yields T Definition 1. − tˆ−(k−1)T III): From the left inequalities in (5) and because λ− = ∗ ∗ − Π (tˆ) = (k 1)Π (t0 + T )+ µ[ A(s)]ds Π (t0 + T )/T by definition, it follows that the norm of each − ∗ nonzero solution of (3) converges to infinity as t , or tZ0 alternatively, analogously as above, → ∞ ∗ ∗ ˆ = (k 1)Π (t0 + T ) + Π (t (k 1)T ), =+, . − − − − − ∗ − X(t)X−1(τ) eΠ (τ)−Π (t) Now, because tˆ (k 1)T [t0,t0 + T ), we have that ≥ − − − ∈ t T − − ∗ ∗ ∗ ∗ − Π ( 0 + ) ( t−τ) δ −δ Π (tˆ) (k 1)Π (t0 + T )+ λ (tˆ (k 1)T t0)+ δ e T e L U ≤ − − − − U ≥ → ∞ ∗ Π (t0 + T ) ∗ ˆ ∗ as t for every fixed τ t0. = (k 1)Π (t0 + T )+ (t (k 1)T t0)+ δU − T − − − Remark→ ∞ 3: Combining Lemma≥ 1 (P2) with [28, Lemma 2 = λ∗(tˆ t )+ δ∗ , what we had to prove. The extension of (Item 3)] and [28, Lemma 5] for µ(t) , µ[A(t)] we get − 0 U the inequality from the interval [t0,t0 + T ] on the whole time another justification of the sufficient condition for uniform ∗ interval [t0, ) for the lower bounds of Π can be proved in exponential stability in Part I of theorem with the difference a similar manner.∞ that we have also derived the values K and α˜ from Definition 1 β Lemma 5: Let us consider (3) with an initial state x(t0) (e and α in [28]) and which are generally classified as Rn. Then ∈ ”difficult to obtain”. − − − −λ (t−t )−δ −Π (t) Remark 4: The connection of Theorem 2 with LTI systems x(t ) e 0 U x(t ) e x(t) k 0 k ≤k 0 k ≤k k x˙ = Ax, which can be considered as LPTV systems with any Π+(t) λ+(t−t )+δ+ x(t ) e x(t ) e 0 U (5) period T > 0 : ≤k 0 k ≤k 0 k I) Remark 1 implies for a Hurwitz matrix A for all t t . ≥ 0 Proof: The claim of lemma follows from Lemma 1 (P3) t0+T + and Lemma 4. Lemma 3 guarantees that the inequalities in (5) Π (t0 + T )= µH [A]ds = T/λmax(H) < 0 make sense. − tZ0 IEEE TRANSACTIONS ON AUTOMATIC CONTROL 4

+ 3.5 This shows that the condition Π (t0 + T ) < 0 is at the same time also necessary condition to be the LTI system 3 x˙ = Ax UES. 2.5

(t) 2 II) Let a nonsingular n n real matrix P and matrix A + × −1 e 1.5 are such that JA , P AP has a Jordan normal form, a b 1 where each block is of the form J = or 0.5 1 b a  − n 0 1 2 3 4 5 J2 = [λ˜] with a 0,b =0 and λ˜ 0. Define on R the ≤ 6 ≤ t T x P T P (= P x 2). Then, from the equality µP P [A]= t k k k k R µ1[A(s)]ds µ2[JA] [10] we get µP T P [A] =0 for block diagonal Π+(t) Fig. 1. The functions e = e0 with µ1[A(t)] = 11/2+ matrix JA with at least one J1 or J2 with a = 0 or (15/2) sin 12t + cos 12t (the solid line) and the lower and upper− barrier +| | + λ˜ =0, that is, λ+t+δ −0.7253t−0.0364 λ+t+δ −0.7253t+1.2872 functions e L = e
, e U = e (the dashed lines). t0+T + Π (t0 + T )= µP T P [A]ds = 0; π/6 tZ0 6 λ+ = [ 11/2+(15/2)(sin 12s + cos12s )] ds + Thus, the condition Π (t0 +T )=0 is also necessary for π − | | Z0 the systems with Jordan normal form described above and which are US but not UES. = 0.7253 and δ+ =1.2872. Analogously, − U III) If all eigenvalues of have positive real part then A ( A) π/6 is a Hurwitz matrix because σ = σ as follows from− −A A − 6 the equality det(A λI ) = ( 1)n det((− A) ( λ)I ) λ = [41/2+(15/2)(sin 12s + cos12s )] ds − n − − − − n π | | [14, p. 524]. Remark 1 yields that Z0

t +T − 0 = 25.2747, and δU =1.2871. For the LN µ2 we analogously − Π (t + T )= µ ˜ [ A]ds = T/λ (H˜ ) < 0. obtain: 0 H − − max tZ0 15√2√sin 12t +1 µ2[A(t)] = 13 [ 13, 2], − 2 − ∈ − Thus, Π (t0 + T ) < 0 establishes also a necessary condition for instability of LTI systems x˙ = Ax, with π/6 √ ( A) being a Hurwitz matrix. + 6 15 2 √sin 12s +1 − λ = 13 ds π " 2 − # Revisiting Example 1 in the light of Theorem 2 we see Z0 2π 2π = 3.4507, and δ+ =0.9337; + U 1 T − Πβ (2π)= µ2[Aβ(s)]ds = max σ (Aβ (s)+A (s))ds 2 β π/6 Z0 Z0 6 15 √2 √sin 12s +1 λ− = + 13 ds =2π max 1,β 1 , π " 2 # {− − } Z0 2π 2π = 22.5493, and δ− =1.0441. − U 1 T Π (2π)= µ2[ Aβ(s)]ds = max σ (−Aβ (s)−A (s))ds In this example, the use of the Euclidean vector µ [A(t)] pro- β − 2 β 2 Z0 Z0 vides the better information regarding the position of the FCEs (= σ ) in the complex-plane as those given by octahedral =2π max 1, 1 β , R { − } norm µ1[A(t)], the vertical strip z C : 22.5493 { ∈ − ≤ + + z 3.4507 µ and z C : 25.2747 z and so Πβ (2π) < 0 if β < 1 (UES system), Πβ (2π)=0 if 2 ℜ{0.7253}≤− , respectively.} The{ result∈ of− simulation≤ is in ℜ{ Fig.} ≤2. β =1 (US system). The sufficient condition for instability is − }µ1 not fulfilled because there is also exponentially stable mode in the system, not influenced by the parameter β. D. Note regarding the robustness of exponentially stable LPTV Example 2: As an illustrative example let us consider for systems against disturbances t 0 the LPTV system with ≥ In this section we will analyze the stability properties of the 11/2+(15/2) sin 12t (15/2)cos12t LPTV systems affected by an external disturbance d(t). Let us A(t)= − consider that the unperturbed system x˙ = A(t)x is UES. What " (15/2)cos12t 41/2 (15/2) sin 12t# − − (6) can we say about the asymptotic behavior of its perturbation For comparison purpose we calculate the barrier functions for x˙ = A(t)x + d(t)? This question represents one of the funda- mental problems on the field of robust stability. The robustness the two vector norms, 1 and 2 . Meaning the lower and upper barrier functionk·k is obviousk·k from Fig. 1. We have that of the systems’ stability is not usually analyzed together with establishing the sufficient conditions ensuring the stability of µ [A(t)] = 11/2+(15/2)(sin12t+ cos12t ) (= µ [A(t)]) some kind for the LTV systems. Among these include, e. g., 1 − | | ∞ IEEE TRANSACTIONS ON AUTOMATIC CONTROL 5

30 14

25 12 10 20

1 2 8 15 ||x|| ||x|| 6 10 4

5 2

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t t (a) Simulation for (b) Simulation for k·k1 k·k2 Fig. 2. The solution of (6) with the initial state x(0) = ( 4, 3)T (the solid line); the bounds for x(t) : ( 4 + 3 )e−0.7253t+1.2872 , ( 4 + − k k1 |− | | | |− | 3 )e−25.2747t−1.2871 (the dashed lines); the bounds for x(t) : √25e−3.4507t+0.9337 , √25e−22.5493t−1.0441 (the dashed lines). | | k k2

[16], [21], [27]–[29]. We have the following result regarding Remark 5: The perturbation d(t) could have been replaced asymptotic behavior of perturbed LPTV systems as t . by a perturbation d˜(x, t) which satisfies d˜(x, t) d(t) → ∞ Theorem 3: Let us consider the perturbed LPTV system, n ≤ k k for all x R . Since this would have introduced no new ideas, we chose∈ to present the notationally simpler case. x˙ = A(t)x + d(t), A(t + T )= A(t), t t0 ( 0), (7) ≥ ≥ The class of allowable disturbances of the form d(t) preserving where A( ) : [0, ) Rn×n and d( ) : [0, ) Rn are · ∞ → · ∞ → the convergence to 0 of the solutions for the UES LPTV continuous and let unperturbed systems is a little wider [25] and contains also + A1) Π (t0 + T ) < 0, that is, the unperturbed system is UES, the functions that do not vanish at infinity. and Theorem 4: Let the unperturbed system is UES. Then all A2) d(t) 0 as t . solutions of perturbed LPTV system x˙ = A(t)x + d(t), A(t + k k → → ∞ Then all solutions of (7) converge to 0 as t (not T )= A(t) and perturbed LTI system x˙ = Ax + d(t) converge necessarily exponentially). → ∞ to 0 as t if and only if Proof: The solution of (7) is given by the Lagrange’s → ∞ variation of constants formula, t+η t sup d(τ)dτ 0 as t . (8) 0≤η≤1 → → ∞ −1 −1 Z x(t)= X(t)X (t )x(t )+ X(t) X (τ)d(τ)dτ t 0 0 Z t0 By [25, Corollary 4.6], the class of allowable perturbations and so x(t) d(x, t) contains also the functions of the form D(t)k(x), k k where D(t) is an n n bounded matrix on [0, ) whose t columns satisfy (8) and× k : Rn Rn is continuous.∞ X(t)X−1(t ) x(t ) + X(t)X−1(τ) d(τ) dτ. → ≤ 0 k 0 k k k Z t0 III. CONCLUSION + t T −1 Π ( 0+ ) (t−τ) δ+−δ+ Using the inequality X(t)X (τ) e T e U L from the proof of Theorem 2, we have≤ We derived, in Theorem 2, the new criterion for uniform

+ and uniform exponential stability of the linear periodic time- Π ( t0 +T ) + (t−t0) δ x(t) x(t ) e T e U varying (LPTV) systems x˙ = A(t)x, t t0, A(t+T )= A(t), k k≤k 0 k ≥ t T > 0 without direct computing of the Floquet character- + t T δ+−δ+ Π ( 0+ ) (t−τ) istic exponents (FCEs). We have shown that the FCEs lie +e U L e T d(τ) dτ k k in the vertical strip z C : λ− z λ+ tZ0 { ∈ − t0≤+T ℜ{ } ≤ } + 1 + of the complex-plane. Here λ = µ[A(s)]ds and Π (t0 +T ) + T (t−t0) δ t0 = x(t0) e T e U t0+T R k k − 1 t + t T λ = µ[ A(s)]ds, where µ[ ] denotes the logarithmic − Π ( 0+ ) τ T e T d(τ) dτ t0 − · + + t0 k k norm (LN)R of matrix associated with an appropriately chosen δU −δL +e + t T . R − Π ( 0+ ) t vector norm. We also briefly discussed the persistence of the e T stability properties for the perturbed LPTV systems under the Applying the L’Hospital rule to the second term we get external disturbances d(t). The fundamental advantage of the

t Π+(t +T ) approach based on the use of LN is the fact that to estimate the − 0 τ e T d(τ) dτ norm of state-transition matrix Φ(t, τ) for system x˙ = A(t)x t0 k k d(t) lim + t T = lim −k k =0, we do not need to know the fundamental matrix solution and t→∞ R − Π ( 0+ ) t t→∞ Π+(t + T )/T e T 0 all necessary estimates are based purely on the matrix A(t) and so x(t) 0 as t . entries. k k → → ∞ IEEE TRANSACTIONS ON AUTOMATIC CONTROL 6

REFERENCES [1] V.N. Afanas’ev, V.B. Kolmanovskii, and V.R. Nosov, Mathematical The- ory of Control Systems Design. Springer, 1996. [2] R. Bellman, Stability theory of differential equations, McGraw-Hill Book Company, Inc., 1953. [3] R.W. Brockett, Finite dimensional linear systems, John Wiley & Sons, Inc., 1970. [4] C. Chicone, Ordinary Differential Equations with Applications, Springer, New York, 1999. [5] P. Colaneri, Theoretical aspects of continuous-time periodic systems, Annual Reviews in Control 29, 205-215 (2005). [6] W.A. Coppel, Dichotomies in Stability Theory, Springer-Verlag Berlin Heidelberg, 1978. [7] W.A. Coppel, Stability and Asymptotic Behavior of Differential Equa- tions, D. C. Heath and Company Boston, 1965. [8] J.J. DaCunha, J.M. Davis, A unified Floquet theory for discrete, contin- uous, and hybrid periodic linear systems, J. Differential Equations 251 2987-3027 (2011). [9] C.A. Desoer, Slowly varying system x˙ = A(t)x, IEEE Transactions on Automatic Control, vol.14, 780–781 (1969). [10] C.A. Desoer, H. Haneda, The Measure of a Matrix as a Tool to Analyze Computer Algorithms for Circuit Analysis IEEE Transactions on Circuits Theory 19, 5, 480–486 (1972). [11] C.A. Desoer, M. Vidyasagar, Feedback Systems: Input-output Properties. Society for Industrial and Applied Mathematics, Philadelphia, 2009. [12] W. Gao, Z-P. Jiang, F.L. Lewis, and Y. Wang, Leader-to-Formation Sta- bility of Multi-Agent Systems: An Adaptive Optimal Control Approach, IEEE Transactions on Automatic Control 63, 10, 3581–3587 (2018). [13] G. Garcia, P.L.D. Peres, and S. Tarbouriech, Necessary and sufficient numerical conditions for asymptotic stability of linear time-varying sys- tems, Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, 5146–5151 (2008). [14] D.A. Harville, Matrix Algebra From a Statisticians Perspective. Springer, New York, 2008. [15] G.-D. Hu, M. Liu, The weighted logarithmic matrix norm and bounds of the matrix exponential, Linear Algebra and its Applications 390, 145-154 (2004). [16] A. Ilchmann, D.H.Owens, and D. Pr¨atzel-Wolters, Sufficient conditions for stability of linear time-varying systems, Systems & Control Letters 9, 157–163 (1987). [17] A. Ioannou and J. Sun, Robust Adaptive Control. Prentice-Hall, Upper Saddle River, NJ, 1996. [18] N.A. Izobov, Lyapunov Exponents and Stability, Cambridge Scientific Publishers, Cambridge, 2012. [19] H.H. Rosenbrock, The stability of linear timedependent control systems, Int. Journal Electr. Control, vol.15, 73–80 (1963). [20] W. J. Rugh, Linear system theory (2nd ed.), Prentice-Hall, Inc., 1996. [21] S. Safavi, U.A. Khan, Asymptotic Stability of LTV Systems With Applications to Distributed Dynamic Fusion, IEEE Transactions on Automatic Control 62(11), 5888–5893 (2017). [22] J.S. Shamma and M. Athans, Guaranteed properties of gain scheduled control for linear parameter-varying plants, Automatica 27(3), 559-564 (1991). [23] G. S¨oderlind, The logarithmic norm. History and modern theory, BIT Numerical Mathematics 46, 631-652 (2006). [24] G. S¨oderlind, R.M.M. Mattheij, Stability and asymptotic estimates in nonautonomous linear differential systems. SIAM J. Math. Anal. 16, No. 1, 69–92 (1985). [25] A. Strauss, and J.A. Yorke, Perturbing uniform asymptotically stable nonlinear systems, Journal of Differential Equations 6, 452-483 (1969). [26] J.-P. Tian, J. Wang, Some results in Floquet theory, with application to periodic epidemic models, Applicable Analysis 94, 1128–1152 (2015). [27] M.Y. Wu, A note on stability of linear time-varying systems, IEEE Transactions on Automatic Control 19, 162–163 (1974). [28] B. Zhou, On asymptotic stability of linear time-varying systems, Auto- matica, 68, 266–276 (2016). [29] J.-M. Wang, Explicit Solution and Stability of Linear Time-varying Differential State Space Systems, International Journal of Control, Au- tomation and Systems 15(4), 1553–1560 (2017).