Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations
EExact stability and instability regions fortwo-dimensional linear autonomous systems offractional-order differential equations
Oana Brandibur, Eva Kaslik
Department of Mathematics and Computer ScienceWest University of Timi¸soara, RomaniaE-mail: [email protected], [email protected]
Abstract:
Necessary and sufficient conditions are explored for the asymptoticstability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependentand fractional-order-independent stability and instability properties are fully char-acterised, in terms of the main diagonal elements of the systems’ matrix, as well asits determinant.
Within the past decades, a growing number of scientific papers debated the perti-nence of fractional calculus in the mathematical modeling of real world phenomena,suggesting that fractional-order systems are capable of delivering more realistic re-sults in a large number of practical applications [8, 14, 16, 17, 24] compared to theirinteger-order counterparts. The main justification of this fact is that fractional-orderderivatives provide for the incorporation of both memory and hereditary properties.Indeed, [13] endorses the index of memory as a plausible physical interpretation ofthe order of a fractional derivative.As in the case of classical dynamical systems theory, stability analysis playsa leading role in the qualitative theory of fractional-order systems. Two surveys[22, 29] have recently summarized the main results that have been obtained withrespect to the stability properties of fractional-order systems. Nevertheless, it hasto be emphasized that most results have been obtained in the framework of linearautonomous commensurate fractional-order systems. In this context, it is importantto note that a generalization of the well-known stability theorem of Matignon [25]has been recently obtained [30]. Furthermore, linearization theorems for fractional-order systems have been presented in [21, 32], providing analogues of the classicalHartman-Grobman theorem.On the other hand, the stability analysis of incommensurate fractional-ordersystems has received significantly less attention throughout the years. Stability a r X i v : . [ m a t h . D S ] O c t roperties of linear incommensurate fractional-order systems with rational ordershave been investigated in [26]. Oscillatory behaviour in two-dimensional incommen-surate fractional-order systems has been explored in [9, 28]. Bounded input boundedoutput stability of systems with irrational transfer functions has been recently an-alyzed in [31]. The asymptotic behavior of the solutions of some classes of linearmulti-order systems of fractional differential equations (such as systems with blocktriangular coefficient matrices) has been investigated in [11].Multi-term fractional-order differential equations [1] and their stability proper-ties are closely related to multi-order systems of fractional differential equations.Very recently, the stability of two-term fractional-order differential and differenceequations has been analyzed in [6, 7, 18].Taking into account the above mentioned developments in the theory of fractional-order systems, necessary and sufficient stability and instability conditions havebeen explored in the case of linear autonomous two-dimensional incommensuratefractional-order systems [3, 4]. In the first paper [3], we have investigated stabil-ity properties of two-dimensional systems composed of a fractional-order differentialequation and a classical first-order differential equation. These results have been ex-tended in [4] for the case of general two-dimensional incommensurate fractional-ordersystems with Caputo derivatives. Specifically, for fractional orders 0 < q < q ≤ O ( t − q )-asymptoticstability of the trivial equilibrium, in terms of the determinant δ of the linear sys-tem’s matrix, as well as the elements a and a of its main diagonal. Moreover,sufficient conditions have also been investigated which guarantee the stability andinstability of the fractional-order system, regardless of the fractional orders.The aim of this work is to complete the stability analysis of two-dimensionalincommensurate fractional-order systems with Caputo derivatives, by extending theresults presented in [4, 5]. On one hand, we fully characterize the fractional-orderdependent stability and instability properties of the considered system, by exploringcertain symmetries related to the characteristic equation associated to our stabilityproblem. On the other hand, we obtain necessary and sufficient conditions forthe stability and instability of the system, regardless of the choice of fractionalorders, in terms of the characteristic parameters a , a and δ mentioned previously.These latter results are particularly useful in practical applications where the exactfractional orders are not precisely known.The paper is structured as follows. Section 2 is dedicated to presenting some pre-liminary results and important definitions. The main results are included in section3 as follows: we first present the statements of the main fractional-order-independentstability and instability theorems, then we prove fractional-order-dependent stabilityand instability results, followed by the proofs of the main theorems. For the sake ofcompleteness, all proofs are presented in detail. Finally, we draw some conclusionsand suggest several directions for future research in section 4.2 Preliminaries
Let us consider the n -dimensional fractional-order system with Caputo derivatives[19, 20, 27]: c D q x ( t ) = f ( t, x ) (1)where q = ( q , q , ..., q n ) ∈ (0 , n and f : [0 , ∞ ) × R n → R n is a continuous functionon the whole domain of definition, Lipschitz-continuous with respect to the secondvariable, such that f ( t,
0) = 0 for any t ≥ . Let ϕ ( t, x ) denote the unique solution of (1) satisfying the initial condition x (0) = x ∈ R n . The existence and uniqueness of the initial value problem asso-ciated to system (1) is guaranteed by the previously mentioned properties of thefunction f [10].It is important to emphasize that in general, due to the presence of the memoryeffect, the asymptotic stability of the trivial solution of system (1) is not of exponen-tial type [7, 15]. Hence, the notion of Mittag-Leffler stability has been introducedfor fractional-order differential equations [23], as a special type of non-exponentialasymptotic stability concept. In this work, we focus on O ( t − α )-asymptotic stability,reflecting the algebraic decay of the solutions. Definition 2.1.
The trivial solution of (1) is called stable if for any ε > thereexists δ = δ ( ε ) > such that for every x ∈ R n satisfying (cid:107) x (cid:107) < δ we have (cid:107) ϕ ( t, x ) (cid:107) ≤ ε for any t ≥ .The trivial solution of (1) is called asymptotically stable if it is stable and t hereexists ρ > such that lim t →∞ ϕ ( t, x ) = 0 whenever (cid:107) x (cid:107) < ρ .Let α > . The trivial solution of (1) is called O ( t − α )-asymptotically stable ifit is stable and there exists ρ > such that for any (cid:107) x (cid:107) < ρ one has: (cid:107) ϕ ( t, x ) (cid:107) = O ( t − α ) as t → ∞ . In this paper, we consider the following two-dimensional linear autonomous incom-mensurate fractional-order system: (cid:26) c D q x ( t ) = a x ( t ) + a y ( t ) c D q y ( t ) = a x ( t ) + a y ( t ) (2)where A = ( a ij ) is a real two-dimensional matrix and q , q ∈ (0 ,
1] are the fractionalorders of the Caputo derivatives. The following characteristic equation is obtainedby means of the Laplace transform method:det (diag( s q , s q ) − A ) = 03hich is equivalent to s q + q − a s q − a s q + det( A ) = 0 . (3)It is important to emphasize that in the characteristic equation (3), s q and s q represent the principal values (first branches) of the corresponding complex powerfunctions [12].By means of asymptotic expansion properties and the Final Value Theorem ofthe Laplace transform [2, 3, 12], necessary and sufficient conditions for the globalasymptotic stability of system (2) have been recently obtained [4]: Proposition 3.1.
1. Denoting q = min { q , q } , system (2) is O ( t − q ) -globally asymptotically stableif and only if all the roots of the characteristic equation (3) are in the openleft half-plane.2. If det( A ) (cid:54) = 0 and the characteristic equation (3) has a root in the open righthalf-plane, system (2) is unstable. The aim of this paper is to analyze the distribution of the roots of the character-istic equation (3) with respect to the imaginary axis of the complex plane. In whatfollows, we denote det( A ) = δ and we consider the complex-valued function∆( s ; a , a , δ, q , q ) = s q + q − a s q − a s q + δ which gives the left-hand side of the characteristic equation (3). Remark 3.1.
The analysis of the roots of the characteristic function ∆( s ; a , a , δ, q , q ) is also encountered in the investigation of the stability properties of the three-termfractional-order differential equation c D q + q x ( t ) − a c D q x ( t ) − a c D q x ( t ) + δx ( t ) = 0 . (4) Therefore, the results presented in this paper are also applicable in the framework ofequation (4) . The statements of the main results are presented below, followed by detailedproofs in the upcoming sections.
Obtaining fractional-order-independent necessary and sufficient conditions forthe asymptotic stability or instability of system (2) are particularly useful in prac-tical applications where the exact values of the fractional orders used in the math-ematical modeling are not precisely known. In this section, we only state the mainresults, giving their complete proofs in section 3.3, due to their complexity.4 heorem 3.1 (Fractional-order independent instability results) . i. If det( A ) < , system (2) is unstable, regardless of the fractional orders q and q .ii. If det( A ) > , system (2) is unstable regardless of the fractional orders q and q if and only if one of the following conditions holds: (cid:40) a + a ≥ det( A ) + 1 or a > , a > , a a ≥ det( A ) . Theorem 3.2 (Fractional-order-independent stability results) . System (2) is asymp-totically stable, regardless of the fractional orders q , q ∈ (0 , if and only if thefollowing inequalities are satisfied: a + a < < det( A ) and max { a , a } < min { , det( A ) } . Remark 3.2.
In the classical integer order case (i.e. q = q = 1 ), it is well-knownthat a two-dimensional linear autonomous system of the form x (cid:48) = A x , with constantmatrix A ∈ R × is asymptotically stable if and only if Tr ( A ) < and det( A ) > .Based on Theorem 3.2, a supplementary inequality max { a , a } < min { , det( A ) }} is required to guarantee that system (2) is asymptotically stable, regardless of thechoice of fractional orders q , q ∈ (0 , . Based on the previous theorems, as the case det( A ) < q , q ∈ (0 , A ) = δ > a , a )-plane: R u ( δ )= { ( a , a ) ∈ R : a + a ≥ δ + 1 or a > , a > , a a ≥ δ } R s ( δ )= { ( a , a ) ∈ R : a + a < { a , a } < min { , δ }} An example is presented for the particular case δ = 4 in Figure 1. Remark 3.3.
Due to Theorem 3.1, when det( A ) = δ > is arbitrarily fixed, system (2) is unstable for any choice of the fractional orders q , q ∈ (0 , if and only if ( a , a ) ∈ R u ( δ ) . On the other hand, based on Theorem 3.2, system (2) is asymp-totically stable for any q , q ∈ (0 , if and only if ( a , a ) ∈ R s ( δ ) . Therefore, if ( a , a ) ∈ R \ ( R s ( δ ) ∪ R u ( δ )) (e.g. white region in Fig. 1), the stability propertiesof system (2) depend on the considered fractional orders. - - - a a det ( A )= Figure 1: The red/blue shaded regions represent the sets R s ( δ ) and R u ( δ ), respec-tively, for δ = det( A ) = 4. The aim of this section is to characterize the stability properties of system (2)when det( A ) = δ > q , q ∈ (0 ,
1] are arbitrarily fixed. The case δ < q , q ∈ (0 , Lemma 3.1.
Let δ > , q , q ∈ (0 , and consider the smooth parametric curve inthe ( a , a ) -plane defined by Γ( δ, q , q ) : (cid:40) a = δ q q q h ( ω, q , q ) a = δ q q q h ( − ω, q , q ) , ω ∈ R , where: h ( ω, q , q ) = (cid:40) ρ ( q , q ) e q ω − ρ ( q , q ) e − q ω , if q (cid:54) = q cos qπ − ω, if q = q := q with the functions ρ ( q , q ) and ρ ( q , q ) defined for q (cid:54) = q as ρ k ( q , q ) = sin q k π sin ( q − q ) π , for k = 1 , . The following statements hold: . The curve Γ( δ, q , q ) is the graph of a smooth, decreasing, concave bijectivefunction φ δ,q ,q : R → R in the ( a , a ) -plane.ii. The curve Γ( δ, q , q ) lies outside the third quadrant of the ( a , a ) -plane.Proof. Let δ > q , q ∈ (0 ,
1] arbitrarily fixed.
Proof of statement (i).
The real-valued function ω (cid:55)→ h ( ω, q , q ) is bijective andmonotonous on R : strictly decreasing if q ≤ q and strictly increasing otherwise.Therefore, the particular form of the parametric equations implies that the curveΓ( δ, q , q ) is the graph of a smooth decreasing bijective function φ δ,q ,q : R → R inthe ( a , a )-plane.If q (cid:54) = q , using the chain rule, we compute: d a da = δ q − q q q · ρ ρ q q ( q − q ) (cid:2) e ( q + q ) ω + e − ( q + q ) ω (cid:3) + 2( q ρ − q ρ )( q ρ e q ω + q ρ e − q ω ) Assuming that q < q , the expression above is strictly negative, as ρ > ρ > q q ≤ ρ ρ ≤ x (cid:55)→ sin xx is decreasing on (0 , π )). A similarargument holds in the case q > q as well. Hence, φ δ,q ,q is a concave function. Proof of statement (ii).
Assume the contrary, i.e. that there exists ( a , a ) ∈ Γ( δ, q , q ) such that a < a <
0, or equivalently, that there exists ω ∈ R such that h ( ± ω, q , q ) <
0. As the case q = q is trivial, we assume without lossof generality that q < q . The inequalities h ( ± ω, q , q ) < ρ ( q , q ) e ± ( q + q ) ω < ρ ( q , q )which leads to ρ ( q , q ) < ρ ( q , q ), or equivalently to q < q , which is absurd.Hence, the curve Γ( δ, q , q ) does not have any points in the third quadrant. Remark 3.4. If q = q := q , Γ( δ, q , q ) represents the straight line: a + a = 2 √ δ cos qπ . In the following, we will denote by N ( a , a , δ, q , q ) the number of unstableroots ( (cid:60) ( s ) ≥
0) of the characteristic function ∆( s ; a , a , δ, q , q ), including theirmultiplicities. The following lemma shows that the function N ( a , a , δ, q , q ) iswell-defined and establishes important properties which will be useful in the proofof the main results. Lemma 3.2.
Let δ > , q , q ∈ (0 , be arbitrarily fixed. The following statementshold:i. The characteristic function ∆( s ; a , a , δ, q , q ) has at most a finite numberof roots satisfying (cid:60) ( s ) ≥ . i. The function ( a , a ) (cid:55)→ N ( a , a , δ, q , q ) is continuous at all points ( a , a ) that do not belong to the curve Γ( δ, q , q ) . Consequently, N ( a , a , δ, q , q ) is constant on each connected component of R \ Γ( δ, q , q ) .Proof. The first step of the proof (see Appendix A.1.) consists of showing thatthere exist a strictly decreasing function l δ,q ,q : R + → R + and a strictly increasingfunction L δ,q ,q : R + → R + such that any unstable root of ∆( s ; a , a , δ, q , q ) isbounded by l δ,q ,q ( (cid:107) a (cid:107) p ) ≤ | s | ≤ L δ,q ,q ( (cid:107) a (cid:107) p ) (5)where p = q + q { q ,q } ≥ a = ( a , a ). Moreover, (cid:107) · (cid:107) p denotes the p -norm in R . Proof of statement (i).
Assuming the contrary, that there exists an infinitenumber of unstable roots, the Bolzano-Weierstrass theorem implies that there existsa convergent sequence of unstable roots ( s j ) with the limit s (cid:54) = 0 (since δ > (cid:60) ( s ) ≥
0. As the function ∆( s ; a , a , δ, q , q ) is analytic in C \ R − ,by the principle of permanence it follows that it is identically zero, which is absurd.Therefore, we obtain that N ( a , a , δ, q , q ) is finite. Proof of statement (ii).
Let a = ( a , a ) ∈ R \ Γ( δ, q , q ) and consider r > B r ( a ) = { a = ( a , a ) ∈ R : (cid:107) a − a (cid:107) p < r } ofthe point a is included in R \ Γ( δ, q , q ).For any a = ( a , a ) ∈ B r ( a ), we have: (cid:107) a (cid:107) p ≤ (cid:107) a − a (cid:107) p + (cid:107) a (cid:107) p < r + (cid:107) a (cid:107) p and hence, inequality (5) implies that any root s of ∆( s ; a , a , δ, q , q ) such that (cid:60) ( s ) ≥ l δ,q ,q ( r + (cid:107) a (cid:107) p ) < | s | < L δ,q ,q ( r + (cid:107) a (cid:107) p ) . Denoting m = l δ,q ,q ( r + (cid:107) a (cid:107) p ) and M = L δ,q ,q ( r + (cid:107) a (cid:107) p ), let us consider in thecomplex plane the simple closed curve ( γ ), oriented counterclockwise, bounding theopen set D = { s ∈ C : (cid:60) ( s ) > , m < | s | < M } . The above construction shows that for any a = ( a , a ) ∈ B r ( a ) all unstable rootsof ∆( s ; a , a , δ, q , q ) are inside the open set D .As ∆( s ; a , a , δ, q , q ) (cid:54) = 0 for any s ∈ ( γ ), it is easy to see that d = min s ∈ ( γ ) | ∆( s ; a , a , δ, q , q ) | > . Moreover, we consider q ≥ p + q = 1 and denote: r (cid:48) = min (cid:26) r, d (cid:107) ( M q , M q ) (cid:107) q (cid:27) . s ∈ ( γ ) and for any a ∈ B r (cid:48) ( a ) ⊂ B r ( a ), we have: | ∆( s ; a , a , δ, q , q ) − ∆( s ; a , a , δ, q , q ) | == | ( a − a ) s q + ( a − a ) s q | ≤≤ | a − a | M q + | a − a | M q ≤≤ (cid:107) a − a (cid:107) p (cid:107) ( M q , M q ) (cid:107) q << r (cid:48) (cid:107) ( M q , M q ) (cid:107) q ≤ d ≤ | ∆( s ; a , a , δ, q , q ) | . Rouch´e’s theorem implies ∆( s ; a , a , δ, q , q ) and ∆( s ; a , a , δ, q , q ) havethe same number of roots in the domain D , and hence N ( a , a , δ, q , q ) = N ( a , a , δ, q , q ) for any a ∈ B r (cid:48) ( a ) . Therefore, the function ( a , a ) (cid:55)→ N ( a , a , δ, q , q ) is continuous on R \ Γ( δ, q , q ),and as it is integer-valued, it follows that it is constant on each connected componentof R \ Γ( δ, q , q ).The following theorem represents the main result characterizing fractional-order-dependent stability and instability properties of system (2). Theorem 3.3 (Fractional-order-dependent stability and instability results) . Let det( A ) = δ > and q , q ∈ (0 , arbitrarily fixed. Consider the curve Γ( δ, q , q ) and the function φ δ,q ,q : R → R given by Lemma 3.1.i. The characteristic equation (3) has a pair of pure imaginary roots if and onlyif ( a , a ) ∈ Γ( δ, q , q ) .ii. System (2) is O ( t − q ) -asymptotically stable (with q = min { q , q } ) if and onlyif a < φ δ,q ,q ( a ) . iii. If a > φ δ,q ,q ( a ) , system (2) is unstable.Proof. Assume that δ > q , q ∈ (0 ,
1] are arbitrarily fixed.
Proof of statement (i).
It is easy to see that the characteristic equation (3)has a pair of pure imaginary roots if and only if there exists ω ∈ R such that∆( iδ q q e ω ; a , a , δ, q , q ) = 0. As i q = cos qπ + i sin qπ , taking the real and theimaginary parts of the previous equation, one obtains: (cid:40) a δ − q q q e q ω cos q π + a δ − q q q e q ω cos q π = e ( q + q ) ω cos ( q + q ) π + 1 a δ − q q q e q ω sin q π + a δ − q q q e q ω sin q π = e ( q + q ) ω sin ( q + q ) π (6)If q (cid:54) = q , solving this system for a and a shows that the characteristicequation (3) has a pair of pure imaginary roots if and only if ( a , a ) belongs tothe curve Γ( δ, q , q ) given by Lemma 3.1.9n the particular case q = q := q , system (6) is compatible if and only if ω = 0.Moreover, the set of solutions of (6) is the straight line a + a = 2 √ δ cos qπ δ, q, q ) given by Lemma 3.1 (see Remark 3.4). Proof of statement (ii).
Choosing a = a = −
1, we argue that ∆( s ; − , − , δ, q , q )does not have any roots in the right half plane. Indeed, assuming that there exists s ∈ C such that (cid:60) ( s ) ≥ s q + q + s q + s q + 1 = 0 , it follows by division by s q that s q + s q − q + 1 + s − q = 0 . As q ∈ (0 , q − q ∈ [ − ,
1] and − q ∈ [ − , N ( − , − , δ, q , q ) = 0. From Lemma 3.1 (ii) and Lemma3.2 it follows that N ( a , a , δ, q , q ) = 0, for any ( a , a ) from the region belowthe curve Γ( δ, q , q ), which leads to the desired conclusion. Proof of statement (iii).
Let s ( a , a , δ, q , q ) denote the root of ∆( s ; a , a , δ, q , q )satisfying s ( a (cid:63) , a (cid:63) , δ, q , q ) = iβ , with β = δ q q e ω as in the proof of statement(i), where ( a (cid:63) , a (cid:63) ) ∈ Γ( δ, q , q ). Taking the derivative with respect to a in theequation s q + q − a s q − a s q + δ = 0we obtain( q + q ) s q + q − ∂s∂a − s q − a q s q − ∂s∂a − a q s q − ∂s∂a = 0 . We deduce: ∂s∂a = s q ( q + q ) s q + q − − a q s q − − a q s q − and therefore ∂ (cid:60) ( s ) ∂a = (cid:60) (cid:18) ∂s∂a (cid:19) = (cid:60) (cid:18) s q ( q + q ) s q + q − − a q s q − − a q s q − (cid:19) . We have ∂ (cid:60) ( s ) ∂a (cid:12)(cid:12)(cid:12) ( a (cid:63) ,a (cid:63) ) = (cid:60) (cid:18) ( iβ ) q P ( iβ ) (cid:19) = β q (cid:60) (cid:32) i q P ( iβ ) | P ( iβ ) | (cid:33) = β q | P ( iβ ) | · (cid:60) (cid:16) i q P ( iβ ) (cid:17) P ( s ) = ( q + q ) s q + q − − a (cid:63) q s q − − a (cid:63) q s q − . A simple computation leadsto ∂ (cid:60) ( s ) ∂a (cid:12)(cid:12)(cid:12) ( a (cid:63) ,a (cid:63) ) = δ q q q · β q + q − | P ( iβ ) | · sin ( q − q ) π (cid:0) q ρ e q ω + q ρ e − q ω (cid:1) == δ q q q · β q + q − | P ( iβ ) | · sin ( q − q ) π · ∂h∂ω ( − ω, q , q ) . In a similar way, we compute ∂ (cid:60) ( s ) ∂a (cid:12)(cid:12)(cid:12) ( a (cid:63) ,a (cid:63) ) and we finally obtain the gradient vector ∇(cid:60) ( s )( a (cid:63) , a (cid:63) ) = (cid:18) ∂ (cid:60) ( s ) ∂a , ∂ (cid:60) ( s ) ∂a (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ( a (cid:63) ,a (cid:63) ) == β q + q − | P ( iβ ) | · sin ( q − q ) π · (cid:18) δ q q q ∂h∂ω ( − ω, q , q ) , δ q q q ∂h∂ω ( ω, q , q ) (cid:19) . From the parametric equations of the curve Γ( δ, q , q ) and the properties of thefunction h it is easy to deduce that the gradient vector ∇(cid:60) ( s )( a (cid:63) , a (cid:63) ) is in fact anormal vector to the curve Γ( δ, q , q ) that points outward from the region belowthe curve. We deduce that the following transversality condition is fulfilled for thedirectional derivative: ∇ u (cid:60) ( s )( a (cid:63) , a (cid:63) ) = (cid:104)∇(cid:60) ( z )( a (cid:63) , a (cid:63) ) , u (cid:105) > , for any vector u which points outward from the region below the curve Γ( δ, q , q ).Therefore, as the parameters ( a , a ) cross the curve Γ( δ, q , q ) into the regionabove the curve, (cid:60) ( s ) becomes positive and the pair of conjugated roots ( s, s ) crossesthe imaginary axis from the open left half-plane to the open right half-plane. Hence, N ( a , a , δ, q , q ) = 2 for any ( a , a ) from the region above the curve Γ( δ, q , q ),and the system (1) is unstable. Remark 3.5.
In Fig. 2, several curves Γ( δ, q , q ) have been plotted for δ = 4 , q = 0 . and q ∈ (0 , , together with the fractional-order-independent stability re-gions R s ( δ ) , R u ( δ ) . The regions below and above each curve represent the asymptoticstability region and instability region, respectively, provided by Theorem 3.3. Lightershades of red and blue have been used to plot the parts of these regions for which sys-tem (2) is asymptotically stable / unstable for the particular values of the fractionalorders q , q which have been chosen, but not for all ( q , q ) ∈ (0 , . - - - a a det ( A )=
4, q1 = = - - - - a a det ( A )=
4, q1 = = - - - - a a det ( A )=
4, q1 = = - - - - a a det ( A )=
4, q1 = = - - - - a a det ( A )=
4, q1 = = - - - - a a det ( A )=
4, q1 = = Figure 2: Curves Γ( δ, q , q ) (black) for δ = 4, q = 0 . q ∈{ . , . , . , . , . , } . Shades of red / blue represent the asymptotic stability/ instability regions, lighter shades being associated to fractional-order-dependentregions and darker shades to fractional-order-independent regions R s ( δ ) and R u ( δ ).12 .3 Proofs of the fractional-order independent stability andinstability results We are now ready to prove the main results presented in section 3.1. Throughoutthis section, we assume det( A ) = δ >
0, unless stated otherwise and we use thenotations R u ( δ ) and R s ( δ ) introduced in section 3.1 for the instability and stabilityregions, respectively.The following lemma provides a sufficient result for the instability of system (2),regardless of the fractional orders q and q . Lemma 3.3. If ( a , a ) ∈ R u ( δ ) , then system (2) is unstable, regardless of thefractional orders q and q .Proof. Let ( a , a ) ∈ R u ( δ ) and ( q , q ) ∈ (0 , arbitrarily fixed. We will showthat the characteristic function ∆( s ; a , a , δ, q , q ) has at least one positive realroot.First, it is easy to see that ∆( s ; a , a , δ, q , q ) → ∞ as s → ∞ .On one hand, let us notice that if a + a ≥ δ + 1, it follows that∆(1; a , a , δ, q , q ) = 1 − a − a + δ ≤ . Hence, the function s (cid:55)→ ∆( s ; a , a , δ, q , q ) has at least one positive real root inthe interval [1 , ∞ ). Therefore, the system (2) is unstable.On the other hand, if a > a > a a ≥ δ , as∆( s ; a , a , δ, q , q ) = ( s q − a )( s q − a ) + δ − a a we see that for s = ( a ) /q >
0, we have∆( s ; a , a , δ, q , q ) = δ − a a ≤ . Hence, the function s (cid:55)→ ∆( s ; a , a , δ, q , q ) has at least one strictly positive realroot. It follows that system (2) is unstable.The following lemma provides a sufficient result for the asymptotic stability ofsystem (2), regardless of the fractional orders q and q . Lemma 3.4. If ( a , a ) ∈ R s ( δ ) then system (2) is asymptotically stable, regardlessof the fractional orders q and q .Proof. Let ( a , a ) ∈ R s ( δ ) and ( q , q ) ∈ (0 , arbitrarily fixed. As a + a < a < s ; a , a , δ, q , q ) has a root s = re iθ in theright half-plane, where r > θ ∈ (cid:2) , π (cid:3) .Multiplying the characteristic equation by s − q , we get: s q − a s q − q − a + δs − q = 0 . s q , s q − q and s − q arein the right half-plane, we obtain: a = (cid:60) ( s q ) − a (cid:60) ( s q − q ) + δ (cid:60) ( s − q ) ≥ min { , − a , δ } (cid:2) (cid:60) ( s q ) + (cid:60) ( s q − q ) + (cid:60) ( s − q ) (cid:3) = min { , − a , δ } (cid:2) r q cos( q θ ) + r q − q cos(( q − q ) θ ) + r − q cos( q θ ) (cid:3) . It is important to remark that for any r > q , q ∈ (0 ,
1] and θ ∈ (cid:2) , π (cid:3) , thefollowing inequality holds: r q cos( q θ ) + r q − q cos(( q − q ) θ ) + r − q cos( q θ ) ≥ . Indeed, denoting q θ = x ∈ (cid:2) , π (cid:3) , q θ = y ∈ (cid:2) , π (cid:3) and r θ = α > α y cos y + α y − x cos( y − x ) + α − x cos x ≥ , ∀ x, y ∈ (cid:104) , π (cid:105) , α > , (7)which is proved in the Appendix A.2. It follows that: a ≥ min { , − a , δ } > . On the other hand, as ( a , a ) ∈ R s ( δ ), we have a < − a and a < min { , δ } .Hence, a < min { , − a , δ } , which leads to a contradiction. Therefore, we deducethat all the roots of the characteristic function ∆( s ; a , a , δ, q , q ) are in the openleft half-plane, and hence, system (2) is asymptotically stable.As sufficiency in Theorems 3.1 and 3.2 has been proved in the previous two lem-mas, the next part of this section is devoted to proving necessity in both theorems.With this aim in mind, in what follows, we will denote by Q ( δ ) the region of the( a , a )-plane which is covered by the curves Γ( δ, q , q ) defined in Lemma 3.1, i.e.: Q ( δ ) = { ( a , a ) ∈ R : ∃ ( q , q ) ∈ (0 , s.t. ( a , a ) ∈ Γ( δ, q , q ) } . The following lemma is the key result which allows us to prove necessity in Theorems3.1 and 3.2.
Lemma 3.5.
The following holds: Q ( δ ) = R \ ( R s ( δ ) ∪ R u ( δ )) . Proof.
A proof by double inclusion is presented below.
Step 1. Proof of the inclusion Q ( δ ) ⊆ R \ ( R s ( δ ) ∪ R u ( δ )) . In the case q = q = q , elementary inequalities and Remark 3.4 provide thatΓ( δ, q, q ) are straight lines which are included in R \ ( R s ( δ ) ∪ R u ( δ )).14et us now consider q < q (the opposite case is treated similarly) and showthat Γ( δ, q , q ) ⊂ R \ ( R s ( δ ) ∪ R u ( δ )). Considering an arbitrary point ( a , a ) ∈ Γ( δ, q , q ), it follows that there exists ω ∈ R such that a = δ q q q h ( ω, q , q ) and a = δ q q q h ( − ω, q , q ), where the function h is given in Lemma 3.1.Let us first show that ( a , a ) / ∈ R u ( δ ). On one hand, one can write: a + a = u ( t ) + δu ( t − )where t = δ q q e ω > u ( t ) = ρ t q − ρ t q , where the arguments of the functions ρ , ρ have been dropped for simplicity. The function u ( t ) reaches its maximal valueat the point t max = (cid:16) q ρ q ρ (cid:17) q − q and a straightforward calculation leads to: u max = u ( t max ) = (cid:18) sin q π q (cid:19) q q − q · (cid:18) q sin q π (cid:19) q q − q · q − q sin ( q − q ) π . We will next show that u max <
1. Indeed, as the function v ( x ) = x ln (cid:0) x sin x (cid:1) is posi-tive and convex on [0 , π ] with lim x → v ( x ) = 0, it follows that v ( x ) is superadditive,and hence: v ( x ) + v ( y − x ) < v ( y ) , for any 0 < x < y ≤ π . Considering x = q π and y = q π in the previous inequality, we obtain ln( u max ) < u max < < q < q ≤
1. Therefore: a + a = u ( t ) + δu ( t − ) ≤ u max ( δ + 1) < δ + 1 . (8)On the other hand, a a = δh ( ω, q , q ) h ( − ω, q , q ) == δ ( ρ e q ω − ρ e − q ω )( ρ e − q ω − ρ e q ω ) == δ (cid:2) ρ + ρ − ρ ρ (cid:0) e ( q + q ) ω + e − ( q + q ) ω (cid:1)(cid:3) << δ (cid:0) ρ + ρ − ρ ρ (cid:1) = δ ( ρ − ρ ) == δ (cid:32) cos ( q + q ) π cos ( q − q ) π (cid:33) < δ and hence, combined with inequality (8) it follows that ( a , a ) / ∈ R u ( δ ).Moreover, assuming by contradiction that ( a , a ) ∈ R s ( δ ), Lemma 3.4 impliesthat all roots of the characteristic function ∆( s ; a , a , δ, q , q ) are in the open lefthalf-plane, and hence, by Theorem 3.3 we obtain that ( a , a ) / ∈ Γ( δ, q , q ), whichis absurd. Therefore, ( a , a ) / ∈ R s ( δ ).Hence, the proof of the inclusion Q ( δ ) ⊆ R \ ( R s ( δ ) ∪ R u ( δ )) is now complete.15 tep 2. Proof of the inclusion R \ ( R s ( δ ) ∪ R u ( δ )) ⊆ Q ( δ ) . Considering the function F δ : R × (0 , × (0 , → R defined by F δ ( ω, q , q ) = (cid:16) δ q q q h ( ω, q , q ) , δ q q q h ( − ω, q , q ) (cid:17) , it is easy to see that Q ( δ ) represents the image of the function F δ , i.e.: Q ( δ ) = F δ ( R × (0 , × (0 , R e ( δ ) = { ( a , a ) ∈ R : 0 ≤ a + a < √ δ } ⊂ Q ( δ ) . Moreover, as h ( − ω, q , q ) = h ( ω, q , q ) for any q (cid:54) = q , it follows that Q ( δ )is symmetric with respect to the first bisector a = a of the ( a , a )-plane.Therefore, in order to determine Q ( δ ) it suffices to find its intersection with anarbitrary straight line l m : a − a = m , m ∈ R , which is parallel to the firstbisector of the ( a , a )-plane. First, Lemma 3.1 implies that each curve Γ( δ, q , q )is the graph of a smooth, decreasing, concave, bijective function in the ( a , a )-plane, and hence, it intersects the line l m exactly in one point. In other words, forarbitrarily fixed q , q ∈ (0 ,
1] and m ∈ R , the equation δ q q q h ( ω, q , q ) − δ q q q h ( − ω, q , q ) = m (9)has a unique solution ω (cid:63)m ( q , q ). From the implicit function theorem and the prop-erties of the function h it follows that the function ω (cid:63)m is continuously differentiableon the open sets S − = { ( q , q ) ∈ (0 , : q < q } S + = { ( q , q ) ∈ (0 , : q > q } Therefore, the abscissa of the point of intersection Γ( δ, q , q ) ∩ l m is a m ( q , q ) = δ q q q h ( ω (cid:63)m ( q , q ) , q , q ) . The function a m is continuously differentiable on S − and S + , and hence, a m ( S ± ) areintervals. The problem of determining these intervals reduces to finding the extremevalues of the function a m over the sets S − and S + , respectively.Defining the functions α ( ω, q , q ) = δ q q q h ( ω, q , q ) and α ( ω, q , q ) = δ q q q h ( − ω, q , q ) , from (9) and the implicit function theorem it follows that ∂ω (cid:63)m ∂q k ( q , q ) = − ∂α ∂q k − ∂α ∂q k ∂α ∂ω − ∂α ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ω (cid:63)m ( q ,q ) ,q ,q ) , k = 1 , .
16n what follows, we will show that the function a m does not have any criticalpoints inside S ± . Indeed, assuming that ∇ a m ( q , q ) = 0 for ( q , q ) ∈ S ± , takinginto account that a m ( q , q ) = α ( ω (cid:63)m ( q , q ) , q , q ), a simple application of the chainrule leads to: ∂α ∂ω ( ω (cid:63)m ( q , q ) , q , q ) · ∂ω (cid:63)m ∂q k ( q , q ) + ∂α ∂q k ( ω (cid:63)m ( q , q ) , q , q ) = 0 , k = 1 , . Combining the last two relations, it follows that: ∂α ∂ω · ∂α ∂q k = ∂α ∂q k · ∂α ∂ω , k = 1 , , where the arguments have been dropped for simplicity. Plugging in the expressionof the function h given in Lemma 3.1 and eliminating δ from the previous systemleads to a quadratic equation in ξ = e ( q + q ) ω (cid:63)m which has a negative discriminant: − ( q ρ − q ρ ) , and hence, does not admit real roots.Therefore, the extreme values of the function a m are reached on the boundariesof the sets S ± , respectively. This is equivalent to the fact that the boundary ∂Q ( δ ) iscomposed of points belonging to Γ( δ, q , q ) when ( q , q ) ∈ ∂S ± . Hence, it remainsto show that ∂R s ( δ ) ∪ ∂R u ( δ ) = ∂Q ( δ ).On one hand, due to the fact that h ( ω, q , q ) → q →
0, for any ω ∈ R and q ∈ (0 , q , q ) → (0 , q ), with q ∈ (0 , δ, q , q ) approaches the union of half-lines given parametrically by H : (cid:40) a = 1 + min { , t } a = δ (1 − max { , t } ) , t ∈ R . Similarly, due to the property h ( − ω, q , q ) = h ( ω, q , q ) which holds for any q (cid:54) = q , we obtain that as ( q , q ) → ( q, q ∈ (0 , δ, q , q )approaches the union of half-lines H : (cid:40) a = δ (1 + min { , t } ) a = 1 − max { , t } , t ∈ R . Moreover, Remark 3.4 provides thatΓ( δ, ,
1) : a + a = 0 . Therefore, a simple geometric analysis of the relative positions of the half-lines H and H given above and the line a + a = 0 shows that ∂R s ( δ ) ⊂ ∂Q ( δ ).On the other hand, considering δ (cid:54) = 1 and choosing ω = 0 in the parametricequations of the curve Γ( δ, q , q ), q (cid:54) = q , given by Lemma 3.1, it follows that thepoints ( a ( q , q ) , a ( q , q )) = (cid:16) δ q q q ( ρ − ρ ) , δ q q q ( ρ − ρ ) (cid:17) Q ( δ ). Let us also notice that the point (cid:16) δ q q q , δ q q q (cid:17) belongs to thearc of the parabola P : a a = δ , considered between the points (1 , δ ) and ( δ, d (( a , a ) , P ) ≤ d (cid:16) ( a , a ) , (cid:16) δ q q q , δ q q q (cid:17)(cid:17) == | ρ − ρ − | (cid:113) δ q q q + δ q q q ≤≤ √ , δ ) | ρ − ρ − | == √ , δ ) (cid:32) − cos ( q + q ) π cos ( q − q ) π (cid:33) −→ , as either q → q →
0. Therefore, P ⊂ ∂Q ( δ ).In a similar manner, considering ω = − ln( δ ) q + q in the parametric equations ofthe curve Γ( δ, q , q ), q (cid:54) = q , given by Lemma 3.1, it follows that the points( a δ ( q , q ) , a δ ( q , q )) = ( ρ − δρ , δρ − ρ ) ∈ Q ( δ ). For an arbitrary µ > µ (cid:54) = 1, let us consider the sequence of points M n = (cid:18) a δ (cid:18) n , µn (cid:19) , a δ (cid:18) n , µn (cid:19)(cid:19) ∈ Q ( δ ) , n ∈ Z + , n > [ µ ] . Applying L’Hospital’s rule results inlim n →∞ M n = (cid:18) µ − δµ − , δµ − µ − (cid:19) := M µ . It is now easy to deduce that the set of limit points M µ , with µ > µ (cid:54) = 1, is infact the straight line a + a = δ + 1, except the segment joining the points ofcoordinates (1 , δ ) and ( δ, a + a = δ + 1 withoutthe segment between (1 , δ ) and ( δ,
1) is also included in ∂Q ( δ ). Combined with theprevious result concerning the arc of parabola P , it follows that ∂R u ( δ ) ⊂ ∂Q ( δ ).The case δ = 1 is trivial, as the boundary ∂R u ( δ ) becomes the whole straightline a + a = 2, which is the limit of Γ(1 , q, q ) as q → Remark 3.6.
In Fig. 3, for δ = 4 , we exemplify the set Q ( δ ) and the resultspresented in Lemma 3.5, by plotting a large number of curves Γ( δ, q , q ) for ( q , q ) = (cid:0) j , k (cid:1) , with j, k = 1 , . The union of all these curves fills in the white regionrepresented in Fig. 1, which separates the stability region R s ( δ ) and the instabilityregion R u ( δ ) . We finally present the proofs of the main theorems.18 - - - a a det ( A )= Figure 3: Curves Γ( δ, q , q ) given by Lemma 3.1, for det( A ) = δ = 4 and q i ∈ (cid:8) k , k = 1 , (cid:9) , i = 1 , q q . The red/blue shaded regions represent the sets R u ( δ ) and R s ( δ ), respectively. Proof of Theorem 3.1.Proof of statement (i).
Because ∆(0) = δ < ∞ ) = ∞ , due to the factthat ∆ is continuous on (0 , ∞ ), it results that it has at least one strictly positivereal root. Therefore, based on Proposition 3.1, it follows that system (2) is unstable. Proof of statement (ii).
If ∆(0) = δ >
0, sufficiency is provided by Lemma 3.3.For the proof of necessity, assuming that system (2) is unstable, regardless of thefractional orders q and q , and assuming by contradiction that ( a , a ) / ∈ R u ( δ ),using Lemma 3.5 it follows that there exist q ∗ , q ∗ ∈ (0 ,
1] (not unique) such that( a , a ) is in the connected component of R \ Γ( δ, q ∗ , q ∗ ) which includes R s ( δ ), i.e.( a , a ) is below the curve Γ( δ, q ∗ , q ∗ ). Hence, based on Theorem 3.3, it followsthat system (2) with the particular fractional orders q ∗ , q ∗ is asymptotically stable,which is absurd. Proof of Theorem 3.2.
Sufficiency is provided by Lemma 3.4. As for the proof ofnecessity, let us assume that system (2) is asymptotically stable, regardless of thefractional orders q and q , and assume by contradiction that ( a , a ) / ∈ R a ( δ ).Lemma 3.5 provides that there exist q ∗ , q ∗ ∈ (0 ,
1] (not unique) such that ( a , a )is in the connected component of R \ Γ( δ, q ∗ , q ∗ ) which includes R u ( δ ), i.e. ( a , a )is above the curve Γ( δ, q ∗ , q ∗ ). Hence, based on Theorem 3.3, it follows that sys-tem (2) with the particular fractional orders q ∗ , q ∗ not asymptotically stable, whichcontradicts the initial hypothesis. 19 Conclusions
In this work, a complete characterization of fractional-order-independent stabilityand instability properties of two-dimensional incommensurate linear fractional-ordersystems has been achieved. Moreover, necessary and sufficient conditions have alsobeen presented for the stability and instability of two-dimensional fractional-ordersystems, depending on the choice of the fractional orders of the Caputo derivatives.These results provide comprehensive practical tools for a straightforward stabilityanalysis of two-dimensional fractional-order systems encountered in real world ap-plications.Extension of these results to the case of two-dimensional systems of fractional-order difference equations requires further investigation. A possible generalizationto higher-dimensional fractional-order systems is still an open question which willbe addressed in future research, taking into account the increasing complexity of theproblem.
Appendix
A.1. Boundedness of the set of unstable roots of ∆( s ; a , a , δ, q , q ) The characteristic equation of system (2) is s q + q − a s q − a s q + δ = 0 . Denoting α = q + q β = q − q ≤ β < α ≤
1, the characteristicequation can be written as s α − a s α + β − a s α − β + δ = 0 . Dividing by √ δs α , we obtain: s α √ δ + √ δs α = a √ δ s β + a √ δ s − β . (10)Denoting z = s α √ δ , c = a ( √ δ ) βα − , c = a ( √ δ ) − βα − and c = ( c , c ), equation(10) becomes z + z − = c z βα + c z − βα . p = q + q { q , q } ≥ q = α | β | = q + q | q − q | ≥ p + 1 q = 1, and Young’s inequality provides: (cid:12)(cid:12) | z | − | z | − (cid:12)(cid:12) ≤ | z + z − | ≤ | c | · | z | βα + | c | · | z | − βα ≤ p ( | c | p + | c | p ) + 1 q (cid:0) | z | + | z | − (cid:1) = (cid:18) − q (cid:19) (cid:107) c (cid:107) pp + 1 q (cid:0) | z | + | z | − (cid:1) . (11)On one hand, if | z | > | z | − , or equivalently | z | >
1, inequality (11) can be writtenas the quadratic inequality | z | − (cid:107) c (cid:107) pp · | z | − γ ≤ , where γ = q + 1 q − | z | ≤ (cid:107) c (cid:107) pp + √ γ. (12)On the other hand, if | z | < | z | − , or equivalently | z | <
1, inequality (11) leads tothe quadratic inequality γ | z | + (cid:107) c (cid:107) pp · | z | − ≥ , and hence: | z | ≥ −(cid:107) c (cid:107) pp + (cid:113) (cid:107) c (cid:107) pp + 4 γ γ (13)In the above calculations, (cid:107) c (cid:107) pp = | c | p + | c | p = | a | p ( √ δ ) p ( βα − ) + | a | p ( √ δ ) − p ( βα +1 ) . Furthermore, as p (cid:18) βα − (cid:19) = q + q { q , q } · (cid:18) q − q q + q − (cid:19) = − q min { q , q }− p (cid:18) βα + 1 (cid:19) = − q + q { q , q } · (cid:18) q − q q + q + 1 (cid:19) = − q min { q , q } . we have: (cid:107) c (cid:107) pp = | a | p ( √ δ ) − q { q ,q } + | a | p ( √ δ ) − q { q ,q } ≤ D ( δ, q , q ) · (cid:107) a (cid:107) pp , where D ( δ, q , q ) = max (cid:110) ( √ δ ) − q { q ,q } , ( √ δ ) − q { q ,q } (cid:111) .21herefore, inequalities (12) and (13) provide that −(cid:107) c (cid:107) pp + (cid:113) (cid:107) c (cid:107) pp + 4 γ γ ≤ | z | ≤ (cid:107) c (cid:107) pp + √ γ (14)Considering the decreasing function f : R + → R + defined by f ( u ) = − u + (cid:112) u + 4 γ γ and the increasing function F : R + → R + F ( u ) = u + √ γ, inequality (11) becomes f ( (cid:107) c (cid:107) pp ) ≤ | z | ≤ F ( (cid:107) c (cid:107) pp ) . Taking into consideration that (cid:107) c (cid:107) pp ≤ D ( δ, q , q ) (cid:107) a (cid:107) pp and z = s α √ δ , the previousinequality implies f (cid:0) D ( δ, q , q ) (cid:107) a (cid:107) pp (cid:1) ≤ f ( (cid:107) c (cid:107) pp ) ≤ | s | α √ δ ≤ F ( (cid:107) c (cid:107) pp ) ≤ F (cid:0) D ( δ, q , q ) (cid:107) a (cid:107) pp (cid:1) , and thus: (cid:16) √ δf ( D (cid:0) δ, q , q ) (cid:107) a (cid:107) pp (cid:1)(cid:17) α ≤ | s | ≤ (cid:16) √ δF (cid:0) D ( δ, q , q ) (cid:107) a (cid:107) pp (cid:1)(cid:17) α . Therefore, considering the decreasing function l δ,q ,q : R + → R + defined by l δ,q ,q ( v ) = (cid:16) √ δf ( D ( δ, q , q ) v p ) (cid:17) α and the increasing function L δ,q ,q : R + → R + defined by L δ,q ,q ( v ) = (cid:16) √ δF ( D ( δ, q , q ) v p ) (cid:17) α inequality (5) is obtained. A.2. Proof of inequality (7) . Because of symmetry, it suffices to prove inequality (7) for α ≥
1, i.e. α y cos y + α y − x cos( y − x ) + α − x cos x ≥ , ∀ x, y ∈ (cid:104) , π (cid:105) , α ≥ . h ( x ) = α y − x cos( y − x ) + α − x cos x , its derivative is h (cid:48) ( x ) = − α y − x ln( α ) cos( y − x ) + α y − x sin( y − x ) − α − x ln( α ) cos x − α − x sin x The equation h (cid:48) ( x ) = 0 is equivalent totan x = α y sin y − α y ln( α ) cos y − ln( α )1 + α y ln( α ) sin y + α y cos y which has a solution x ∗ ( y ) on the interval (cid:2) , π (cid:1) if and only if the numerator of theright-hand term of the above equations positive, i.e. α y (sin y − ln( α ) cos y ) ≥ ln( α ) . (15)If inequality (15) does not hold, it means in fact that h (cid:48) (0) <
0, which implies h (cid:48) ( x ) <
0, for any x ∈ (cid:0) , π (cid:1) . Therefore the function h is decreasing and its minimalvalue is h (cid:0) π (cid:1) = α y − π sin y .Otherwise, if inequality (15) holds, i.e. h (cid:48) (0) ≥
0, it turns out that x ∗ ( y ) is amaximum point of h ( x ) and the function h is increasing on the interval (0 , x ∗ ( y )) anddecreasing on the interval (cid:0) x ∗ ( y ) , π (cid:1) . Therefore, the minimal value of the function h is either h (0) = α y cos y + 1 or h (cid:0) π (cid:1) = α y − π sin y . However, it is easy to see that α y − π sin y ≤
1, for any y ∈ (cid:0) , π (cid:1) , and hence, the minimal value of the function h is h (cid:0) π (cid:1) = α y − π sin y .Therefore, we obtain that h ( x ) ≥ α y − π sin y, ∀ x, y ∈ (cid:104) , π (cid:105) , α ≥ , which leads to: α y cos y + α y − x cos( y − x ) + α − x cos x ≥ α y cos y + α y − π sin y. (16)Considering the function g ( y ) = α y cos y + α y − π sin y and its derivative g (cid:48) ( y ) = α y ln( α ) cos y − α y sin y + α y − π ln( α ) sin y + α y − π cos y, we obtain that g (cid:48) ( y ) = 0 if and only if y = y ∗ = arctan (cid:18) ln( α ) + α − π − ln( α ) α − π (cid:19) . It can be easily seen that y ∗ is a local maximum point for the function g on theinterval (cid:0) , π (cid:1) , and hence, the minimal values of g are reached in g (0) = g (cid:0) π (cid:1) = 1.Therefore, g ( y ) ≥
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