Stability criterion for linear systems of ordinary differential equations
aa r X i v : . [ m a t h . C A ] F e b MSC 34D20
A new stability criterion for systems of two first-orderlinear ordinary differential equationsG. A. Grigorian
1. Introduction . Let a ( t ) , b ( t ) , c ( t ) and d ( t ) be complex-valued locally integrableand locally bounded functions on [ t , + ∞ ) . Consider the linear system φ ′ = a ( t ) φ + b ( t ) ψ,ψ ′ = c ( t ) φ + d ( t ) ψ, t ≥ t . (1 . By a solution of this system we mean an ordered pair ( φ ( t ) , ψ ( t )) of absolutely continuousfunctions φ ( t ) and ψ ( t ) , satisfying (1.1) almost everywhere on [ t , + ∞ ) . Definition 1.1 . The system (1.1) is called Lyapunov (asymptotically) stable if its allsolutions are bounded on [ t , + ∞ ) (vanish at + ∞ ). Study of the question of stability of the system (1.1), in general, of linear systems ofordinary differential equations, is an important problem of qualitative theory of differentialequations. Being of interest not only in theory but also for applications it is the subjectof numerous investigations (see e. g., [1 - 12]). There exist many methods of estimationof solutions of linear systems of ordinary differential equations allowing to describe (todetect) wide classes of stable and (or) unstable systems of ordinary differentia; equations.Among them the main ones include the Lyapunov’s, Bogdanov’s, Lozinski’s estimate1ethods and the freezing method (see [4], pp. 40 -98, 132 -145). The fundamental methodof Lyapunov characteristic exponents allows to describe the asymptotic growth of solutionsof linear systems of ordinary differential equations via these exponents and thereforecarrying out the stability behavior of solutions of the system. However the application ofthis method has some difficulties, arising in the calculation process of Lyapunov characte-ristic exponents. There exist also other estimation methods for special classes of linearsystems of ordinary differential equations (see e. g., [5 - 10]), allowing to describe wideclasses of stable and (or) unstable linear systems of ordinary differential equations. Hoveverthese, indicated above and other methods cannot completely describe the stable andunstable linear systems of ordinary differential equations (in terms of their coefficients).In this paper we use the Riccati equation method to establish a new stability criterionfor the system (1.1). By two examples we compare the obtained result with the resultsobtained by the Lyapunov and Bogdanov methods, by a method involving estimates ofsolutions in the Lozinskii logarithmic norms and by the freezing method.
2. Auxiliary propositions . Let f ( t ) , g ( t ) , h ( t ) , f ( t ) , g ( t ) , h ( t ) be real-valuedlocally integrable functions on [ t , + ∞ ) . Consider the Riccati equations y ′ + f ( t ) y + g ( t ) y + h ( t ) = 0 , t ≥ t . (2 . y ′ + f ( t ) y + g ( t ) y + h ( t ) = 0 , t ≥ t , (2 . j = 1 , , and the differential inequalities η ′ + f ( t ) η + g ( t ) η + h ( t ) ≥ , t ≥ t . (2 . η ′ + f ( t ) η + g ( t ) η + h ( t ) ≥ , t ≥ t , (2 . By a solution of Eq. (2.1) (Ineq. (2.3)) on an interval [ t , t ) ( t ≤ t < t ≤ + ∞ ) we mean an absolutely continuous function y ( t ) ( η ( t )) on [ t , t ) , satisfying (2.1) ((2.3))almost everywhere on [ t , t ) . Note that for f ( t ) ≥ f ( t ) ≥ , t ≥ t every solutionof the linear equation η ′ + g ( t ) η + h ( t ) = 0 ( η ′ + g ( t ) η + h ( t ) = 0) on [ t , + ∞ ) is alsoa solution of the inequality (2,3) ((2.4)). Therefore if f ( t ) ≥ f ( t ) ≥ , t ≥ t , thenthe inequality (2.3) ((2.4)) has a solution on [ t , + ∞ ) , satisfying any initial condition. Theorem 2.1 . Let y ( t ) be a solution of Eq. (2 . on [ t , + ∞ ) , and let η ( t ) , η ( t ) be solutions of Ineq. (2.3) and Ineq. (2.4) respectively with η k ( t ) ≥ y ( t ) , k = 0 , . Let f ( t ) ≥ ,λ − y ( t ) + t Z t exp (cid:26) τ Z t [ f ( ξ )( η ( ξ ) + η ( ξ )) + g ( ξ )] (cid:27) × [( f ( t ) − f ( t )) y ( t ) + ( g ( t ) − g ( t )) y ( t ) + h ( t ) − h ( t )] dτ ≥ , t ≥ t , for some λ ∈ [ y ( t ) , η ( t )] . Then for each y (0) ≥ y ( t ) Eq. (2.1) has the solution y ( t ) on [ t , + ∞ ) , satisfying the initial condition y ( t ) = y (0) , moreover y ( t ) ≥ y ( t ) , t ≥ t . Proof. By analogy of the proof of Theorem 3.1 from [13].
Definition 2.1.
A solution of Eq. (2.1) is called t -regular if it exists on [ t , + ∞ ) . Definition 2.2. A t -regular solution y ( t ) of Eq. (2.1) is called t -normal if thereexists a δ -neighborhood U δ ( y ( t )) ≡ ( y ( t ) − δ, y ( y ) + δ ) ( δ > of y ( t ) such thatevery solution y ( t ) of Eq. (2.1) with y ( t ) ∈ U δ ( y ( t )) is also t -regular, otherwise y ( t ) iscalled t -extremal. Let y ( t ) be a t -regular solution of Eq. (2.1). We can interpret y ( t ) as a solution of thelinear equation y ′ + G ( t ) y + h ( t ) = 0 , t ≥ t , where G ( t ) ≡ f ( t ) y ( t ) + g ( t ) , t ≥ t . Then by Cauchy formula we hve y ( t ) ≡ exp (cid:26) − t Z i [ f ( τ ) y ( τ ) + g ( τ )] dτ (cid:27)(cid:20) y ( t ) − t Z t exp (cid:26) τ Z t g ( s ) ds (cid:27) h ( τ ) φ ( τ ) dτ (cid:21) , t ≥ t , where φ ( t ) ≡ exp (cid:26) t R t f ( τ ) y ( τ ) dτ (cid:27) , t ≥ t . From here it follows y ( t ) φ ( t ) = y ( t ) exp (cid:26) − t Z t g ( τ ) dτ (cid:27) − t Z t exp (cid:26) − t Z τ g ( s ) ds (cid:27) h ( τ ) φ ( τ ) dτ, t ≥ t . (2 . Lemma 2.1.
Let y ( t ) be a t -regular solution of Eq. (2.1) and let f ( t ) ≥ , t ≥ t . Then t Z t f ( τ ) y ( τ ) dτ ≤ y ( t ) t Z t f ( τ ) exp (cid:26) − τ Z t g ( s ) ds (cid:27) dτ − t Z t f ( τ ) dτ τ Z t exp (cid:26) − τ Z ξ g ( s ) ds (cid:27) h ( ξ ) dξ,t ≥ t . Proof. By analogy of the proof of Lemma 2.2 from [11].
Lemma 2.2.
Let the following conditions be satisfied. + ∞ Z t g ( τ ) dτ = + ∞ , t Z t exp (cid:26) − t Z τ g ( s ) ds (cid:27) | h ( τ ) | dτ is bounden on [ t , + ∞ ) . hen for every continuous function φ ( t ) , vanishing at + ∞ , the relation lim t → + ∞ t Z t exp (cid:26) − t Z τ g ( s ) d ( s ) (cid:27) | h ( τ ) | φ ( τ ) dτ = 0 is valid. Proof. By analogy of the proof of Lemma 2.5 from [11].Along with the system (1.1) consider the following one φ ′ = Re a ( t ) φ + | b ( t ) | ψ,φ ′ = | c ( t ) | φ + Re d ( t ) ψ, t ≥ t . (2 . Lemma 2.3.
If the system (2.6) is Lyapunov (asymptotically) stable, then the system(1.1) is also Lyapunov (asymptotically) stable.
Proof. Let ( φ ( t ) , ψ ( t )) be a solution of the system (1.1). We can interpret φ ( t ) as asolution of the linear equation φ ′ = a ( t ) φ + L ( t ) , t ≥ t , where L ( t ) ≡ b ( t ) ψ ( t ) , t ≥ t . Then by Cauchy formula we have φ ( t ) = exp (cid:26) t Z t a ( τ ) dτ (cid:27)(cid:20) φ ( t ) + t Z t exp (cid:26) − τ Z t a ( s ) ds (cid:27) L ( τ ) dτ (cid:21) , t ≥ t or φ ( t ) = exp (cid:26) t Z t a ( τ ) dτ (cid:27)(cid:20) φ ( t ) + t Z t exp (cid:26) − τ Z t a ( s ) ds (cid:27) b ( τ ) ψ ( τ ) dτ (cid:21) , t ≥ t . (2 . By analogy for ψ ( t ) we can derive the equality ψ ( t ) = exp (cid:26) t Z t d ( τ ) dτ (cid:27)(cid:20) ψ ( t ) + t Z t exp (cid:26) − τ Z t d ( s ) ds (cid:27) c ( τ ) φ ( τ ) dτ (cid:21) , t ≥ t . (2 . Substitute in place of ψ ( t ) the right hand part of the last equality into (2.7). After somesimplifications we obtain φ ( t ) = F ( t ) + t Z t K ( t, ξ ) φ ( ξ ) dξ, t ≥ t , (2 . F ( t ) ≡ φ ( t ) exp (cid:26) t Z t a ( τ ) dτ (cid:27) + ψ ( t ) t Z t exp (cid:26) τ Z t d ( s ) ds + t Z τ a ( s ) ds (cid:27) b ( τ ) dτ, t ≥ t ,K ( t, ξ ) ≡ c ( ξ ) exp (cid:26) − ξ Z t d ( s ) ds (cid:27) t Z ξ exp (cid:26) t Z τ a ( s ) ds (cid:27) b ( τ ) dτ, t ≥ ξ ≥ t . By analogy substituting in place of φ ( t ) the right hand part of the equality (2.7) into (2.8)we arrive at the equality ψ ( t ) = G ( t ) + t Z t L ( t, ξ ) ψ ( ξ ) dξ, t ≥ t , (2 . where G ( t ) ≡ ψ ( t ) exp (cid:26) t Z t d ( τ ) dτ (cid:27) + φ ( t ) t Z t exp (cid:26) τ Z t a ( s ) ds + t Z τ d ( s ) ds (cid:27) c ( τ ) dτ, t ≥ t ,L ( t, ξ ) ≡ b ( ξ ) exp (cid:26) − ξ Z t a ( s ) ds (cid:27) t Z ξ exp (cid:26) t Z τ d ( s ) ds (cid:27) c ( τ ) dτ, t ≥ ξ ≥ t . Let ( φ ( t ) , ψ ( t )) be a solution of the system (2.6). By (2.9) and (2.10) we have respectively φ ( t ) = F ( t ) + t Z t K ( t, ξ ) φ ( ξ ) dξ, t ≥ t , (2 . ψ ( t ) = G ( t ) + t Z t L ( t, ξ ) ψ ( ξ ) dξ, t ≥ t , (2 . where F ( t ) ≡ φ ( t ) exp (cid:26) t Z t Re a ( τ ) dτ (cid:27) + ψ ( t ) t Z t exp (cid:26) τ Z t Re d ( s ) ds + t Z τ Re a ( s ) ds (cid:27) | b ( τ ) | dτ, ≥ t , K ( t, ξ ) ≡ | c ( ξ ) | exp (cid:26) − ξ Z t Re d ( s ) ds (cid:27) t Z ξ exp (cid:26) t Z τ Re a ( s ) ds (cid:27) | b ( τ ) | dτ, t ≥ ξ ≥ t ,G ( t ) ≡ ψ ( t ) exp (cid:26) t Z t Re d ( τ ) dτ (cid:27) + φ ( t ) t Z t exp (cid:26) τ Z t Re a ( s ) ds + t Z τ Re d ( s ) ds (cid:27) | c ( τ ) | dτ,t ≥ t , L ( t, ξ ) ≡ | b ( ξ ) | exp (cid:26) − ξ Z t Re a ( s ) ds (cid:27) t Z ξ exp (cid:26) t Z τ Re d ( s ) ds (cid:27) | c ( τ ) | dτ, t ≥ ξ ≥ t . By (2.9) and (2.10) we can represent φ ( t ) and ψ ( t ) respectively via the following seriesexpansion φ ( t ) = F ( t ) + t Z t K ( t, ξ ) F ( ξ ) dξ + t Z t K ( t, ξ ) dξ ξ Z t K ( ξ, ζ ) F ( ζ ) dζ + ..., t ≥ t , (2 . ψ ( t ) = G ( t ) + t Z t L ( t, ξ ) G ( ξ ) dξ + t Z t L ( t, ξ ) dξ ξ Z t L ( ξ, ζ ) G ( ζ ) dζ + .... t ≥ t , (2 . Assume φ ( t ) = | φ ( t ) | , ψ ( t ) = | ψ ( t ) | . Then by (2.11) and (2.12) from (2.13) and(2.14) we obtain respectively: | φ ( t ) | ≤ F ( t ) + t Z t K ( t, ξ ) F ( ξ ) dξ + t Z t K ( t, ξ ) dξ ξ Z t K ( ξ, ζ ) F ( ζ ) dζ + ... = φ ( t ) , (2 . t ≥ t , | ψ ( t ) | ≤ G ( t ) + t Z t L ( t, ξ ) G ( ξ ) dξ + t Z t L ( t, ξ ) dξ ξ Z t L ( ξ, ζ ) G ( ζ ) dζ + ... = ψ ( t ) , (2 . t ≥ t . Assume the system (2.6) is Lyapunov (asymptotically) stable Then the estimates(2.15) and (2.16) imply that the system (1.1) is also Lyapunov (asymptotically) stable.The lemma is proved.
3. Main result.
Set E ( t ) ≡ Re a ( t ) − Re d ( t ) , t ≥ t . heorem 3.1. Let the following conditions be satisfied:1) sup t ≥ t t R t exp (cid:26) t R τ Re d ( s ) ds (cid:27) | c ( τ ) | dτ < + ∞ ;2) sup t ≥ t t R t (cid:20) Re a ( τ ) + | b ( τ ) | τ R t exp (cid:26) − τ R ξ E ( s ) ds (cid:27) | c ( ξ ) | dξ (cid:21) dτ < + ∞ ; (cid:18) ′ ) + ∞ R t E ( τ ) dτ = + ∞ , lim t → + ∞ t R t (cid:20) Re a ( τ ) ++ | b ( τ ) | τ R t exp (cid:26) − τ R ξ E ( s ) ds (cid:27) | c ( ξ ) | dξ (cid:21) dτ = −∞ . (cid:19) Then the system (1.1) is Lyapunov (asymptotically) stable.
Proof. Consider the Riccati equations y ′ + | b ( t ) | y + E ( t ) y − | c ( t ) | = 0 , t ≥ t , (3 . y ′ + | b ( t ) | y + E ( t ) y = 0 , t ≥ t . Applying Theorem 2.1 to these equations we conclude that for every t ≥ t and γ ≥ Eq. (3.1) has a solution y ( t ) on [ t , + ∞ ) with y ( t ) = γ and y ( t ) ≥ , t ≥ t . (3 . All solutions y ( t ) of Eq. (3.1), existing on [ t , + ∞ ) , are connected with solutions ( φ ( t ) , ψ ( t )) of the system (2.6) by relations (see [11]): φ ( t ) = φ ( t ) exp (cid:26) t Z t [ | b ( τ ) y ( τ ) + Re a ( τ )] dτ (cid:27) , φ ( t ) = 0 , ψ ( t ) = y ( t ) φ ( t ) , t ≥ t . (3 . Let y ( t ) be the solution of Eq. (3.1) with y ( t ) = 0 . By already proven y ( t ) exists on [ t , + ∞ ) and is non negative. Set φ ( t ) ≡ exp (cid:26) t R t | b ( τ ) | y ( τ ) dτ (cid:27) , t ≥ t . By (2.5) we have y ( t ) φ ( t ) = t Z t exp (cid:26) − t Z τ E ( s ) ds (cid:27) | c ( τ ) | φ ( τ ) dτ, t ≥ t . (3 . Let ( φ ( t ) , ψ ( t )) be the solution of the system (2.6) with φ ( t ) = 1 , ψ ( t ) = 0 . Then by(3.3) we have φ ( t ) = exp (cid:26) t Z t [ | b ( τ ) | y ( τ ) + Re a ( τ )] dτ (cid:27) , ψ ( t ) = y ( t ) φ ( t ) , t ≥ t . (3 . ψ ( t ) = t Z t exp (cid:26) − t Z τ Re d ( s ) ds (cid:27) | c ( τ ) | φ ( τ ) dτ, t ≥ t . (3 . In virtue of Lemma 2.1 from the first equality of (3.5) it follows < φ ( t ) ≤ exp (cid:26) t Z t h Re a ( τ ) + | b ( τ ) | τ Z t exp (cid:26) − τ Z ξ E ( s ) ds (cid:27) | c ( ξ ) | dξ i dτ (cid:27) , t ≥ t . (3 . Show that ( φ ( t ) , ψ ( t )) is bounded (vanish at + ∞ ). The condition 2) (the condition2’)together with (3.7) implies that φ ( t ) is bounded (vanish at + ∞ ). From here and from(3.6) (by Lemma 2.2 from here and from (3.6)) it follows that ψ ( t ) is also bounded (vanishat + ∞ ). Therefore ( φ ( t ) , ψ ( t )) is bounded (vanish at + ∞ ). Since c ( t ) there exists t ≥ t such that y ( t ) > , t ≥ t . Hence by the second equality of (3.5) ψ ( t ) > , t ≥ t . (3 . Let ( φ ( t ) , ψ ( t )) be a solution of the system (2.6) such that φ ( t ) > , ψ ( t ) > and det (cid:18) φ ( t ) ψ ( t ) φ ( t ) ψ ( t ) (cid:19) = 0 . Then ( φ ( t ) , ψ ( t )) and ( φ ( t ) , ψ ( t )) are linearly independent.Taking into account (3.5) and (3.8) we have ψ ( t ) φ ( t ) > , ψ ( t ) φ ( t ) > . (3 . Let y ( t ) be the solution of Eq. (3.1) with y ( t ) = ψ ( t ) φ ( t ) . Then by the second inequalityof (3.9) and by the already proven y ( t ) is t -normal. By the first inequality of (3.9) andby the already proven y ( t ) is also t -normal. Therefore (see [14]) M ≡ sup t ≥ t (cid:12)(cid:12)(cid:12)(cid:12) t Z t | b ( τ ) | ( y ( τ ) − y ( τ )) dτ (cid:12)(cid:12)(cid:12)(cid:12) < + ∞ . (3 . By (3.3) we have φ ( t ) = φ ( t ) exp (cid:26) t Z t [ | b ( τ ) | y ( τ )+ Re a ( τ )] dτ (cid:27) = φ ( t ) φ ( t ) exp (cid:26) t Z t [ | b ( τ ) | y ( τ )+ Re a ( τ )] dτ (cid:27) × exp (cid:26) t Z t | b ( τ ) | [ y ( τ ) − y ( τ )] dτ (cid:27) , t ≥ t . This together with (3.10) implies < φ ( t ) < φ ( t ) φ ( t ) exp { M } φ ( t ) , t ≥ t . Therefore φ ( t ) is bounded (vanish at + ∞ ). Hence since ( φ ( t ) , ψ ( t )) and ( φ ( t ) , ψ ( t )) arelinearly independent to complete the proof of the theorem it is enough to show that ψ ( t ) is bounded (vanish at + ∞ ). Let z ( t ) and z ( t ) be the solutions of the Riccati equation z ′ + | c ( t ) z − E ( t ) z − | b ( t ) | = 0 , t ≥ t with z ( t ) = φ ( t ) ψ ( t ) > , z ( t ) = φ ( t ) ψ ( t ) > . Then by already proven z ( t ) and z ( t ) are t -normal, and therefore (see [14]) M ≡ sup t ≥ t (cid:12)(cid:12)(cid:12)(cid:12) t Z t | c ( τ ) | ( z ( τ ) − z ( τ )) dτ (cid:12)(cid:12)(cid:12)(cid:12) < + ∞ . (3 . By (3.3) we have ψ ( t ) = ψ ( t ) exp (cid:26) t Z t [ | c ( τ ) | z ( τ )+ Re d ( τ )] dτ (cid:27) == ψ ( t ) ψ ( t ) ψ ( t ) exp (cid:26) t Z t [ | c ( τ ) | z ( τ ) + Re d ( τ )] dτ (cid:27) exp (cid:26) t Z t | c ( τ ) | ( z ( τ ) − z ( τ )) dτ (cid:27) , t ≥ t . This together with (3.11) implies that < ψ ( t ) ≤ ψ ( t ) ψ ( t ) exp { M } ψ ( t ) , t ≥ t . Hence ψ ( t ) is bounded (vanish at + ∞ ). The theorem is proved.Let p ( t ) , q ( t ) and r ( t ) be complex-valued continuous functions on [ t , + ∞ ) and let p ( t ) = 0 , t ≥ t . Consider the second order linear ordinary differential equation ( p ( t ) φ ′ ) ′ + q ( t ) φ ′ + r ( t ) φ = 0 , t ≥ t . (3 . ψ = p ( t ) φ ′ we reduce this equation to the system φ ′ = p ( t ) ψ,ψ ′ = − r ( t ) φ − q ( t ) p ( t ) ψ, t ≥ t . (3 . Definition 3.1.
Eq. (3.12) is called Lyapunov (asymptotically) stable if the system(3.13) is Lyapunov (asymptotically) stable.
From Theorem 3.1 we immediately get
Corollary 3.1.
Let the functions I ( t ) ≡ t Z t exp (cid:26) − t Z τ Re q ( s ) p ( s ) ds (cid:27) | r ( τ ) dτ,I ( t ) ≡ sup t ≥ t t Z t dτ | p ( τ ) | τ Z t exp (cid:26) − τ Z ξ Re q ( s ) p ( s ) (cid:27) | r ( ξ ) | dξ, t ≥ t be bounded. Then Eq. (3.12) is Lyapunov stable. (cid:4) Remark 3.1.
In the case p ( t ) > , r ( t ) ≤ and q ( t ) is real-valued the condition ofCorollary 3.1 is also necessary for Lyapunov stable of Eq. (3.12) (see [12]). In this sensethe conditions 1) and 2) of Theorem 3.1 are sharp. Remark 3.2.
It is not difficult to verify that in the case p ( t ) > , r ( t ) ≤ and q ( t ) is real-valued the condition 2’) of Theorem 3.1 for the system (3.13) is not satisfiable. Onthe other hand using Theorem 2.1 to the pair of equations y ′ + 1 p ( t ) y + q ( t ) p ( t ) y + r ( t ) = 0 , t ≥ t ,y ′ + 1 p ( t ) y + q ( t ) p ( t ) y = 0 , t ≥ t one can easily show that in this case Eq. (3.12) cannot be asymptotically stable (it has apositive and non decreasing solution). In This sence the condition 2’) of Theorem 3.1 issharp. Remark 3.3.
Theorem 3.1 is an improvement of results (Theorem 3.1 andTheorem 3.2) of the work [11]. xample 3.1. Consider the system φ ′ = ν ( t ) φ + µ ( t ) t ln tz ψ,ψ ′ = µ ( t ) φ + ( ν ( t ) − ψ, t ≥ e, (3 . where ν ( t ) and µ ( t ) are some real-valued continuous functions on [ e, + ∞ ) and µ ( t ) isbounded on [ e, + ∞ ) . Assume t R e ν ( τ ) dτ is upper bounded on [ e, + ∞ ) ( + ∞ R e ν ( τ ) dτ = −∞ and t R e ( ε − ν ( τ )) dτ is upper bounded on [ e, + ∞ ) for some ε ∈ (0 , ). Then it is notdifficult to verify that the conditions 1) and 2) (2’)) of Theorem 3.1 for the system (3.14)are satisfied. Therefore under the indicated restrictions the system (3.14) is Lyapunov(asymptotically) stable. Since at least one of the integrals + ∞ R e | ν ( τ ) | dτ, + ∞ R e | ν ( τ ) − | dτ diverges to + ∞ the application of the estimates of Lyapunov and Yu. S. Bogdanov ([4],p. 133) to the system (3.14) gives no result. Let A ( t ) be the matrix of the coefficients ofthe system (3.14). Then γ ± ( t ) ≡ ν ( t ) − ± q µ ( t ) t ln t , t ≥ e are its eigenvalues. Therefore if sup t ≥ e ν ( t ) ≥ , then the application of the freezing method([4], p. 139, Theorem 4.6.4) to the system (3.14) gives no result. Let us now discuss theapplicability of estimates of solutions via logarithmic norms γ I ( t ) , γ II ( t ) and γIII ( t ) ofS. M. Lozinski ([4], pp. 135, 136). From the Lozinski’s theorem ([4], p. 137) it follows thatif one of the integrals t R e γ i ( τ ) dτ, i = I, II, III is upper bounded then the correspondinglinear system is Lyapunov stable. For the system (3.14) we have γ I ( t ) ≥ ν ( t ) + | µ ( t ) | , t ≥ e. Therefore if sup t ≥ e t R e ( ν ( τ ) + | µ ( τ ) | ) dτ = + ∞ , then the application of γ I ( t ) to the system(3.14) gives no result. If | µ ( t ) | ≥ ee − , t ≥ e then the logarithmic norm γ II ( t ) of thesystem (3.14) satisfies to the inequality γ II ( t ) ≥ ν ( t ) , t ≥ e. ence if sup t ≥ e t R e (1 + ν ( τ )) dτ = + ∞ , then the application of γ II ( t ) to the system (3.14)gives no result. Finally the logarithmic norm γ III ( t ) for the system (3.14) is γ III ( t ) = 2 ν ( t ) − q µ ( t )(1 + t ln t )2 , t ≥ e. Therefore if + ∞ R e µ ( τ ) dτ = + ∞ and t R e ν ( τ ) dτ is bounded from below then + ∞ R e γ III ( τ ) dτ = + ∞ and, hence the application of γ III ( t ) to the system (3.14) givesalso no result. Thus if t R e ν ( τ ) dτ is bounded and | µ ( t ) | ≥ ee − , t ≥ e then none of thelogarithmic norms γ I ( t ) , γ II ( t ) and γ III ( t ) is applicable to the system (3.14). Example 3.2.
Consider the system φ ′ = ( λ − C sin t ) φ + µ ψ,ψ ′ = µ φ + λ ψ, t ≥ , (3 . where λ k , µ k , k = 1 , , C are some real constants, µ k > , k = 1 , , C > . It is notdifficult to verify that under the restrictions λ k < , k = 1 , , λ − λ > , λ + µ µ λ − λ ≤ <
0) (3 . the conditions 1) and 2) (2’)) of Theorem 3.1 for the system (3.15) are satisfied. Thereforeunder these restrictions the system (3.15) is Lyapunov (asymptotically) stable. Since + ∞ R | λ − C sin t | dt = + ∞ the application of estimates of Lyapunov and Yu. S. Bogdanovgives no result. Let A ( t ) be the matrix of coefficients of (3.15). Then γ ( t ) ≡ λ − λ − C sin t + p ( λ − λ + C sin t ) + 4 µ µ , t ≥ is its greatest eigenvalue. Hence, if C ≥ | λ + λ | then sup t ≥ γ ( t ) ≥ , and in this case thefreezing method is not applicable to the system (3.15). For (3.15) we have the followinglogarithmic norms: γ I ( t ) = max { λ + µ − C sin t, λ + µ } ; II ( t ) = max { λ + µ − C sin t, λ + µ } ; γ III ( t ) = λ + λ − C sin t + p ( λ − λ + C sin t ) + ( µ + µ ) , t ≥ . The set of parameters µ k , λ k for which at least one of norms γ I ( t ) , γ II ( t ) is applicableto (3.15) does not include the set, defined by (3.16). For example for λ = − , λ = − ,µ = 2 , µ = the application of γ ( t ) and γ II ( t ) to (3.15) gives no result, whereasfor these values of λ k , µ k , k = 1 , the conditions (3.16) are satisfied. It is not difficultto verify that for all enough large (with respect to λ k | , µ k , k = 1 , ) C the equality + ∞ R γ III ( t ) dt = + ∞ is satisfied. Therefore for all enough large C the application of γ III ( t ) to the system (3.15) gives no result. References
1. L. S. Pontriagin, Obyknovennye differential’nye uravneniya (Ordinary differentialequations) Moskaw, Nauka, 1974.2. L. Cezary, Asymptotic Behavior and Stability Problems in Ordinary DifferentialEquations, Berlin, 1959.3. N. W. Mac Lachlan, Theory and application of Mathieu Functions, Oxford, ClarendonPress, 1947.4. L. Y. Adrianoba, Introduction to the theory of linear systems of differential equations.S. Peterburg, Publishers of St. Petersburg University, 1992.5. V. A. Yakubovich, V. M. Starzhinsky, Linear differential equations with periodiccoefficients and their applications. Moscow, ”Nauka”, 1972.6. R. Bellman, Stability theory of differential equations. Moscow, Foreign LiteraturePublishers, 1954.7. V. I. Burdina, On boundedness of solutions of systems of differential equations, Dokl.AN SSSR, 93:4 (1953), 603–606.8. I. M. Sobol. Study of the asymptotic behaviour of the solutions of the linearsecond order differential equations wit the aid of polar coordinates. "Matematicheskijsbornik vol. 28 (70), N ◦
3, 1951, pp. 707 - 714.9. M. V. Fedoriuk. Asymptotic methods for linear ordinary differential equations.Moskow, ”Nauka”, 1983.10. Ph. Hartman, Ordinary differential Equations. Second Edition, SIAM, 2002.11. G. A. Grigorian, On the Stability of Systems of Two First - Order Linear OrdinaryDifferential Equations, Differ. Uravn., 2015, vol. 51, no. 3, pp. 283 - 292.132. G. A. Grigoryan, Stability Criterion for Systems of Two First-Order Linear OrdinaryDifferential Equations. Math. Notes, 103:6 (2018), 892–900.13. G. A. Grigorian. On Two Comparison Tests for Second-Order Linear OrdinaryDifferential Equations (Russian) Differ. Uravn. 47 (2011), no. 9, 1225 - 1240; translationin Differ. Equ. 47 (2011), no. 9 1237 - 1252, 34C10.14. G. A. Grigorian, On some properties of solutions of the Riccati equation. Izvestiya NASof Armenia, vol. 42, N ◦◦