6 1 . LYAPUNO V STABILIT Y THEOR Y O F DIFFERENTIA L EQUATION S works o f Lyapuno v an d Perron . Som e o f it s mai n result s ar e presente d i n this chapter . 1.1. Lyapuno v Exponent s fo r Differentia l Equation s Consider a linea r differentia l equatio n i) = A(t)v, (1.1.1 ) where v{t) G Cn an d A(t) i s a n x n matri x wit h comple x entrie s dependin g continuously o n t G 1R. W e assum e tha t th e matri x functio n A(t) i s bounded , i.e., s u p ( P ( t ) | | :t GM } O C . (1.1.2 ) It follow s tha t fo r ever y VQ G Cn ther e exist s a uniqu e solutio n v(t) v(t: vo) of Equatio n (1.1.1 ) tha t i s define d fo r ever y t G IR and satisfie s th e initia l condition ^(0,^o ) = i?o - Consider th e trivia l solutio n v{t) = 0 fo r t 0 . If th e matri x functio n A(t) i s constant , i.e. , A(t) A fo r al l t 0 , the n th e trivia l solutio n i s exponentially stabl e i f an d onl y i f th e rea l par t o f ever y eigenvalu e o f th e matrix A i s negative . A simila r resul t hold s i n th e cas e whe n th e matri x function A(t) i s periodic . In orde r t o characteriz e th e stabilit y o f th e trivia l solutio n i n th e gen - eral cas e w e introduc e th e Lyapunov exponent x + : C n R U {—oo} o f Equation (1.1.1 ) b y th e formul a X+(^) = limsup^log||i (t)|| , (1.1.3 ) for eac h v G C n , wher e v(t) i s th e uniqu e solutio n o f (1.1.1 ) satisfyin g th e initial conditio n v(0) = v. I t follow s immediatel y fro m (1.1.3 ) tha t th e Lyapunov exponen t x + satisfies : 1. x + {av) X^iv) r e a c n v £ C n an d a ^ 0 2. x + ( ^ + w ) ^ m a x {x + (^) 5 X + (^) } r eac h v, w £ C n 3. X + (0 ) = - o o . The functio n x + attain s onl y finitely man y distinc t value s x~\ ' *' xf o n C n \ {0 } where s n (th e proo f o f thi s statemen t i s give n i n Sectio n 1. 2 be - low). Eac h numbe r \ t occur s wit h som e multiplicit y hi s o that YH=\ ^ ~ n - Note tha t fo r ever y e 0 ther e exist s C £ 0 suc h tha t fo r ever y solutio n v(t) o f (1.1.1 ) an d an y t 0 w e hav e IK*)|| C e e^ + £ )*|ji (0)|| . (1.1.4 ) It follow s fro m (1.1.4 ) tha t i f X f0 (1.1.5 ) then fo r an y sufficientl y smal l 0 , ever y solutio n v(t) 0 a s t +c o with a n exponentia l rate . I n othe r word s th e trivia l solutio n v{t) 0 i s exponentially stable .
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