CHAPTER 1 Lyapunov Stabilit y Theor y o f Differentia l Equations The stabilit y theor y o f differential equation s i s centered aroun d th e prob - lem whethe r a give n solutio n x(t) £ R n o f th e equatio n x = F(x) (1.0.1 ) is stable unde r a smal l perturbatio n o f eithe r th e initia l conditio n xo = x(0) or th e functio n F (i n a n appropriat e topolog y i n th e spac e o f continuou s functions i t i s assume d tha t th e solutio n x(t) i s well-define d fo r al l t 0) . The latte r i s importan t i n "practice " sinc e th e functio n F i s usuall y know n only u p t o a give n precision . Stability unde r smal l perturbations o f initial condition s mean s tha t ever y solution y(£) , whos e initia l conditio n y(0 ) lie s i n th e 5-bal l aroun d XQ, stay s within th e ^-neighborhoo d o f th e solutio n x(t) fo r al l sufficientl y smal l e and 5 = S(s) chose n appropriatel y (i n particular , i t i s assume d tha t y(t) i s well-defined fo r al l t 0) . On e say s tha t stabilit y i s exponential i f solution s y(t) approac h x(t) wit h a n exponentia l rate , an d i t i s uniforml y exponentia l if th e convergenc e i s unifor m ove r th e initia l conditions . Th e exponentia l stability usuall y survive s unde r smal l perturbation s o f th e right-han d sid e function F an d therefore , i s "observabl e i n practice" . In orde r t o stud y th e stabilit y o f a solutio n x(t) unde r smal l perturba - tions o f initia l condition s on e linearize s th e syste m (1.0.1 ) alon g #(£) , i.e. , considers the linea r syste m o f differential equation s (know n a s the variationa l system o f equations ) v = A{t)v, (1.0.2 ) where A(t) = F f (x(t)). Passin g bac k fro m th e linea r syste m (1.0.2 ) t o th e nonlinear syste m (1.0.1 ) ca n b e viewe d a s solvin g th e proble m o f stabilit y under smal l perturbations o f the right-han d sid e function i n (1.0.2 ) an d thus , reducing th e stud y o f on e typ e o f stabilit y t o th e othe r one . One o f th e mai n result s o f th e classica l stabilit y theor y assert s tha t i f the solutio n v(t) = 0 o f th e linea r syste m (1.0.2 ) i s uniforml y exponentiall y stable the n s o i s th e solutio n x(t) o f th e nonlinea r syste m (1.0.1) . If th e solution v(t) = 0 i s onl y (nonuniformly ) exponentiall y stabl e th e situatio n becomes muc h mor e subtl e an d require s a n additiona l an d somewha t sophis - ticated assumptio n (know n a s Lyapuno v regularity ) t o hold . Thi s situatio n is the subjec t o f stud y i n th e Lyapuno v stabilit y theory , whic h originate d i n 5 http://dx.doi.org/10.1090/ulect/023/02
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