By William T. Reid (Eds.)

ISBN-10: 0125862504

ISBN-13: 9780125862509

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**Example text**

3M) determined by the initial conditions --f then 47 7. 3,) determined by the initial conditions If then ~ , ~ (=t W(t)U,,(t) ) on (a, b,). 6) For r W ( t )= V(t)U-l(t), E for t E (a, b,). 4 implies that and since d*(s)[V8,,(s)- W(s)Q(s,r ) ] = 0 we have that - - As Q(s, I) is bounded for r E (c, b,), if F # ( r , s I W ) 0 as I -+ b, then Var(s)- W(s)Q(s,I ) 0 as r + b,. +00, and for any such sequence we have that {Vsrj(s)} P(s) = W(s)Q(s)as j - w. 4. 4) we have that VSz(t) = W(t)Us2(t). and consequently for Y ( t )= ( U ( t ) ; V ( t ) )defined as in the statement of the theorem we have V ( t )= W ( t ) U ( t ) for t E (a, b,).

26’) --A(s), F ( s : s, t ) + B(s), W(s:s, t ) + C(s), G ( s :S, t ) -+ --D(s). Now if a matrix function N ( t ) is nonsingular throughout a neighborhood of t = s, and has a finite derivative Nf(s)= lim { ( t - s ) - l [ N ( t )- N ( s ) ] } l+a the relation N ( t ) - N(s) = N ( t ) hence has a derivative “(s) equal to -fl(s)N’(s)fl(s). Consequently, if A(s:t ) = H-’(s: t ) and e(s: t ) = G-’(s: t). 26”) s, then N ( t ) = N - l ( t ) satisfies - N ( t ) ] A ( s ) ,and A ( $ )= A&: s), D ( s ) = C,(s: s).

BASIC PROPERTIES OF SOLUTIONS rb in terms of which we have the following result. 6) (a) J [ u , v ; e, (71 = 0 for e E 9 * o [ a , bl : u ; (b) there exists a constant vector y and a v o ( t )E b%l,[a,b] such that B ( t ) [ v ( t )- v O ( t ) ]= 0 on [a, b ] , and for t E [a, b ] . 6b) holds, then for e 9 * o [ a ,b ] : u we have Ja = e"(t)vo(t) I t-b t-a E = O? 6a) holds also. 6a) is satisfied, for w(t)=SZ(a, t I u, v ) we have that w ( ~ )BBm[a, E b] and if e E 9 * o [ a ,b ] : u then Now if Y ( t )is the solution of the differential system Y ( t ) = D"(t)Y(t), Y ( a )= E m , then ( e ( t ) ,u ( t ) ) E%,[u, b] x Cn2[a,b] with L2*[e,a](t)=O on [a, b ] , and e ( b ) = 0 if and only if 53 8.

### Riccati Differential Equations by William T. Reid (Eds.)

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