Consider the folloing ODE system: x ′ ( t ) = A ( t )...

Grimanijd

Grimanijd

Answered

2022-07-03

Consider the folloing ODE system:
x ( t ) = A ( t ) x ( t ) + b ( t )
x ( t 0 ) = x 0
Show that the solutions of the system are globally defined.

Answer & Explanation

soosenhc

soosenhc

Expert

2022-07-04Added 16 answers

We assume that b , A are continuous over R and that (1) t R | t 0 t b | is bounded by K and (2) t R | t 0 t | | A | | | is bounded by L.
Then x ( t ) = t 0 t A x + t 0 t b + x 0 and | | x ( t ) | | t 0 t | | A | | | | x | | + K + | | x 0 | |; according to Gronwall, | | x ( t ) | | ( K + | | x 0 | | ) exp ( t 0 t | | A | | ) ( K + | | x 0 | | ) exp ( L ). Thus the solution x ( t ) is bounded over any segment and, consequently, the maximal solution is defined over whole R.
EDIT. We can do better (although I feel that my post does not interest the OP). The conditions (1),(2) are so that the solution exists and is bounded over R. If you want only the existence, then (1),(2) are useless.
Proof. Let I be a segment, K I = sup t I | t 0 t b | , L I = sup t I | t 0 t | | A | | |. Then sup t I | | x ( t ) | | ( K I + | | x 0 | | ) exp ( L I ) and we are done.

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