Question

Consider the nonhomogenous system a' (t) = A(t)x(t) + B(t), where A(t) and B(t) are continuous on 0 < to < t < 0, prove that if all solutions are bounded, then they are stable.

Answer 1

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It date of chong dy dr =0, Than, - B() A (D) X(t) B (+) AO is called X Lt) a critical point de Given that oltolth2 2 for no culto) -4(o). Hence Given that solutions bounded. We know that from Baundry value theoren, solutions ore bounded then they are equibonded. Moreover stability and boundedriens if all are equivalent. Cat xi= bild) are solution x = delt) प Difforential equation then, of giron - Critical point is then there exist -Bit) Alt) 110) -x/l <8.5 a 870 such that, lin 0 (t) = Critical point tys Ø (+) B(+) AlD um to Csolutions are bounded)