### Problem 2: Lyapunov Stability**Consider the nonlinear system**\[\dot{x} = Ax + g(x) \tag{2}\]where $ g(x) $ is a nonlinear function of the state $ x $. Prove that there exists an $ \epsilon > 0 $ such that if\[\|g(x)\|_2 < \epsilon \|x\|_2\]and the system\[\dot{x} = Ax \tag{3}\]is asymptotically stable, then system (2) is asymptotically stable.

given a non linear system x·=Ax+gx ...i where gx is a non linear function of the state x. to prove…

Problem 2 Lyapunov stability Consider the non linear system i = Ax + g(r) (2) where g(x) is a non linear function of the state x. Prove that there exists an e > 0 such that if ||g(x)||2 < e ||1||2 and the system i = Ax (3) is asymptotically stable then system (2) is asymptotically stable.

Problem 2 Lyapunov stability Consider the non linear system i = Ax + g(r) (2) where g(x) is a non linear function of the state x. Prove that there exists an e > 0 such that if ||g(x)||2 < e ||1||2 and the system i = Ax (3) is asymptotically stable then system (2) is asymptotically stable.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Problem 2: Lyapunov Stability

**Consider the nonlinear system**

\[
\dot{x} = Ax + g(x) \tag{2}
\]

where $ g(x) $ is a nonlinear function of the state $ x $. Prove that there exists an $ \epsilon > 0 $ such that if

\[
\|g(x)\|_2 < \epsilon \|x\|_2
\]

and the system

\[
\dot{x} = Ax \tag{3}
\]

is asymptotically stable, then system (2) is asymptotically stable.

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