### 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 if gx2<εx2 and the system x·=Ax ...ii is asymptotically stable then the system i is also asymptotically stable. definition a system is called asymptotically stable if all the roots of system matrix has negative real parts.proof let X0=x0 solution of system i is of the form xt=eAtx0+∫0teAt-sgxsds as we know system ii is asymptotically stable at x=0 this implies, by using definition of asymtotic stability , we can say eigenvalue of A has negative real parts so, let the eigenvalue is of the form λk=αk+iβk where αk<0 choose some α=minαk now, let any positive number σ such that σ<α thus, eAt2≤Me-σt ...iii apply norm over the solution of system i we get xt=eAtx0+∫0teAt-sgxsds using equation iii xt≤Me-σtx0+∫0teAt-sgxsds using gx2<εx2 xt≤Me-σtx0+∫0tεMeAt-Asxds ≤Me-σtx0+ε∫0teσsxds multiply eσt both sides eσtXt≤Mx0+εM∫0teσsxdsusing gronwall's inequality eσtxt≤Mx0eMεt⇒xt≤Mx0etMε-σ choose some ε such that Mε-σ<ε ε<σM xt≤Mx0etMε-σ<ε thus for ε<σM system i is asymptotically stable.

Math

Advanced Math

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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