Prove 0 is an exponentially stable equilibrium of the system x ′ = f ( x ) + g ( x ) if f ( 0 ) = g ( 0 ) = 0Besides the conditions in the title, we have:1. 0 is an exponential equilibrium of the system y ′ = f ( y )2. | g ( x ) | ≤ μ | x | , ∀ x ∈ R n 3. μ is sufficiently small!What I have tried so far is that:1. | | y ( 0 ) | | &lt; δ 1 ⇒ | | y ( t ) | | &lt; α 1 | | y ( 0 ) | | e − β 1 t , t ≥ 02. We want to prove: ∃ δ , α , β &gt; 0 , such that | | x ( 0 ) | | &lt; δ ⇒ | | x ( t ) | | &lt; α | | x ( 0 ) | | e − β t , t ≥ 0Considering the following system:1. x 1 ′ ( t ) = f ( x 1 )2. x 2 ′ ( t ) = g ( x 2 )Then we will have:1. ∃ α 1 , β 1 , δ &gt; 0 such that | | x 1 ( 0 ) | | &lt; δ 1 ⇒ | | x 1 ( t ) | | &lt; α 1 | | x 1 ( 0 ) | | e − β 1 t , t ≥ 02. x 2 ′ ( t ) ≤ μ | x 2 ( t ) |, and by Gronwall inequality x 2 ( t ) ≤ x 2 ( 0 ) e μ t I have a sense that it might be somewhat close to what we want to get. And I am stuck at this stage.

Question

Prove   0 is an exponentially stable equilibrium of the system       x    ′    =  f  (  x  )  +  g  (  x  ) if   f  (  0  )  =  g  (  0  )  =  0Besides the conditions in the title, we have:1.   0 is an exponential equilibrium of the system       y    ′    =  f  (  y  )2.   |  g  (  x  )  |  ≤  μ  |  x  |  ,  ∀  x  ∈        R    n  3.   μ is sufficiently small!What I have tried so far is that:1.   |  |  y  (  0  )  |  |  &amp;lt;      δ    1    ⇒  |  |  y  (  t  )  |  |  &amp;lt;      α    1    |  |  y  (  0  )  |  |      e          −              β        1            t        ,  t  ≥  02. We want to prove:   ∃  δ  ,  α  ,  β  &amp;gt;  0  , such that  |  |  x  (  0  )  |  |  &amp;lt;  δ  ⇒  |  |  x  (  t  )  |  |  &amp;lt;  α  |  |  x  (  0  )  |  |      e          −      β      t        ,  t  ≥  0Considering the following system:1.       x    1    ′    (  t  )  =  f  (      x    1    )2.       x    2    ′    (  t  )  =  g  (      x    2    )Then we will have:1.   ∃      α    1    ,      β    1    ,  δ  &amp;gt;  0 such that  |  |      x    1    (  0  )  |  |  &amp;lt;      δ    1    ⇒  |  |      x    1    (  t  )  |  |  &amp;lt;      α    1    |  |      x    1    (  0  )  |  |      e          −              β        1            t        ,  t  ≥  02.       x    2    ′    (  t  )  ≤  μ  |      x    2    (  t  )  |, and by Gronwall inequality       x    2    (  t  )  ≤      x    2    (  0  )      e          μ      t      I have a sense that it might be somewhat close to what we want to get. And I am stuck at this stage.

Elyse Huff · Accepted Answer

If   f  (  x  )  =  −      1    2    μ  x, then the equation       y    ′    =  f  (  y  ) has   0 as an exponentially stable fixed point. Now consider   g  (  x  )  =  μ  x. Then       x    ′    =  f  (  x  )  +  g  (  x  )  =      1    2    μ  x has   0 as an unstable fixed point. What am I missing?

Prove 0 is an exponentially stable equilibrium of the system x ′ </msup>

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New Questions in Algebra I