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1

$begingroup$

$B=C^-1AC=left[
beginarraycc
P & 0\
0 & Q
endarray
right]$
consider the system y'=By+G(y).

Let $U(t)=left[
beginarraycc
e^Pt & 0\
0 & 0
endarray
right] $ and $V(t)=left[
beginarraycc
0 & 0\
0 & e^Qt
endarray
right] $ where $P$ is $ktimes k$ has eigenvalues with negative real part and $Q$ is $(n-k) times (n-k)$ has eigenvalues with positive real part. For $alpha>0$ sufficiently small that for $j=1,..,k$ we choose $Re(lambda_j)<-alpha<0$. We have $e^Bt=U(t)+V(t)$

Why in the proof it was written that for $K>0$ and $delta <0$ we have
$$
||U(t)||le Ke^-(alpha+delta)t text for all $tge 0$
$$
and
$$
||V(t)||le Ke^delta t text for all $tle 0$
$$

ordinary-differential-equations matrix-exponential

share|cite|improve this question

edited Mar 29 at 13:54

Speed22

asked Mar 29 at 9:33

Speed22Speed22

63

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You should give more context.
  $endgroup$
  – user539887
  Mar 29 at 12:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Still the context is missing: for instance, does your last sentence mean "there exist $K>0$ and $delta,colorred<,0$ such that $lVert V(t)rVertle Ke^delta t$ for all $t,colorredle,0$"? If so, this follows in a straightforward way from the standard estimates of the exponential matrix, independent of the signs of the real parts of its eigenvalues. Please take more care in formulating your question, otherwise it can be downvoted and eventually closed.
  $endgroup$
  – user539887
  Mar 29 at 19:06
  
  

  
  
  
  

add a comment | 

1

$begingroup$

$B=C^-1AC=left[
beginarraycc
P & 0\
0 & Q
endarray
right]$
consider the system y'=By+G(y).

Let $U(t)=left[
beginarraycc
e^Pt & 0\
0 & 0
endarray
right] $ and $V(t)=left[
beginarraycc
0 & 0\
0 & e^Qt
endarray
right] $ where $P$ is $ktimes k$ has eigenvalues with negative real part and $Q$ is $(n-k) times (n-k)$ has eigenvalues with positive real part. For $alpha>0$ sufficiently small that for $j=1,..,k$ we choose $Re(lambda_j)<-alpha<0$. We have $e^Bt=U(t)+V(t)$

Why in the proof it was written that for $K>0$ and $delta <0$ we have
$$
||U(t)||le Ke^-(alpha+delta)t text for all $tge 0$
$$
and
$$
||V(t)||le Ke^delta t text for all $tle 0$
$$

ordinary-differential-equations matrix-exponential

share|cite|improve this question

edited Mar 29 at 13:54

Speed22

asked Mar 29 at 9:33

Speed22Speed22

63

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You should give more context.
  $endgroup$
  – user539887
  Mar 29 at 12:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Still the context is missing: for instance, does your last sentence mean "there exist $K>0$ and $delta,colorred<,0$ such that $lVert V(t)rVertle Ke^delta t$ for all $t,colorredle,0$"? If so, this follows in a straightforward way from the standard estimates of the exponential matrix, independent of the signs of the real parts of its eigenvalues. Please take more care in formulating your question, otherwise it can be downvoted and eventually closed.
  $endgroup$
  – user539887
  Mar 29 at 19:06
  
  

  
  
  
  

add a comment | 

$begingroup$

$B=C^-1AC=left[
beginarraycc
P & 0\
0 & Q
endarray
right]$
consider the system y'=By+G(y).

Let $U(t)=left[
beginarraycc
e^Pt & 0\
0 & 0
endarray
right] $ and $V(t)=left[
beginarraycc
0 & 0\
0 & e^Qt
endarray
right] $ where $P$ is $ktimes k$ has eigenvalues with negative real part and $Q$ is $(n-k) times (n-k)$ has eigenvalues with positive real part. For $alpha>0$ sufficiently small that for $j=1,..,k$ we choose $Re(lambda_j)<-alpha<0$. We have $e^Bt=U(t)+V(t)$

Why in the proof it was written that for $K>0$ and $delta <0$ we have
$$
||U(t)||le Ke^-(alpha+delta)t text for all $tge 0$
$$
and
$$
||V(t)||le Ke^delta t text for all $tle 0$
$$

ordinary-differential-equations matrix-exponential

share|cite|improve this question

edited Mar 29 at 13:54

Speed22

asked Mar 29 at 9:33

Speed22Speed22

63

$endgroup$

share|cite|improve this question

edited Mar 29 at 13:54

Speed22

asked Mar 29 at 9:33

Speed22Speed22

63

share|cite|improve this question

edited Mar 29 at 13:54

Speed22

asked Mar 29 at 9:33

Speed22Speed22

63

asked Mar 29 at 9:33

Speed22Speed22

63

asked Mar 29 at 9:33

Speed22Speed22

63