Timeline for Adjoint of differential equation
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 8, 2020 at 21:44 | comment | added | Michael Renardy | In general, consider a non-autonomous system $y'=A(t)y$. If $\Phi(t)$ denotes the solution operator (i.e. $y(t)=\Phi(t)y(0)$, then $\Phi'(t)=A(t)\Phi(t)$, so $(\Phi^*)'(t)=\Phi^*(t)A^*(t)$. In the autonomous case, the operators $\Phi^*$ and $A^*$ commute, but you cannot expect this in a non-autonomous system. | |
| Feb 8, 2020 at 13:33 | history | edited | YCor |
edited tags
|
|
| Apr 13, 2018 at 21:12 | comment | added | user539887 | Do you know Evolution Semigroups in Dynamical Systems and Differential Equations by Chicone and Latushkin? | |
| Apr 12, 2018 at 9:14 | history | edited | Umberto | CC BY-SA 3.0 |
added 297 characters in body
|
| Apr 11, 2018 at 19:29 | comment | added | Umberto | @user539887 in your terminology what I call the flow is $\Phi(t,0).$ | |
| Apr 11, 2018 at 19:22 | comment | added | user539887 | A terminological remark: since the linear equation $$y'(t)=Ay(t)+\kappa(t)By(t)$$ is nonautonomous, one cannot speak of the flow generated by it. The object generated by it should be termed process (or, better, semiprocess): $\Phi(s;s)=\mathrm{Id}_H$ for any $s \in \mathbb{R}$, $\Phi(u,s)=\Phi(u,t)\circ\Phi(t,s)$ for any $s \le t \le u$. | |
| Apr 11, 2018 at 18:54 | history | edited | Umberto | CC BY-SA 3.0 |
edited title
|
| Apr 11, 2018 at 16:03 | history | asked | Umberto | CC BY-SA 3.0 |