Writing PDEs as differential equations on Hilbert space

[agahlawa]

Hi,

I was reading a paper on control of the 1-D heat equation with boundary control, the equation being
$\frac{\partial u(x,t)}{\partial x}$= $\frac{\partial^2 u(x,t)}{\partial x^2}$ with boundary conditions:

$u(0,t)=0$ and $u_x(1,t)=w(t)$, where $w(t)$ is the control input.

The authors then proceed to write the equation as:

$v(t)=Av(t)+\delta_1(x)w(t)$ where $A$ is the double differential operator, $v(t)=u(\cdot,t)$ is the state with $L_2(0,1)$ being the state space and $\delta_1$ being the dirac delta distribution centered at $x=1$.

I understand the idea behind putting $A$ but I do not understand how the two equations are same.

Any help is much appreciated.
Thanks.

 

[matematikawan]

I will be following this thread.

Hi,

$\frac{\partial u(x,t)}{\partial x}$= $\frac{\partial^2 u(x,t)}{\partial x^2}$
Any help is much appreciated.
Thanks.

For a diffusion equation, the first derivative is wrt to time.
$$\frac{\partial u(x,t)}{\partial t}= \frac{\partial^2 u(x,t)}{\partial x^2}$$

 

[agahlawa]

Hi,

I was reading a paper on control of the 1-D heat equation with boundary control, the equation being
$\frac{\partial u(x,t)}{\partial t}$= $\frac{\partial^2 u(x,t)}{\partial x^2}$ with boundary conditions:

$u(0,t)=0$ and $u_x(1,t)=w(t)$, where $w(t)$ is the control input.

The authors then proceed to write the equation as:

$v(t)=Av(t)+\delta_1(x)w(t)$ where $A$ is the double differential operator, $v(t)=u(\cdot,t)$ is the state with $L_2(0,1)$ being the state space and $\delta_1$ being the dirac delta distribution centered at $x=1$.

I understand the idea behind putting $A$ but I do not understand how the two equations are same.

Any help is much appreciated.
Thanks.

Sorry, was a typo. Fixed. Thank you.