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.