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Writing PDEs as differential equations on Hilbert space

  1. Oct 4, 2011 #1

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

    [itex] u(0,t)=0[/itex] and [itex] u_x(1,t)=w(t)[/itex], where [itex] w(t)[/itex] is the control input.

    The authors then proceed to write the equation as:

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

    I understand the idea behind putting [itex] A [/itex] but I do not understand how the two equations are same.

    Any help is much appreciated.
  2. jcsd
  3. Oct 5, 2011 #2
    I will be following this thread. :smile:

    For a diffusion equation, the first derivative is wrt to time.
    [tex] \frac{\partial u(x,t)}{\partial t}= \frac{\partial^2 u(x,t)}{\partial x^2} [/tex]
  4. Oct 6, 2011 #3
    Sorry, was a typo. Fixed. Thank you.
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