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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.

Thanks.

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# Writing PDEs as differential equations on Hilbert space

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