I am working on a problem from a textbook and have run into difficulties on this specific question. Any assistance will be appreciated,
Consider the partial differential equation,
$\frac{\partial z}{\partial t}(x,t) = \frac{\partial^2z}{\partial x^{2}}(x,t) - \alpha\frac{\partial z}{\partial x}(x,t)$
$\frac{\partial z}{\partial x}(0,t) = \frac{\partial z}{\partial x}(1,t) = 0$
$z(x,0) = z_{0}(x)$
a) Formulate this as an abstract system on the state space $L_{2}(0,1)$ with both the usual inner product and the weighted inner product
$\langle z_{1} , z_{2} \rangle_{a} = \int_{0}^{1} z_{1}(x)\overline{z_{2}(x)}\exp(-\alpha x)dx $
For this part, I have this solution (I have confirmed this is correct)
$Ah = \frac{d^{2}h}{dx^{2}} - \alpha\frac{dh}{dx}$
$D(A) = \{ h \in L_{2}(0,1) | h , \frac{dh}{dx} \text{ are absolutely continuous,}$
$\frac{d^{2}h}{dx^{2}} \in L_{2}(0,1) \text{ and } \frac{dh}{dx}(0,t) = \frac{dh}{dx}(1,t) = 0 \}$
b) Show that $A$ generates a contraction semigroup $T(t)$ on $L_{2}(0,1)$ with the weighted inner product.
I am stuck as to how to approach the second part of this question. I will appreciate any guidance or an appropriate source for guidance,
Thank you for your time
This problem appears in:
An Introduction to Infinite-Dimensional Linear Systems Theory Ruth F. Curtain, Hans Zwart