Uniqueness of linear PDE

Question

Is there an energy functional or some other method which can prove the uniqueness of the system $$\rho_t+\varphi_x =0, \qquad\varphi_t+\rho_x=\alpha \rho +\beta \varphi,$$ with initial and boundary conditions $$\rho(0,x)=0, \qquad \varphi(0,x)=0, \\ \rho(t,0)=0, \qquad \varphi(t,L)=0,$$ where $\alpha$ and $\beta$ are constants and $\rho, \, \varphi:[0,T]\times [0,L]\rightarrow \mathbb R$? That is, show $\rho=\varphi=0$? I appreciate any suggestions.