Let us define $$ \mathbb{H}^{1} = H^{1}(-L,0) \times H^{1}(0,L) \ \ \text{and} \ \ \mathbb{L}^{2} = L^{2}(-L,0)\times L^{2}(0,L), $$ where $H^{1}(I) = \big\lbrace u \in L^{2}(I) \ \text{and} \ u_{x} \in L^{2}(I); I = (a,b) \big\rbrace$.

Besides these, $$ \mathbb{M} = \big\lbrace (u,v) \in \mathbb{H}^{1}; u(-L) = v(L) = 0 \ \text{and} \ u(0) = v(0) \big\rbrace . $$ Under the above conditions, we have that the phase space is given by $$ \mathcal{H} = \mathbb{M} \times \mathbb{L}^{2}. $$ Note that this space equipped with the inner product $$ \langle (u_{1},v_{1},w_{1},z_{1}), (u_{2},v_{2},w_{2},z_{2}) \rangle = \int_{-L}^{0}u_{1_{x}}\overline{u}_{2_{x}} + \int_{0}^{L}v_{1_{x}}\overline{v}_{2_{x}} + \int_{-L}^{0}w_{1}\overline{w}_{2} + \int_{0}^{L}z_{1}\overline{z}_{2} $$ is a Hilbert space.