Well-posedness hypothesis for the regularity-class wave equation

Let $\rho$ and $\boldsymbol T$ be the coefficient fields in the boundary-controlled wave equation, let $\boldsymbol z_0=\left(\begin{matrix}\operatorname{grad}(w_0)\\ \rho^{-1}w_1\end{matrix}\right)\in\mathcal Z_\kappa$, and let $u\in\mathcal C^2([0,\infty);\mathcal U_\kappa)$ satisfy

$$u(0)=\gamma_\perp\left(\boldsymbol T\operatorname{grad}(w_0)\right).$$

Here $\mathcal X_\kappa=\mathbf H^\kappa(\operatorname{div};\Omega)\times H^\kappa(\Omega)$ is the energy space, $\mathcal Z_\kappa$ is the solution space, $\mathcal U_\kappa=H^{\kappa-\frac12}(\partial\Omega)$ is the control space, and $\mathcal Y_\kappa=H^{\kappa+\frac12}(\partial\Omega)$ is the observation space.

Well-posedness hypothesis. There exists a unique solution $\boldsymbol z\in\mathcal C^1([0,\infty);\mathcal X_\kappa)\cap\mathcal C([0,\infty);\mathcal Z_\kappa)$ with $y\in\mathcal C([0,\infty);\mathcal Y_\kappa)$.

This regularity assertion supplies the continuous well-posedness needed for the subsequent interpolation and convergence analysis. The supplied text presents it as a formal hypothesis and does not establish its resolution.