System state estimation for gas flows using observers based on density measurements (Q6613567)

The authors consider the 1D barotropic Euler equations: \(\partial _{t}\rho +\partial _{x}m=0\), \(\partial _{t}m+\partial _{x}(\frac{m^{2}}{\rho }+p(\rho ))=-\gamma \frac{\left\vert m\right\vert m}{\rho }\), for \(0<x<l\), \(t>0\), that describe the flow through gas pipes, where \(\rho \) is the density, \( m=\rho v\) the mass flow rate, \(v\) being the velocity, \(\gamma \) a friction parameter and \(p(\rho )\) a smooth and strictly monotone pressure law, so that for given positive constants \(\underline{\rho },\overline{\rho }\) there exist positive constants \(\underline{C}_{p^{\prime }},\overline{C} _{p^{\prime }}\) such that \(0<\underline{C}_{p^{\prime }}\leq p^{\prime }(\rho )\leq \overline{C}_{p^{\prime }}\), \(\forall 0<\underline{\rho }\leq \rho \leq \overline{\rho }\). Introducing the specific enthalpy \(h=\frac{1}{2} v^{2}+P^{\prime }(\rho )\), where \(P\) is a smooth and strictly convex pressure potential that is connected to the pressure law by \(p^{\prime }(\rho )=\rho P^{\prime }(\rho )\), they rewrite the above system as: \( \partial _{t}\rho +\partial _{x}v=0\), \(\partial _{t}v+\partial _{x}h=-\gamma \left\vert v\right\vert v\). Finally introducing the associated energy functional \(\mathcal{H}(\rho ,v)=\int_{0}^{l}(\frac{1}{2}\rho v^{2}+P(\rho ))dx\), the preceding system can be written in the Hamiltonian form as: \( \left( \begin{array}{c} \partial _{t}\rho \\\N\partial _{t}v \end{array} \right) +\left( \begin{array}{cc} 0 & \partial _{x} \\\N\partial _{x} & 0 \end{array} \right) \left( \begin{array}{c} \delta _{\rho }\mathcal{H} \\\N\delta _{v}\mathcal{H} \end{array} \right) =\left( \begin{array}{c} 0 \\\N-\gamma \left\vert v\right\vert v \end{array} \right) \), and they observe that the operator \(A=\left( \begin{array}{cc} 0 & \partial _{x} \\\N\partial _{x} & 0 \end{array} \right) \) is anti-self adjoint with respect to the \(L^{2}\)-scalar product up to boundary terms. The authors also introduce the observer system: \(\partial _{t}\widehat{\rho }+\partial _{x}\widehat{m}=\mu (\rho -\widehat{\rho })\), \( \partial _{t}\widehat{v}+\partial _{x}\widehat{h}=-\gamma \left\vert \widehat{v}\right\vert \widehat{v}\), with nudging parameter \(\mu >0\). The main result of the paper proves that if \(u=(\rho ,v)\) is a solution to the second above system and \(\widehat{u}=(\widehat{\rho },\widehat{v})\) is a solution to the observer system with some \(\mu >0\), satisfying the boundary conditions \(m(t,0)=\widehat{m}(t,0)=m_{b}(t)\), \(h(t,l)=\widehat{h} (t,l)=h_{b}(t)\), \(0\leq t\leq T\), and under hypotheses concerning the existence of classical solutions to the above systems, there exist positive constants \(C_{1},C_{2}\) such that \(\left\Vert u(t,\cdot )-\widehat{u} (t,\cdot )\right\Vert _{L^{2}(0,l)}\leq C_{1}\left\Vert u_{0}-\widehat{u} _{0}\right\Vert _{L^{2}(0,l)}e^{-C_{2}t}\), for all \(0\leq t\leq T\). For the proof, the authors first observe that the relative energy is equivalent to the \(L^{2}\)-norm for subsonic solutions because of the strict convexity of the pressure potential and they prove a bound on the time derivative of the relative energy. In the last part of their paper, the authors extend this main result to the case of star-shaped networks.\N\NFor the entire collection see [Zbl 1539.35003].