An observer for pipeline flow with hydrogen blending in gas networks: exponential synchronization (Q6585235)

The authors consider a pipeline network built as a finite graph \(G=(V,E)\), where \(V\) denotes the set of vertices and \(E\) the set of edges. The pipe \(e\) has a length \(L^{e}>0\), a diameter \(D^{e}>0\) and a friction coefficient \( \lambda _{fric}^{e}>0\). A mixture between natural gas and hydrogen is flowing through the network, its density through the pipe \(e\) is \(\widehat{ \rho }^{e}\), its pressure \(\widehat{p}^{e}\) and its the mass flow rate \( \widehat{q}^{e}\). The authors study a model that is based upon the \(2\times 2 \) Euler equations: \(\widehat{\rho }_{t}^{e}+\widehat{q}_{x}^{e}=0\), \( \widehat{q}_{t}^{e}+(\widehat{p}^{e}+\frac{(\widehat{q}^{e})^{2}}{\widehat{ \rho }^{e}})_{x}=-\frac{1}{2}\theta ^{e}\frac{\widehat{q}^{e}\left\vert \widehat{q}^{e}\right\vert }{\widehat{\rho }^{e}}\), with \(\theta ^{e}=\lambda _{fric}^{e}/D^{e}\). They consider two examples that lead to hyperbolic equations: an isentropic gas law \(p(\widehat{\rho })=a\widehat{ \rho }^{\gamma }\) with \(a>0\) and \(\gamma >1\), and the model of the American Gas Association \(p(\widehat{\rho })=\frac{R_{s}T\widehat{\rho }}{1- \widetilde{\alpha }\widehat{\rho }}\), where \(\mathcal{T}\) denotes the temperature, \(R_{s}\) is the gas constant, and \(\widetilde{\alpha }\leq 0\). For the hydrogen flow, the authors introduce the density \(\rho _{(h)}^{e}\) and the mass flow rate \(q_{(h)}^{e}\). They assume that \(q_{(h)}^{e}=W( \widehat{q}^{e},\widehat{\rho }^{e})\rho _{(h)}^{e}\), with \(W=\frac{\widehat{ q}^{e}}{\widehat{\rho }^{e}+\gamma _{w}}\), \(\gamma _{w}\in \lbrack 0,\infty ) \). The conservation law is \((\widehat{\rho }_{(h)}^{e})_{t}+(\widehat{q} _{(h)}^{e})_{x}=0\). The authors write the problem as: \( V_{t}^{e}+A(V^{e})V_{x}^{e}=G^{e}(V^{e})\), where \(V^{e}=\left( \begin{array}{c} \widehat{q}^{e} \\\N\widehat{\rho }^{e} \\\Nq_{(h)}^{e} \end{array} \right) \).\ They then introduce Riemann invariants \(R^{e}=\left( \begin{array}{c} R_{+}^{e} \\\NR_{-}^{e} \\\NR_{0}^{e} \end{array} \right) \), they rewrite the problem as: \( R_{t}^{e}+D(R^{e})R_{x}^{e}=S^{e}(R^{e})\) and they compute the eigenvalues of the matrix \(D\). They write the node conditions.\ They build the observed system, the observer system, and the system that is satisfied by the observer error. The first main result proves an existence result for a semi-global classical solution to the observed system on a given finite time interval \([0,T]\) in a \(C^{1}\)-neighborhood of a steady reference state. It requires \(C^{1}\)-compatibility of the initial state and the controls. A similar result can be stated for the observer system and these two existence results imply the existence of a unique classical solution to the error system. The second and third results prove that the \(L^{2}\)-norm of differences between the first two components of the state of the observer and the state of the original system decays exponentially fast up to the perturbation level. Here, the authors use appropriate Lyapunov functions with exponential weights and Gronwall's Lemma.