Source reconstruction algorithms for coupled parabolic systems from internal measurements of one scalar state

Abstract

This paper is devoted to the study of source reconstruction algorithms for coupled systems of heat equations, with either constant or spatially dependent coupling terms, where internal measurements are available from a reduced number of observed states. Two classes of systems are considered. The first comprises parabolic equations with constant zero-order coupling terms (through a matrix potential term or via the diffusion matrix). The second type considers parabolic equations coupled by a matrix potential that depends on spatial variables, which leads to the analysis of a non-self-adjoint operator. In all configurations, the source is assumed to be of separate variables, the temporal part is a known scalar function, and the spatial dependence is an unknown vector field. Several numerical examples using the finite element method in 1D and 2D are presented to show the reconstruction of space-dependent sources.

Figures (5)

• Figure 1: Diagram depicting the source reconstruction methodology for coupled parabolic systems in the form \ref{['sys.case1.matrizQ.ctes']}-\ref{['sys.Q.variable']}.
• Figure 2: Source reconstructions when the coupling matrix is given by $q_{11}(x)=q_{12}(x)=q_{22}(x)=0$ and $q_{21}(x)=-x^3 + 4x^2 - 3x + 1$. First column is the reconstruction of $F_1(x)$ and $F_2(x)$ observing both components, i.e., $(y_{1,\text{obs}},y_{2,\text{obs}})^t$. The second column shows the reconstruction of $F_1(x)$ and $F_2(x)$ observing the first component $y_{1,\text{obs}}$, and the last one is the reconstruction observing the second component $y_{2,\text{obs}}$ only. The highlighted gray areas correspond to the observation domain $\mathcal{O}$.
• Figure 3: Source reconstructions when the coupling matrix is given by $q_{11}(x)=q_{22}(x)=0$, $q_{12}(x)=4x-2$, and $q_{21}(x)=-4x+2$. . First column is the reconstruction of $F_1(x)$ and $F_2(x)$ observing both components, i.e., $(y_{1,\text{obs}},y_{2,\text{obs}})^t$. The second column shows the reconstruction of $F_1(x)$ and $F_2(x)$ observing the first component $y_{1,\text{obs}}$, and the last one is the reconstruction observing the second component $y_{2,\text{obs}}$ only. The highlighted gray areas correspond to the observation domain $\mathcal{O}$.
• Figure 4: Source reconstruction for $F_3(x,y)$ using a coupling matrix given by $q_{11} = 1$, $q_{12} = 4$, $q_{21} = 0$, and $q_{22} = 1$. The first column displays the sources and the locations of the observation domains, indicated by a white dashed line. The second column displays the reconstructions from observing both components. The third column shows the reconstruction obtained by observing only the last component.
• Figure 5: Source reconstruction for $F_3(x,y)$ using the coupling matrix $q_{11}=q_{22}=0$, $q_{12} = 4$ and $q_{21}= 2$. The first column displays the sources and the locations of the observation domains, indicated by a white dashed line. The second, third, and fourth columns display the reconstructions with both components, followed by the reconstructions with the first and second components, respectively.

Theorems & Definitions (18)

• lemma 1
• remark 1
• lemma 2
• remark 2
• lemma 3
• lemma 4
• remark 3
• remark 4
• Theorem 1
• proof