Inverse problem for a parabolic system with two components by measurements of one component

By Assia Benabdallah, Michel Cristofol, Patricia Gaitan, Masahiro Yamamoto. 2008

78 citations recorded by the source corpus.

We consider a 2 × 2 system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time θ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-degeneracy condition at θ for the two components and for it, we can approximately control the two components of the 2 × 2 system by inputs to only one component. Such approximate controllability is proved also by our new Carleman estimate. Finally, we establish a Carleman estimate for a 3 × 3 system for parabolic equations with coupling of zeroth-order terms by one component to show the corresponding approximate controllability with a control to one component.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6fc2722bdffbb0b90286e51f470c19caa34866d2

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In Computational Theory and Mathematics

In Control and Systems Engineering

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Citations in the corpus

Carleman estimates for parabolic equations and applications (2009, background): “Then in [12], the following Carleman estimate is proved.”
Controllability to trajectories for some parabolic systems of three and two equations by one control force (2013, methodology): “Furthermore, the Carleman estimate obtained in [8] for a 3× 3 reaction-diffusion system differs from one obtained in this paper by the norm used: W 2,1 2 (ωT ) is the norm used in the first one and a L(2) norm in the second one (see Remark 2.”