Carleman estimates for the structurally damped plate equations with Robin boundary conditions and applications

Abstract

In this paper, we consider Carleman estimates for the damped fourth-order plate operators ${\mathrm{\partial }}_{tt}-\rho {\mathrm{\partial }}_t\mathrm{\Delta }+{\mathrm{\Delta }}^2$ in a bounded smooth domain with Robin boundary conditions. Because the appearance of such kind of structural damping, the speed of propagation for solutions to the plate equation is infinite and the corresponding properties of the solution similar to heat equations, and is significantly different from that of the usual plate equations without damping. As applications, we consider two types of inverse problems in determining source terms for the plate equation with structural damping. The time-dependent measurements can be restricted to an arbitrary small sub-domain or arbitrary sub-boundary, respectively. Under some assumptions on the regularity of the solutions and coefficients, we prove the global stability results for these inverse problems.

Keywords:

• Carleman estimate
• plate equation
• structural damping
• Lipschitz stability

2000 Mathematics Subject Classifications:

• 35R30
• 35G15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China (NSF of China) under grant 11971333,11931011.