A uniqueness theorem for the determination of sources in the Germain–Lagrange plate equation

We prove a uniqueness result related to the Germain–Lagrange dynamic plate differential equation. We consider the equation { ∂ 2 u ∂ t 2 + △ 2 u = g ⊗ f , in  ] 0 , + ∞ ) × R 2 , u ( 0 ) = 0 , ∂ u ∂ t ( 0 ) = 0 , where u stands for the transverse displacement, f is a distribution compactly supported in space, and g ∈ L l o c 1 ( [ 0 , + ∞ ) ) is a function of time such that g ( 0 ) ≠ 0 and there is a T 0 > 0 such that g ∈ C 1 [ 0 , T 0 [ . We prove that the knowledge of u over an arbitrary open set of the plate for any interval of time ] 0 , T [ , 0 < T < T 0 , is enough to determine uniquely f ∈ E ′ ( R 2 ) .