An inverse problem for a wave equation with arbitrary initial values and a finite time of observations

OpenAlex · English

We consider a solution u ( p , g , a , b ) to an initial value–boundary value problem for a wave equation: and we discuss an inverse problem of determining a coefficient p ( x ) and a , b by observations of u ( p , g , a , b )( x , t ) in a neighbourhood ω of ∂Ω over a time interval (0, T ) and ∂ i t u ( p , g , a , b )( x , T 0 ), x ∊ Ω, i = 0, 1, with T 0 < T . We prove that if T − T 0 and T 0 are larger than the diameter of Ω, then we can choose a finite number of Dirichlet boundary inputs g 1 , …, g N , so that the mapping is uniformly Lipschitz continuous with suitable Sobolev norms provided that { p , a j , b j } 1 ⩽ j ⩽ N remains in some bounded set in a suitable Sobolev space. In our inverse problem, initial values are also unknown, and we do not assume any positivity of the initial values. Our key is a Carleman estimate and the exact controllability in a Sobolev space of higher order.