An inverse problem for a wave equation with arbitrary initial values and a finite time of observations. (English) Zbl 1229.35323
Summary: We consider a solution \(u(p,g,a,b)\) to an initial value-boundary value problem for a wave equation:
\[ \begin{alignedat}{3} \partial_t^2u(x,t)&=\Delta u(x,t)+ p(x)u(x,t), &&x\in\Omega, &&\;0<t<T,\\ u(x,0)&= a(x),\;\partial_tu(x,0)=b(x), \quad &&x\in\Omega,&&{}\\ u(x,t)&= g(x,t), &&x\in\partial\Omega, &&\;0<t<T, \end{alignedat} \]
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 \(\omega\) of \(\partial\Omega\) over a time interval \((0,T)\) and \(\partial_t^iu(p,g,a,b)(x,T_0)\), \(x\in\Omega\), \(i=0,1\), with \(T_0<T\). We prove that if \(T-T_0\) and \(T_0\) are larger than the diameter of \(\Omega\), then we can choose a finite number of Dirichlet boundary inputs \(g_1,\dots,g_N\), so that the mapping
\[ \big\{u(p,g_j,a_j,b_j)|_{\omega\times(0,T)}, \partial_t^iu(p,g_j,a_j,b_j)(\cdot,T_0) \big\}_{i=0,1,\,1\leq j\leq N}\longrightarrow \big\{p,a_j,b_j\big\}_{1\leq j\leq N} \]
is uniformly Lipschitz continuous with suitable Sobolev norms provided that \(\{p,a_j,b_j\}_{1\leq j\leq N}\) 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.
\[ \begin{alignedat}{3} \partial_t^2u(x,t)&=\Delta u(x,t)+ p(x)u(x,t), &&x\in\Omega, &&\;0<t<T,\\ u(x,0)&= a(x),\;\partial_tu(x,0)=b(x), \quad &&x\in\Omega,&&{}\\ u(x,t)&= g(x,t), &&x\in\partial\Omega, &&\;0<t<T, \end{alignedat} \]
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 \(\omega\) of \(\partial\Omega\) over a time interval \((0,T)\) and \(\partial_t^iu(p,g,a,b)(x,T_0)\), \(x\in\Omega\), \(i=0,1\), with \(T_0<T\). We prove that if \(T-T_0\) and \(T_0\) are larger than the diameter of \(\Omega\), then we can choose a finite number of Dirichlet boundary inputs \(g_1,\dots,g_N\), so that the mapping
\[ \big\{u(p,g_j,a_j,b_j)|_{\omega\times(0,T)}, \partial_t^iu(p,g_j,a_j,b_j)(\cdot,T_0) \big\}_{i=0,1,\,1\leq j\leq N}\longrightarrow \big\{p,a_j,b_j\big\}_{1\leq j\leq N} \]
is uniformly Lipschitz continuous with suitable Sobolev norms provided that \(\{p,a_j,b_j\}_{1\leq j\leq N}\) 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.
