Identification problems of retarded differential systems in Hilbert spaces (Q501494)

Denote by \(\Omega\) a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial\Omega\), the operator \(A_0=-A(x,D_x)\) where \(A(x,D_x)\) is the elliptic operator in \(L^1(\Omega)\), defined as follows \[ A(x,D_x)=-\sum\limits_{i,j=1}^n\frac{\partial}{\partial x_j}(a_{i,j}(x)\frac{\partial}{\partial x_i})+\sum^n_{i=1}b_i(x)\frac{\partial}{\partial x_i}+c(x). \] The authors consider the inverse problem for the following retarded functional differential equation \[ u'(t)=A_0u(t)+\gamma A_0u(t-h)+\int^0_{-h}a(s)A_0u(t+s)ds \] with initial conditions \[ u(0)=g^0,\,u(s)=g^1(s),\,s\in[-h,0). \] The main task of the paper is to identify the unknown quantities \(A_0\), \(\gamma\) and \(a(\cdot)\). This identifiability can be regarded as a kind of inverse problem in many applications. Sufficient conditions for the identification problem are charaterized in terms of the initial values and eigenvectors of the adjoint operator. The authors use an approach which is a combination of the solution semigroup, the structural operators, the representations of spectral projections and the completeness of generalized eigenspaces.