Inverse Problem for a Fourth-Order Hyperbolic Equation with a Complex-Valued Coefficient

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form of a Fourier series in terms of the eigenfunctions of a non-self-adjoint fourth-order ordinary differential operator. The proofs of the uniform convergence of the Fourier series are based on estimates of the norms of the derivatives of the eigenfunctions of a fourth-order ordinary differential operator and the uniform boundedness of the Riesz bases of the eigenfunctions.