Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

摘要

The problem of determining the pair w:={F(x, t);f(t)} of source terms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) and in the Neumann boundary condition k(0)u_x(0, t) = f(t) from the measured data μ(x):=u(x, T) and/or ν(x):=u_t(x, t) at the final time t = T is formulated. It is proved that both components of the Fréchet gradient of the cost functionals J₁(w) = ‖u(x, t;w) – μ(x)‖₀2 and J₂(w) = ‖u_t(x, T;w) – ν(x)‖₀2 can be found via the solutions of corresponding adjoint hyperbolic problems. Lipschitz continuity of the gradient is derived. Unicity of the solution and ill-conditionedness of the inverse problem are analysed. The obtained results permit one to construct a monotone iteration process, as well as to prove the existence of a quasi-solution.