In the thesis we consider a damped Klein-Gordon equation with a variable diffusion coefficient:[special characters omitted] where the nonlinear term is g(u) = |u|^γu with the constant γ satisfying [special characters omitted] The goal is to derive necessary conditions for the optimal set of parameters q* = (α, β, γ) ∈ P minimizing the objective function J(q ) = ||u(q) − z _d[special characters omitted]. First, we study the nonlinear term (g( u) for the different cases of γ, and derive its properties which are crucial to the entire research. Then we show that the solution maps q → u(q): P → L²(0,T; V) and q → u'(q): P → L²(0, T; H) are continuous. Furthermore, the solution map is shown to be weakly Gâteaux differentiable on the admissible set P, implying the Gâteaux differentiability of the objective function. Finally we study the Fréchet differentiability of J and optimal parameters for these problems. Unlike the sine-Gordon equation, which has a bounded nonlinear term, Klein-Gordon equation requires stronger assumptions on the initial data. The further difficulties in mathematical analysis of the equation arise from the unbounded nonlinear term g(u) = |u| ^γu and the variable diffusion coefficient β( x).