Control problems for vibrating systems with variable distributed parameters are especially important in view of almost-periodicity of that vibrations. Although control problems for nonhomogeneous strings were investigated very rarely (see eg [1]–[5] and the references cited therein). In [1] explicit form of Dirichlet boundary control functions are found, as well as necessary and sufficient conditions on initial and terminal data for boundary controllability of free vibrations of non-homogeneous string controlled on both ends are obtained. In [2, 3] boundary null–and approximate null-controllability problems in modified Sobolev spaces are considered for non–homogeneous string, when control processes are carried out by Neumann and Dirichlet boundary conditions respectively, at that in [2] the deflection of the string is fixed at the left end–point and in [3] the velocity of the string is fixed at the right end–point. In [4, 5] optimal control problems are considered for non-homogeneous string which is fixed at the left end–point and vibrating under impulsive concentrated perturbations, and non-homogeneous rod which is fixed at the left end–point and is free at the right end–point, respectively, while control process is carried out by mixed boundary conditions in [4] and by external control impacts, distributed on the rod arbitrarily, in [5]. The problem is reduced to a truncated system of moments problem and solved explicitly by means of generalized functions. Controllability conditions of the string and rod in the sense of the Lebesgue space L1 are obtained for the initial and terminal data. Here we concern with boundary optimal control problem for non-homogeneous wave equation with variable coefficients, describing forced vibrations of non–homogeneous string, that is fixed at the left end–point, under impulsive concentrated perturbations, at that control process is carried out by Dirichlet boundary controls containing constant delay at the right end–point of the string. A functional describing the summary" linear momentum" of controls in the considered time interval is taken as a control process optimality criterion.

Throughout this paper we shall call a real function admissible control if it satisfies necessary and sufficient conditions of controlled system needed solution existence. We shall say, that a controlled system is fully controllable in a given space of functions, if there exists an admissible control, which resolves posed optimal control problem for the system in terms of that space.