Approximate solution to boundary control and observation problems for the equation describing transverse vibrations of a bar

Mutually dual difference approximations are constructed for the problems of boundary control and observation for the differential equation describing the transverse vibrations of an inhomogeneous bar. (The equation is second-order in time and fourth-order in space.) The controllability and observability condition requiring that the length of the underlying time interval be larger than a certain critical value is assumed to be fulfilled. It is shown how the a priori information available in this case can be used to construct approximate solutions to both difference problems that converge strongly for arbitrary objective and observed states in the classes under consideration.