Threshold optimization in observability inequality for the wave equation with homogeneous Robin-type boundary condition

For the wave equation with variable coefficients, problems with one-side boundary controls of three basic types and a boundary condition of the third kind at the uncontrolled end are considered. For dual problems with one-side boundary observations in the classes of strong generalized solutions, new constructive observability inequalities are obtained that are superior to the earlier known ones in two respects. First, inequalities with an optimal value of the controllability-observability threshold are derived, and second, the value of the final evaluation constant is bounded away from zero on time intervals whose length is close to the critical length. This opens up a possibility of constructing stable approximate solutions to the indicated classes of dual control and observation problems on time intervals not only of an arbitrary supercritical but also of precisely critical length.