Uniqueness theorems for generalized solutions to four mixed problems for the wave equation with nonlocal boundary conditions

Uniqueness theorems for generalized solutions to four mixed problems for the wave equation with nonlocal boundary conditions have been proved. In obtaining integral identities which generalized solutions of such problems must satisfy and conditions on test functions in these identities, it was assumed that, when each sum in the boundary conditions is replaced by a given function, these problems transform into well-studied problems with local boundary conditions, for which the form of the integral identities and the conditions on the test functions in these identities are known. The obtained results are used essentially in optimizing boundary controls at one end of a string by both displacements and elastic boundary forces and transferring the string oscillation process from the state of complete rest into a given final state or from a given initial state into the state of complete rest. The methods used to prove the uniqueness of solutions to the mixed problems were also considered.