On the search for a time-optimal boundary control using the method of moments for systems governed by the diffusion-wave equation

For a system described by a one-dimensional, inhomogeneous diffusion-wave equation on a segment, two types of optimal boundary control problems are considered: the problem of finding a control with a minimum norm for a given control time and the problem of finding a control that brings the system to a given state in a minimum time under a given constraint on the norm of the control. Various ways of specifying conditions on the final state are considered. The finite-dimensional $l$-problem of moments is analyzed, to which the optimal control problem can be reduced. We show that under the conditions of well-posedness and solvability of this problem, the problem of finding a control with a minimum norm always has a solution, while the problem of finding a control with a minimum transition time may not have a solution.