The Optimal Control over Solutions of the Initial-finish Value Problem for the Boussinesque–Löve Equation

Of concern is the optimal control problem for the Sobolev type equation of second order with relatively polynomially bounded operator pencil. The theorem of existence and uniqueness of strong solutions of initial-finish problem for abstract equation is proved. The sufficient and, in the case when infinity is a removable singularity of the $A$-resolvent operator pencil, the necessary conditions for optimal control existence and uniqueness of such solutions are found. The initial-finish problem for the Boussinesque–Löve equation, which describes the longitudinal vibrations of an elastic rod, is investigated. We use the ideas and methods developed by G. A. Sviridyuk and his disciples. The proof of the existence and uniqueness of optimal control theorem is based on the theory of optimal control developed by J.-L. Lions.