Optimal control of solutions to the initial-final problem for the model of linear waves in a plasma

The optimal control problem for a Sobolev type equation of higher order with a relatively polynomially bounded operator pencil is investigated in the paper. The results are applied to the study of the optimal control of solutions to the initial-final problem for the model of linear waves in plasma. The first results on the investigation of equation that describes the linear ion-acoustic waves in an unmagnetized plasma and on the study of some properties of these waves were obtained by Yu.D. Pletner. The initial-final conditions posed for the fourth-order Sobolev type equation are the generalization of the conditions in the Cauchy problem that is unsolvable at the arbitrary initial values. The research is based on the phase space method developed by G.A. Sviridiuk and the theory of relatively polynomially bounded operator pencil developed by A.A. Zamyshlyaeva. The article considers an equation that describes ion-acoustic waves in a plasma in an external magnetic field.