Characteristic determinant in the problem of optimal control synthesis for oil and gas transport through long pipelines

In this paper we consider the characteristic determinant which arises in the problem of optimal control synthesis for oil and gas transport through long pipelines. Expansion of the determinant with traditional methods accounting for three or more tones in expansion of the solution is a very difficult task, but in this paper characteristic determinant is expanded in general way for any number of tones, and we obtained an expression that contains only even powers of the variable.