Optimal control of stationary linear systems with polynomial performance index

A stable stationary linear system is considered in the paper. A quality criterion is represented by a functional in the integral form of the sum of positive polynomial with respect to the state variables and squared control. It is proved that the optimal control is an irrational function or polynomial depending on the state variables. Minimum value of performance index can be in two forms: hyperelliptic integral or polynomial depending on the initial state.