One Class of H∞ Cheap Control Problems: Asymptotic Solution

An infinite-horizon H∞ linear-quadratic control problem is considered. This problem has the following features: (i) the control cost in the cost functional has a positive small coefficient (small parameter), meaning that the control cost is much smaller than the state cost; (ii) the current cost of the fast state variable in the cost functional is a non-zero positive semi-definite quadratic form. These features require developing a significantly novel approach to asymptotic analysis of the matrix Riccati algebraic equation appearing in the solvability conditions of the considered H∞ problem. Using this solution, an asymptotic analysis of the H∞ problem is carried out. This analysis yields parameter-free solvability conditions for this problem and a simplified controller solving this problem. An example illustrating the theoretical results is presented.