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 quadratic form of the control in the integrand of the cost functional is with a positive small multiplier (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 positive semi-definite (but non-zero) quadratic form. These features require developing a significantly novel approach to asymptotic solution of the matrix Riccati algebraic equation associated with the considered H_∞ problem by solvability conditions. 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 illustrative example is presented.