On the stabilizability of linear control systems depending on parameters

Motivated by questions in system reliability and in adaptive control, we consider the existence, for a linear system (A(λ), B(λ)) defined for λ ε U an open subset of RN, of a state feedback K (λ), defined for λεU and possessing the same functional properties as (A(λ), B(λ)), such that the closed loop system is asymptotically stable for alI λεU. It is shown, for example, that if (A(λ), B(λ)) is continuous or Lipschitz continuous on compact subsets of U and controllable for all λ, then the stablilzing steady-state solution of an associated Riccati equation is also continuous, or Lipschitz continuous on compact subsets, respectively. This is not the case if (A(λ), B(λ))is polynomial for λε RN> , but in this case we show that there exists some polynomial K(ε) which stabilizes the system for all ε, provided either (A(ε), B(ε)) is controllable for all λ or in a special case for which the system is scalar and is controllable generically.