Componentwise stabilization of interval systems

The componentwise stability of a linear system is a special type of asymptotic stability induced by the existence of exponentially decreasing rectangular sets that are invariant with respect to the free response. An interval system ( ) ( ) ( ) x t Ax t Bu t = + , [ , ] A A A − + ∈ , [ , ] B B B − + ∈ , is componentwise stabilizable if there exists a constant feedback ( ) ( ) u t Fx t = that ensures the componentwise stability of the whole family of linear systems defined by ( ) ( ) ( ) x t A BF x t = + . The paper formulates computable necessary and sufficient conditions for the componentwise stabilizability of interval systems. It is shown that the componentwise stabilizing feedback matrices define the solution set of two equivalent linear inequalities These results are further exploited to construct a linear programming problem for which (i) the absence of a feasible solution means the componentwise stabilization is not possible, (ii) a feasible solution provides a componentwise stabilizing feedback matrix. The applicability of the theoretical development is illustrated by a numerical example.