Based on the analogy between structural mechanics and optimal control, interval mixed variable energy and the corresponding variational method are introduced to transform the induced norm γ of an ℋ︁∞ filter into the fundamental eigenvalue γ−2cr of a self-adjoint operator, which can itself be expressed as a generalized Rayleigh quotient of the variational principle with two kinds of variable, i.e. the state vector x and the co-state vector λ. These vectors correspond, respectively, to the displacement and internal force vectors in the theory of structural mechanics. The interval matrices Q, G and F of the mixed variable energy are introduced in a natural way, based upon which the precise integration of a time invariant system is proposed and then combined with the extended Wittrick–Williams algorithm to solve the eigen-value problem, enabling the optimal eigenvalue parameter γ−2cr to be computed with high precision. Copyright © 1999 John Wiley & Sons, Ltd.