The paper studies the problem of minimax control design for linear evolution equations in Hilbert spaces with measurement noise and additive exogenous disturbances. The key result of the paper is an algorithm, generating a control in an output-feedback form, which steers the state of the system as close as possible to a given sliding hyperplane, asymptotically as time goes to infinity. The control is designed in the state space of the minimax filter, and guarantees that the state of the filter will be exactly on the sliding surface, and the state of the plant will belong to an ellipsoid centered at the filter's state vector for large enough time. The optimality of the designed feedback and estimation error is proven. The feedback is represented by means of the unique solution of an algebraic Riccati equation. The theory is then applied to design a minimax control for linear hereditary systems subject to noise and disturbances. This is achieved by projecting the hereditary system onto a finite dimensional subspace of the corresponding state space by means of a finite-volume approximation method, designing feedback in the state space of the resulting finite dimensional system. The solution of the operator Riccati equation is obtained using a (modified) Kleinman-Newton method. The efficacy of the proposed algorithm is illustrated by a numerical example for a time-delay linear systems with constant point delays.