Abstract: The robust finite-time control problem under the constraint of energy-to-peak performance index was investigated for linear systems with delays. Norm-bounded uncertainties was included in the coefficient matrices of systems and discrete and distributed delays were included in the states. A Lyapunov functional involving a parameter was introduced. Its exponential growth was checked for bounding the states and outputs. Then， sufficient conditions were derived guaranteeing that the closed-loop systems are finite-time bounded and satisfy prescribed energy-to-peak performance index for all admissible uncertainties. What’s more， the control law was obtained simultaneously. These conditions were formulated in terms of linear matrix inequalities with an eigenvalue constraint. Through transforming the constraint to nonlinear matrix inequalities， a cone complementarity linearization algorithm was designed to solve such matrix inequalities. An example was presented to show the effectiveness of the proposed method.

Key words: distributed delay, finite-time boundedness, energy-to-peak performance, cone complementarity linearization algorithm, linear matrix inequalities