This work presents a novel decentralized control strategy with a guaranteed cost for bilinear multi-agent systems subjected to products between the state and the control input, state constraints, and limitations on the amplitude and total energy of the control action, which can prevent the consensus from reaching the desired value. We propose state feedback and switching control laws to deal with these restrictions. The main objectives of the work are twofold: i) to design control laws that ensure stability and guaranteed cost bounds under constraints, and ii) to determine an estimation of the domain of attraction (DOA) as large as possible characterized by a polyhedral and an ellipsoidal invariant region contained in the space defined by the state constraints.We adopt a convex optimization procedure based on linear matrix inequalities (LMI) to address these objectives. An original approach employing Lyapunov sets is proposed to deal with control energy and state constraints, and positive system properties are used to estimate the DOA in only one orthant of state space. Through numerical examples, we demonstrate the effectiveness of the proposed Lyapunov-based approach, showing its ability to handle complex constraints and large networks.