This paper deals with the problem of designing state or full-order output feedback H-infinity controllers satisfying an α-stability constraint. A descriptor system approach is adopted to derive linear matrix inequality (LMI) conditions of α-stability and H-infinity performance for linear time-invariant (LTI) systems, under which the multiobjective control laws can be constructed with a mix of H-infinity performance and α-stability. Being expressed in terms of a group of bilinear matrix inequalities, the design algorithm is numerically tractable via an LMI optimization method. A numerical example illustrates the feasibility and effectiveness of the proposed approach, and the results are also compared with those obtained by the well-known Lyapunov shaping paradigm (LSP). This paper deals with the problem of designing state or full-order output feedback H-infinity controllers satisfying an α-stability constraint. A descriptor system approach is derived to linear matrix inequality (LMI) conditions of α-stability and H-infinity performance for linear time-invariant (LTI) systems, under which the multiobjective control laws can be constructed with a mix of H-infinity performance and α-stability. Being expressed in terms of a group of bilinear matrix inequalities, the design algorithm is numerically tractable via an LMI optimization method. A numerical example illustrates the feasibility and effectiveness of the proposed approach, and the results are also compared with those obtained by the well-known Lyapunov shaping paradigm (LSP).