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This paper investigates the quadratic stabilizability of Takagi-Sugeno (T-S) fuzzy systems by means of parallel distributed state feedback compensators. Using Finsler's lemma, a new design condition assuring the existence of such a controller is formulated as a parameter-dependent linear matrix inequality (LMI) with extra matrix variables and parameters in the unit simplex. Algebraic properties of the system parameters and recent results of positive polynomials are used to construct LMI relaxations that, differently from most relaxations in the literature, provide certificates of convergence to solve the control design problem. Due to the degrees of freedom obtained with the extra variables, the conditions presented in this paper are an improvement over earlier results based only on Polya's theorem and can be viewed as an alternative to the use of techniques based on the relaxation of quadratic forms. An extension to cope with guaranteed H infin attenuation levels is also given, with proof of asymptotic convergence to the global optimal controller under quadratic stability. The efficiency of the proposed approach in terms of precision and computational effort is demonstrated by means of numerical comparisons with other methods from the literature.