Necessary and sufficient linear matrix inequality (LMI) conditions are provided to compute parameter-dependent state feedback control gains that ensure closed-loop quadratic stability for linear systems affected by arbitrarily fast time-varying parameters inside a polytope. The proposed conditions, based on an extension of Polya's theorem and on the systematic construction of homogeneous polynomial solutions for parameter-dependent LMIs, are written as a sequence of progressively less and less conservative LMI conditions. Necessity is attained as the level of relaxation increases, providing a parameter-dependent state feedback gain that quadratically stabilises the system whenever such a gain exists. Moreover, parameter-dependent gains of arbitrary degree assuring quadratic stability with H(infinity) and H(2) guaranteed costs are also provided. The convergence to the minimum values of the attainable H(infinity) and H(2) guaranteed costs under closed-loop quadratic stability occurs as the degree of the polynomially parameter-dependent gain increases. Numerical results illustrate the efficiency of the proposed conditions.1617261735