This paper proposes an identification-based nonlinear control scheme which casts the plant model as a recurrent high-order neural network. The model thus obtained consists on polynomials of a fixed number of nonlinearities, a fact that is exploited by transforming it into an exact tensor-product representation whose nested convex sums may increase with the network order while preserving the number of different interpolating functions. Convexity is then used along the direct Lyapunov method to find conditions for controller design in the form of linear matrix inequalities or sum-of-squares; thanks to the fixed number of nonlinearities, they can be made progressively more relaxed while preventing the computational burden usually associated with Pólya-like relaxations. The control law thus obtained is a generalization of the well-known parallel distributed compensation; its effectiveness is illustrated in academic examples and an internal combustion engine setup