Structural properties of gradient recurrent high-order neural networks

The structural properties of Recurrent High-Order Neural Networks (RHONN) whose weights are restricted to satisfy the symmetry property, are investigated. First, it is shown that these networks are gradient and stable dynamical systems and moreover, they remain stable when either bounded deterministic or multiplicative stochastic disturbances concatenate their dynamics. Then, we prove that such networks are capable of approximating arbitrarily close, a large class of dynamical systems of the form χ˙=F(χ). Appropriate learning laws, that make these neural networks able to approximate (identify) unknown dynamical systems are also proposed. The learning laws are based on Lyapunov stability theory, and they ensure error stability and robustness