Riccati equations in stability theory of difference equations with memory

This paper defines several Riccati equations that allow checking the stability of difference equations with delay effect as x_i+1 = ^mΣ_j=0 A_j x_i-j (x_i ϵ Rⁿ). These various matrix Riccati equations have the same dimension n than the vector x, whatever the order m may be: this represents an advantage for high orders m when compared to classical matrix Lyapunov equations which should be of order mn. For instance, as a corollary, independent-on-delay (m) conditions are derived in the special case x_i+1 = A x_i + Bx_i-m. All the proposed conditions are sufficient, but tend to necessary-and-sufficient ones if there is no delay effect (Aj = 0 for j ≥ 0).