Construction of the Lyapunov functions for discrete random-parameter systems (Q1287273)

The author considers a discrete random-parameter system \[ x_{n+1}= f(x_n,\xi_{n+ 1}),\quad n= 0,1,\dots, \] where \(x_k(\omega)\) and \(\xi_k(\omega)\) are random vectors, \(\xi_k\) are independent and identically distributed. The asymptotic stability of this system can be investigated with the help of a Lyapunov function \(V(x)\) satisfying the conditions \(V(x)> 0\), \(LV(x)< 0\) in some neighborhood of zero, \(LV(x)= E\{V(x_{n+ 1})/ x_n= x\}- V(x)\). The aim of the paper is to construct a Lyapunov function of the form \(V(x)= V_p(x)= \sum^{N_p}_{\ell= 1} \alpha_1\phi_1(x)\), where \(\alpha_1\) are the coefficients and \(\phi_1(x)\) are standard monomials \(x_ix_j\), \(x_ix_jx_k,\dots\)\ . The Lyapunov function is constructed in some annular domain \(\Gamma= \{x\in R^r:\varepsilon\leq\| x\|< R\}\). A theorem about the system's asymptotic stability in the presence of Lyapunov functions mentioned above is proved. The problem of searching for the coefficients \(\alpha_1\) is reduced to a problem of linear programming. A numerical example is presented for a second-order system.