Solutions of nonlinear matrix equations arising in nonlinear control theory. I

An attempt is made to establish the existence of solutions of nonlinear matrix equations of the form X(z)A( z)X(z)=X(z), where all matrices X(z), A(z) are holomorphic matrices. Such matrix equations arise in nonlinear control theory of nonlinear dynamical systems where A(z) is a given holomorphic matrix. Numerical methods of solutions of such nonlinear matrix equations are given. Use is made of parallel processing and supercomputers to carry out the numerical calculations of such problems. Examples of methods used for problems of the above type are given