Stabilization of feedback control and stabilizability optimal solution for nonlinear quadratic problems

Abstract

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrix, function of the state variable. Dynamic constraints are represented by bilinear differential systems of the form x ˙ = A ( x ) x + B ( x ) u , x ( 0 ) = x 0 . One selects an adequate factorization of A( x) such that the analyzed system should be controllable. Employing the Hamilton–Jacobi equation it results the matrix algebraic equation of Riccati associated to the optimum problem. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, as well as the adjoint system corresponding to those functions. Thus one obtains a matrix differential equation where the solution representing the positive defined symmetric matrix P( x), verifies the Riccati algebraic equation. The stability analysis for the autonomous systems solution resulting for the determined feedback control is performed using the Liapunov function method. Finally we present certain significant cases.