Abstarct: In this thesis the linear matrix inequality analysis of dynamical system investigated.lyapunov method that has been raised by a number of specific practical problems in engineering controls, particulary the stabikity control system have been applied to aparticular function. formulated identically in optimization problem, are intersted in finding conditions for optimal solutions are available. Hence it is natural to resort a branch of analysis that provide these conditions. In this thesis, we have tried solving linear matrix inequality and its application fields have expressed a concrete example,in addition to the charcterization for linear system, descxriptive and design of robust output feedback controller using this method the stability of the linear state feedback control via a linear investigated.