Application of interior point methods in minimization problems of eigenvalues

In this thesis, the robust Hurwitz stability problem of the matrices family is considered. Conditions for the existence of parameter-dependent Lyapunov functions corresponding to the family of affine matrices are given. An algorithm that minimizes the maximal eigenvalue of affine symmetric matrices is constructed. The existence of parameter-dependent Lyapunov functions for a given affine matrices family is examined with this algorithm. The robust stability problems of matrices family can also be generally expressed by linear matrix inequalities. For the solution of such inequalities, a barrier method is examined. An additional barrier function is defined and it is shown that this function has self-condordant property. The modified method by using the new barrier function is applied the existence of the common Lyapunov function of the matrices and the LMI problems. Obtained results are explained numerically with numerous examples.