Investigation of the stability properties of dynamical systems by new methods.

In this thesis, two important problems of the theory of linear systems, the problem of existence of common quadratic Lyapunov functions (CQLF) and the problem of existence of a stable member in uncertain systems that depends on the parameter vector are considered. In a CQLF problem the cutting plane method (Kelley's method) which is a convex optimization method, modified gradient method, weighted common quadratic function method are considered. These methods are examineted over the numerous examples and it is seen that these methods are faster than known methods. In a matrix polytope a problem of the existence of a stable member is very important in stabilization problems of linear systems. In this thesis, this problem is reduced to a new nonconvex optimization problem. Polyak's algorithm where uncertainty parameters vary in whole space is converted to an algorithm where the parameters vary in a box. A sufficient condition with application of Bendixson theorem is obtained. A random selection algorithm is given for an 3×3 dimensional interval family with the combination of division-elimination procedure.