Let y"=f(x,y,y') be a 2nd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates. We are going to solve this problem by an applicable method which has been recognized by R. Gardner, and classify them.

By using the fixed point methods, we prove some generalized Hyers-Ulam stability of homomorphisms for Cauchy and Cauchy-Jensen functional equations on the product algebras and on the triple systems.

Recently, a new data driven controller synthesis is presented for calculating the family of stabilizing first, second and fixed order controllers using frequency respons. However, this method is applicable just for plants that can guarantee some smoothness at the boundary of the resulted high dimension LMI. This paper solve that issue and extends the approach to fixed order controllers guaranteeing some performance criteria which are applicable for the more general types of plants. It is shown that knowing the frequency response of plant is sufficient to calculate the stabilizing fixed order controllers from a set of convex linear inequalities. The H¥ norm on sensitivity and complementary sensitivity functions are satisfied from some frequency domain inequalities (FDI) that could be examined from frequency response data. The usefulness of the proposed approach is illustrated by an academic example.

This paper presents delay-dependent stability analysis for some types of nonlinear time-delay systems. Motivated by Lyapunov-Krasovskii functionals and the related theorems on stability of time-delay systems, we introduce a new procedure for generating such functionals. The main reason for considering Lyapunov-Krasovskii functionals is the less conservatism of this method. The obtained conditions are sufficient and local. Applying the introduced procedure on a typical system with time-invariant parameters derives more results and a numerical example presents the capability of the method.