The global asymptotic stability and stabilization in nonlinear cascade systems with delay

We study certain sufficient conditions for the local and global uniform asymptotic stability, as well as the stabilizability of the equilibrium in cascade systems of delay differential equations. As distinct from the known results, the assertions presented in this paper are also valid for the cases, when the right-hand sides of equations are nonlinear and depend on time or arbitrarily depend on the historical data of the system.
We prove that the use of auxiliary semi-definite functionals and functions with semi-definite derivatives taken by virtue of the system, essentially simplifies the statement of sufficient conditions for the asymptotic stability of a cascade.
We adduce an example which illustrates the use of the obtained results. It demonstrates that the proposed procedure makes the study of the asymptotic stability and the construction of a stabilizing control easier in comparison with the traditional methods.