Homogeneity, Forward Completeness, and Global Stabilization of a Family of Time-Delay Nonlinear Systems by Memoryless Non-Lipschitz Continuous Feedback

For a family of genuinely nonlinear systems with delays in the input and state, global strong stabilization (GSS) in the sense of Kurzweil is shown to be possible by memoryless state and/or output feedback under two conditions: 1) The input delay is within an appropriate range although the state delays can be sufficiently large; 2) the time-delay bounding system is homogeneous of degree zero and has a lower triangular structure. The proof is based on the Lyapunov–Krasovskii functional theorem combined with the homogeneous domination philosophy. Using the emulation approach, we design memoryless homogeneous state and output feedback controllers, respectively, achieving GSS of the time-delay closed-loop systems with severe nonlinearity. Extensions to a wider class of time-delay nonlinear systems in the Hessenberg form are also given. Examples and counter-examples presented in this article illustrate not only the necessity of the homogeneous growth conditions but also the significance of the memoryless non-Lipschitz continuous feedback control strategies.