Robust Semiglobal and Global Stabilization for Nonlinear Normal Form Systems by Time-Varying Feedback

A scalarization approach, integrating a Lyapunov-like lemma and a parametric Lyapunov design, is developed while using time-varying feedback for nonlinear systems. The Lyapunov-like lemma is recalled and updated for stability analysis of time-varying nonlinear systems, with the estimate of the region of attraction (ROA). The parametric Lyapunov design yields an explicit parameterized control law by solving a parametric Lyapunov equation (PLE), which can simplify to a tractable first-order linear matrix equation. Within this framework, the problems of asymptotic, exponential, hyperexponential, and prescribed-time stabilization are transformed into solving PLE and selecting the appropriate (time-varying) parameter of PLE, while concurrently providing an estimation of ROA. Applying this approach, we achieve both semiglobal and global stabilization with exponential, hyperexponential, and prescribed-time performance for uncertain nonlinear systems in the normal form.