Conditions and Synthesis of Adaptive Prescribed-Time Controllers

This article addresses conditions and synthesis of adaptive prescribed-time controllers. The sufficient condition says that a system can achieve prescribed-time stabilization if its certain variant can be made globally bounded (on infinite-time interval). From the condition, one can know far more varieties of (uncertain nonlinear) systems can achieve prescribed-time control, rather than just special varieties in the related literature. The necessary condition states that a system can achieve prescribed-time stabilization attached global uniform boundedness (GUB) on finite-time interval only if it, by some feedback type, can achieve GUB on infinite-time interval. From the condition, or rather, its converse-negative version, one can see that what kind of systems cannot virtually achieve the prescribed-time stabilization attached GUB for some feedback type. The sufficient condition builds on two crucial transformations, which change prescribed-time convergence on finite-time interval into global boundedness on infinite-time interval. With the aid of the conditions and transformations, we conduct the synthesis of adaptive prescribed-time controllers for general uncertain nonlinear systems typically with unknown control directions. By integrating time-varying factors into Lyapunov functions, an adaptive smooth controller is devised such that the system states converge to zero before an arbitrarily prescribed time, while the controller and its dynamic gains are bounded.