# Time-Specified Adaptive Robust Control Framework for Managing Nonlinear System Uncertainties
## 1 Introduction
The prescribed-time stabilization of strict-feedback systems and stochastic systems (where the nonlinear function meets a linear growth condition) was studied in [34; 35], respectively. Prescribed-time control presents a significant advancement over finite-time and fixed-time control methodologies. Unlike finite-time control, where convergence time depends on initial conditions, or fixed-time control, which provides an upper bound independent of initial conditions, prescribed-time control achieves exact convergence at a specified time regardless of initial conditions. This precise temporal specification is particularly valuable in mission-critical applications requiring coordination between multiple systems or processes. Mathematically, prescribed-time control employs time-varying feedback structures that introduce a time-dependent scaling factor, often in the form of \(\frac{1}{(T_{p}-t)}\), where \(T_{p}\) is the prescribed convergence time. This factor ensures that the system states reach equilibrium exactly at \(t=T_{p}\), after which a different control strategy maintains the system at equilibrium. It is widely recognized that precisely modeling a system in practice can be very difficult. Adaptive Control offers a functional avenue to tackle nonlinear systems with uncertain parameters by estimating unknown parameters online [36]. Many well-known adaptive designs for uncertain nonlinear systems have been proposed (see [37; 38; 39]). However, due to estimation errors, [PERSON]'s lemma--commonly employed in Adaptive Control--cannot be directly utilized for the stability analysis of finite/prescribed-time control [36]. Current adaptive finite/prescribed-time control results generally fall into two categories. The first ensures the convergence of system states to a neighborhood of the equilibrium point or to a defined compact set within a finite time (refer to [40; 41]); the second guarantees that the system states converge to the equilibrium point within a finite interval (refer to [36; 42]). Obviously, the latter is more challenging, yet more meaningful. Hence, this article concentrates on the latter issue. In [36], global adaptive finite-time control was investigated for a class of nonlinear systems in p-normal form with parametric uncertainties. From this earlier work, it is evident that the finite/fixed-time control methods based on \"Lyapunov's differential inequality\" in [5; 7] are not immediately applicable to systems with uncertain parameters. Meanwhile, [42; 43; 44] achieved adaptive prescribed-time stabilization for uncertain nonlinear systems using the dynamic high-gain scaling approach. Nevertheless, because the time-varying function is applied to scale the states in all coordinate transformations, the computational load involving the derivative of the scaling function grows considerably [35]. Moreover, prescribed-time stabilization is particularly significant in tackling transient problems; its benefits align well with numerous applications that impose strict time constraints, such as missile interception. Yet the majority of controlled systems demand that the state reaches the origin within the prescribed time and remains there--aircraft autonomous rendezvous and docking, and unmanned aerial vehicles arriving at designated locations, are prime examples. Under these conditions, it is imperative not only to ensure finite-time convergence but also to maintain the system at the origin post-convergence. While the proposed adaptive prescribed-time control strategy effectively handles parameter uncertainties, real-world systems frequently encounter disturbances, unmodeled dynamics, and noise that can degrade performance or even destabilize the closed-loop system if not addressed. Therefore, incorporating robust control principles into this research is essential to further ensure stability and performance within the prescribed time bound, even under significant external perturbations.