# Time-Specified Adaptive Robust Control Framework for Managing Nonlinear System Uncertainties
## 1 Introduction  
Finite-time control has drawn increasing attention from scholars in recent decades owing to its faster convergence rate [1]. The concept of finite-time control first appeared in [2] for the purpose of achieving time-optimal control. Afterward, non-smooth feedback control methods were utilized to ensure finite-time stabilization [3, 4]. Unfortunately, due to the presence of negative fractional powers and discontinuous terms, singularities and chattering may arise in the system's operating process [5], which significantly restricts the practical application of sliding mode finite-time control as, for example, in missile guidance [6]. To address these issues, a classical finite-time control theorem grounded in \"Lyapunov's differential inequality\" was initially developed in [7], and further investigated in [8]. The findings from [7, 8] have been extended to a variety of systems, including double-integrator systems [9], time-varying dynamical systems [10], general nonlinear systems [11], stochastic systems [12], and multiagent systems [13]. The homogeneous approach was introduced in [14, 15]. Additionally, notable achievements in the finite-time stabilization of partial differential equations (PDEs) were presented in [16, 17]. Although finite-time control strategies have been broadly studied, their settling time is dependent on initial conditions [18]. Furthermore, in numerous real-world applications, it can be difficult or even impossible to know the system's initial values in advance, making control of the settling time infeasible. Consequently, the work in [5] developed a fixed-time control algorithm where the upper bound of the settling time does not depend oninitial conditions. Follow-up results for fixed-time control concerning certain linear and nonlinear systems were provided in [1; 19; 20]. The fixed-time stabilization issue for reaction-diffusion PDEs was resolved in [21] by employing a continuous boundary time-varying feedback control strategy. While current fixed-time stabilization results can drive system states to zero within a desired time by selecting appropriate control parameters, achieving convergence at an arbitrary time remains extremely challenging [22]. Recently, the problem of predefining the upper bound of the settling time in fixed-time stabilization, also known as predefined-time stabilization, was examined in [22; 23; 24; 25]. In these references, one can freely choose the upper bound of the settling time by suitably configuring control parameters. Nevertheless, the actual convergence time in predefined-time stabilization still hinges on the initial conditions, which distinguishes it from prescribed-time stabilization results [26], where the actual convergence time is independent of initial conditions. Research on prescribed-time stabilization has been reported in [25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. In [25; 27], through a state-scaling approach (the time-varying function in the scaling goes to infinity as time nears the prescribed end time), the global prescribed-time stabilization problem for nonlinear systems in strict-feedback form was solved. The advantages of prescribed-time regulation over finite-time regulation were highlighted in [28]. In [29], a sufficient condition was presented to guarantee convergence within an arbitrary time, and a novel prescribed-time control strategy for linear systems was introduced. By employing the backstepping technique, [30] successfully tackled the prescribed-time stabilization of reaction-diffusion PDEs. Using a decoupling backstepping approach, [31] accomplished prescribed-time trajectory tracking for triangular systems of reaction-diffusion PDEs. The most recent development on prescribed-time stabilization for reaction-diffusion PDEs can be found in [32]. In [33], several Lyapunov-like prescribed-time stability theorems were provided and compared with certain finite-time stability theorems. Grounded in the Lyapunov-like function, [26] addressed the global prescribed-time stabilization of a class of nonlinear systems.