## 1 Introduction  
With the evolution of manufacturing, flexible structure-type systems are commonly used in a variety of engineering areas such as moving strips, adjustable marine risers, and drill cable strings [1; 2; 3]. The string system is a distributed parameter system (DPS) which is composed of one partial differential equation (PDE) and two ordinary differential equations (ODEs). The presence of external disturbances and system parameters uncertainty in the string system makes the dynamics solution and control design indispensable. Hence, it has become more challenging and received increasing attention [4]. The iterative learning control (ILC) approach has been discussed previously using intelligent control methodologies [5; 6; 7; 8; 9; 10; 11; 12; 13]. ILC is a practical approach to control such systems that execute similar tasks in a repetitive manner [14], such as a robotic arm in industry [15], a chemical reactor production [16], a nonlinear process [17], and position control of nano systems [18]. This method aims to create a repetitive algorithm for an undefined plant by enhancing the control inputs using the data from the previous iterations [19; 20]. ILC plays a vital role in DPS, including a Timoshenko beam [21], a class of mixed DPSs [22], a Euler-Bernoulli beam [23], a non-linear infinite dimensional system [24], and a flexible robotic manipulator system [25]. For the sake of DPS with known parameters, ILC is widely used in the presence of initial errors [26] and tracking of the tip location [27]. Further, the DPS on Hilbert space have been thoroughly investigated by ILC in Reference [28]. In engineering fields, uncertainty is usually measured as a shortage of knowledge that could lead the whole process to an unsafe condition. It has a very important consequence in the system design as the close-loop process should manipulate with the worst-case situation [29]. Hence, the close-loop system could work in a secure region for the full extent of conceivable parameters values. The uncertainty in the process may generate vibrations, errors, imprecision, and instability [30]. Consequently, it is essential to manipulate it efficiently. Until now, few studies have discussed the system parameters uncertainty through ILC. The handling of the uncertain flexible robot with external disturbances is accomplished by implementing an adaptive ILC in Reference [31]. Robust \(H_{\infty}\)ILC algorithm is considered to enhance the performance of the uncertain system with constraints in Reference [32]. In Reference [33], \(H_{\infty}\) optimal ILC is proposed for uncertain discrete systems affected with external disturbances. However, to the best of our knowledge, no study suggests \(H_{\infty}\)ILC to manipulate with vibrations and uncertainties of DPS in the presence of the distributed and boundary disturbances. In order to obtain an accurate approximated solution for the PDE system, an appropriate solution method has to be applied. Otherwise, it is a possibility to find inaccuracies due to, truncation errors, deviation in real values, and drawbacks between all the sets of data [34]. Several numerical methodologies can be applied for the dynamic of a PDE system to get the estimated solution, such as the method of line (MOL) [35; 36; 37], the finite difference (FD) [38; 39; 40], and the finite element (FE) [41; 42] techniques. MOL has been used to address the drawbacks of the FD method. In this paper, the system parameter uncertainties, external disturbances, and the MOL were considered for the string system. Due to the flexibility features of the string system with external disturbances, finding the dynamic solution and control design were crucial. In order to address such a challenge, by integrating the \(H_{\infty}\) control with iterative learning boundary control (ILBC), an \(H_{\infty}\)ILBC-based MOL scheme was considered for vibrations damping of the certain and uncertain vibrating string system in the presence of iteration-varying distributed/boundary disturbances.