## 1 Introduction  
In nonlinear control theory, control Lyapunov functions (CLFs) [1] and CLF-based controller designs [2; 3] are one of the standard approaches for asymptotic stabilization. Once a CLF of a controlled system is obtained, we can constructively design asymptotically stabilizing state feedback controllers. Moreover, the designed controllers minimize meaningful cost functionals and guarantee robustness in the sense of stability margins. When applying control theory to real-world control systems, it is critical to consider existing input constraints due to physical limitations or safety reasons. In the CLF-based framework, asymptotic stabilization of input-constrained systems is also well-studied. Specifically, for systems with norm input constraints, some stabilizing state feedback controllers have been proposed [4; 5; 6; 7; 8]. Although the norm input constraints are an important class, they are too restrictive for a controller design that utilizes input transformations, including trajectory tracking control, as described later. As a more general class containing the norm input constraints, convex input constraints, where the control inputs are constrained to known compact convex sets, are also of interest [9]. In [10; 11; 12], global asymptotic stabilization of nonlinear systems with convex input constraints based on CLFs is studied. Since bounded controllers may not achieve global stabilization, a locally asymptotically stabilizing controller is also proposed in [13]. The basic idea of the controller design of [13] is to find the state feedback that minimizes the time derivative of a given CLF, which we call the minimizing input. Under the convex input constraints, the minimizing input is characterized as the optimal solution of a convex optimization problem. This approach is also extended to a discontinuous stabilizing controller design [14] and an inverse optimal controller design [15]. Trajectory tracking is an essential control problem in many control applications, including robotics. Specifically, trajectory tracking of autonomous vehicles has been extensively studied, and many tracking controller design methods have been proposed (see,e.g., [16; 17; 18; 19; 20; 21]). Moreover, tracking controller design for input-constrained vehicles is also discussed [22; 23; 24]. From the viewpoint of nonlinear control theory, trajectory tracking is studied mainly for specified classes of nonlinear control systems, such as feedback linearizable systems [25], port Hamiltonian systems [26; 27], and incrementally passive systems [28]. Although nonlinear model predictive control (NMPC) can provide a general framework for constrained trajectory tracking [29], computational costs for real-time implementation are still an issue. Consequently, a general low-computational cost framework for constrained tracking is required. Recently, tracking control Lyapunov functions (TCLFs) [16; 30; 31], an extension of CLFs to trajectory tracking control, have attracted attention. When there are no input constraints, some TCLF-based tracking controllers have been proposed by extending the CLF-based stabilizing feedback controllers [32; 33; 34]. However, we cannot directly extend the CLF-based constrained stabilizing feedback controllers for input-constrained systems. This is due to the input transformations used to transform the original nonlinear systems into the corresponding error systems. In particular, the norm constraints are not invariant under the input transformations, i.e., even if the original systems have norm input constraints, the input constraints for the transformed error systems are no longer simple norm constraints. Therefore, to design a constrained trajectory-tracking controller, we need to consider a class of input constraints that are invariant under input transformations. Based on the above background, this article proposes a general controller design framework for trajectory tracking of nonlinear systems with convex input constraints based on TCLFs by extending the controller design method of [13].