## 1 Introduction  
The phenomenon of saturation affects all real-world systems due to the inherent physical limitations of actuation devices. Designing a control system without taking into account the presence of saturation may lead to severe deterioration of the performance and even to instability of the closed-loop system. For this reason, this issue has been investigated by several control theorists, with recent results such as those reported in References [1; 2; 3; 4; 5; 6; 7]. Existing solutions may be divided into two categories--anti-windup compensation [8], where a compensator is added to an already designed controller in order to handle the saturation constraints and direct control design [9], in which the input constraints are considered at the controller design stage. In the vast literature addressing the saturation problem, the assumption that the saturation limits are constant in time is made, to the best of the authors' knowledge. For example, this assumption can be found in Reference [10], where the region of attraction of a saturated linear parameter varying (LPV) system with bounded parameter variations is optimized by means of parameter-dependent Lyapunov functions, generalized sector conditions and a static output-feedback controller. The same assumption holds in Reference [11], where the idea is to approximate the region of attraction using the concept of quadratic boundedness, such that off-line optimization algorithms are presented to design a saturated dynamic output feedback controller for an LPV system with bounded disturbance. On other hand, in Reference [12] the saturation phenomenon is included in the design problem of an \(\mathcal{H}_{\infty}\) dynamic output-feedback controller for a class of uncertain discrete stochastic nonlinear time-varying systems using the recursive linear matrix inequality (RLMI) approach, thus obtaining a suitable algorithm for online applications. However, from a practical viewpoint, it makes sense to consider time-varying saturation limits. They could arise in control systems due to several reasons, such as the natural wear of engines and devices, that would provide a progressively decreasing actuation signal or temporary shortages in the availability of electrical or pneumatic power. Moreover, in trajectory tracking problems, the control signal is usually obtained as the sum of a feedforward and a feedback component. When a time-varying trajectory is considered, the feedforward component changes in time, which would be perceived by the feedback controller as a time-varying saturation. The main goal of this paper is to propose a methodology for designing state-feedback controllers that take into account time-varying input saturations. It makes sense that a change in the saturation function should be tied to a change in the performance achieved by the closed-loop control system (e.g., if the maximum possible input decreases, the system's response should become slower). For this reason, the time-varying saturation limits are addressed using _shifting specifications_, following some ideas found, for example, in References [13; 14]. This means that some parameters are introduced which, on the one hand, they are scheduled by the time-varying saturations and, on the other hand, they schedule the performance criteria in such a way that different values of these parameters imply different performances (in this paper, we will consider the guaranteed decay rate but the developed results can be extended straightforwardly to other criteria, for example, pole clustering or \(\mathcal{H}_{\infty}/\mathcal{H}_{2}\) guaranteed bounds). The direct consequence of introducing the above mentioned scheduling parameters is that the closed-loop system becomes a parameter-varying system and the design conditions can be determined within the framework of linear parameter varying (LPV) systems [15; 16].