## 2 State of the Art
### Energy Control Models and Optimization Models  
Reducing energy consumption has become a critical issue across all sectors. Optimization is an indispensable tool for saving energy, and the choice of the cost function is crucial. Various mathematical models are defined for mobile robots to address different objectivesof the problem. The power function is frequently used because it depends on simple parameters such as speed and motor force [27; 28]. Both reducing and increasing speed have a significant impact on energy consumption, and variations in motor force also affect energy usage. In this paper, we are interested in an optimal control problem, and the speed profile generator is determined by the motor force, which depends on the acceleration (or deceleration) and the mass of mobile robots. Optimizing energy consumption involves finding an optimal control strategy modeled by the motor force. Several techniques have been employed to tackle the optimization problem in mobile robots. The authors [29] propose a genetic algorithm to reduce the energy consumption of a mobile robot. The cost function is the sum of the linear velocity squared and angular velocity squared. This algorithm relies on an optimal fuzzy logic controller, which enables the robot to track different paths (e.g., zigzags and sharp turns). The fuzzy logic Mamdani method is employed in [30] to determine three outputs: curve speed, energy consumption, and time information. Through the use of optional curved path data (radius, center angle, load, and ground friction), three fuzzy rules are defined to identify these outputs. The total energy consumed is the sum of the power sensor, control, and motion. A trapezoidal speed profile is utilized to create an optimal profile based on fixed times or distances for straight paths, involving three segments: acceleration, constant speed, and deceleration. Alternatively, a triangular speed profile can be used, requiring only acceleration and deceleration phases. Minimum energy consumption is achieved from these speed profiles for straight paths. For curved paths, the speed profile is derived from fuzzy logic, incorporating the maximum and minimum values of the trapezoidal speed profile. The integration of these methods highlights the importance of optimizing speed and motor force to minimize energy consumption in mobile robots. In the railway field, the most commonly used technique is based on braking distance to calculate a speed profile that respects the maximum speed across the path. In the paper by [PERSON] et al. [31], the problem of train speed trajectory optimization is treated as an optimal control problem. The cost function represents the net electrical energy, which depends on the maximum electrical traction and braking forces, as well as the efficiency of electric motors during traction and braking operations. The authors propose a numerical algorithm based on PMP to find the optimal speed trajectory. An in-depth study using this principle identified conditions for the traction and braking phases. Based on the initial and final train speeds on a segment, and through the use of a numerical algorithm based on distance, the traction and braking values at each stage converge towards cruising speed. Various comparisons with other numerical optimization methods demonstrate the effectiveness of this approach. To address an optimal control problem for a mobile robot operating in an airport environment, the goal is to solve an energy optimization problem specific to this setting. The cost function is given by the integral of the power delivered via the robot's motor, which provides the absolute work. This refinement of the cost function offers a more accurate reflection of the energy consumption during the AIV mission. Additionally, it enables the AIV to perform three types of movements simultaneously: acceleration, deceleration, and constant speed.