# General Periodic Cruise Guidance Optimization for Hypersonic Vehicles
## 1 Introduction  
Nowadays, hypersonic vehicles are attractive in many countries [1; 2; 3]. In addition to aerodynamic hypersonic gliders, powered hypersonic vehicles primarily operate at constant velocities at specific altitudes because this strategy can be easily implemented in practice. The cruising altitude and velocity were determined according to the principle of fuel efficiency maximization to achieve a long flight range. In other words, the flight range, with respect to the fuel consumption, should be as large as possible. In the future, there is a need to further improve the fuel efficiency of the cruising phase to make room for useful payloads and reduce the weight and volume of the launch vehicle. Therefore, a more advantageous flight mode should be developed instead of the traditional steady-state cruise, to increase the efficiency-cost of fuel consumption. This kind of research is particularly important for inter-continental hypersonic vehicles. The conventional steady-state cruise is the simplest paradigm in the sense of optimality, which is an achievable 2 degree-of-freedom (DOF) trajectory guidance law in terms of altitude and velocity at the expense of a certain flight range. The theoretically optimal trajectory is an infinite DOF curve, which cannot be implemented in practice. In addition to the steady-state cruise, a periodic cruise of more than 2 DOF may improve the fuel efficiency in hypersonic flight. Periodic control is a repetition of the same control actions in each cycle, and the results of optimal periodic control (OPC) have appeared intermittently since the 1950s. Early OPC germinated from the chemical engineering process to achieveeconomic benefits [4; 5; 6; 7; 8; 9]. Since then, once it was properly developed, the OPC concept was suitable for handling a much wider variety of dynamical systems. Besides chemical processes, cruise control of automobiles and aircrafts is a typical application of periodic control during long-period normal operations because of the economic reasons [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. According to [11; 12; 13; 14; 15; 16; 17; 18; 19; 20], a flight vehicle cruising at non-constant altitudes and velocities with proper power switches may save the fuel consumption to some extent, or it can fly at a longer range. For example, the thrust can be switched off during the descending phase in a cycle such that the gravitational force can be fully utilized to reduce the velocity loss, while the thrust can be switched on again at a certain time in the ascending phase to compensate for the velocity loss. This conversion between the potential energy and kinetic energy has the possibility of enhancing the fuel efficiency. This is quite attractive in the aerospace industry. Compared to the application of OPC, the theoretical analysis of OPC is quite difficult. In the theoretical research, the main focuses are the sufficient and necessary conditions for OPC [21; 22]. The sufficient condition aims to give a criterion for improving the optimal steady-state control through cyclic operations [23]. The existing sufficient conditions are based on the Hamilton-Jacobi theory [24]. The state-of-the-art sufficient condition is the so-called \(\pi\)-test in the frequency domain [25; 26; 27]. However, the sufficient condition is only a guarantee of existence, which can slightly improve the steady-state performance with slight perturbation, without considering optimality. We must resort to the necessary conditions for finding the optimal periodic solution. The necessary conditions are almost unexceptionally based on the maximum principle [28], which provides a unique basis for the subsequent optimization of characteristic parameters with the help of Hamiltonian functions by using gradient-based methods, including direct and indirect schemes [29]. The indirect philosophy analytically derives the necessary condition and then obtains the optimal solution. These methods use the calculus of variations to establish a set of first-order optimality equations.