## AEROSPACE TRAJECTORIES

**Credits**: 12

**Learning objectives***Part 1 - Optimization methods in orbital mechanics (6 credits)*

The determination of a nominal trajectory is an essential premise for the realization of a space mission, and leads to defining the performance attainable from a given space system, with reference to the operational scenario. Hence, trajectory optimization is a central topic in the context of aerospace mission analysis. This course provides the students with the theoretical foundations and the algorithmic techniques tailored to solving aerospace trajectory optimization problems and in particular: launcher trajectories for orbit injection and finite-thrust and impulsive transfers between two arbitrary Keplerian orbits.

As most space trajectory optimization problems are not amenable to an analytical solution, numerical techniques are usually employed. Indirect deterministic methods use the analytical conditions arising from the calculus of variations.

Direct approaches convert the optimal control problem into a parameter optimization problem. Most recently, heuristic methods have been introduced. They emulate the stochastic behavior of a population of individuals, under the assumption that each of them represents a possible solution to the problem of interest. An overview on all these techniques is given and at the end of the course, the student will possess the theoretical principles and the practical techniques to solve trajectory optimization problems, as well as the capability of choosing the appropriate method of solution with regard to the problem of interest. The course is organized as it follows:

The phases of a launcher trajectory, launcher guidance laws, parameter optimization, optimal control theory, Primer vector theory. Performances of a launcher. Unperturbed and perturbed keplerian orbits, globally optimal impulsive transfers, optimization of low-energy interplanetary orbit transfers.*Part 2 - Dynamic game theory in flight mechanics (6 credits)*

Aerospace vehicles can be involved in challenging scenarios in which two or more competing objectives exist and drive the overall dynamics. Noncooperative motion of two distinct vehicles is to be investigated through dynamic game theory, a multi-agent decision theory that addresses the situations in which two or more competing actors are involved. Dynamic game theory has application in economics, computer science, operations research, environmental science, and engineering. In particular, this course covers the theoretical foundations of dynamic game theory, regarded as an extension of optimal control theory, which pertains to a single actor. Nash and saddle-point equilibrium solutions, associated with the use of optimal strategies by all the players, are defined, and the related analytical conditions are derived. Possible applications in flight mechanics range from the optimal interception of optimally evasive targets to the worst-case performance analysis of aerospace vehicles in uncertain environments, by assuming that the second player is the hostile environment. More specifically, several scenarios exist that can be modeled as zero-sum dynamic games: (a) fighter aircraft evasion maneuvers against an intercepting missile, (b) air combats involving fighter aircraft and unmanned aerial vehicles, (c) noncooperative spacecraft interception, and (d) noncooperative orbit maneuvers of two space vehicles in close proximity. These scenarios are associated with distinct dynamical frameworks, and the numerical solution of the related games indicates the optimal maneuvers that each vehicle must perform in order to penalize the opponent player as much as possible. The course is organized as follows: Fundamentals of optimal control theory, Introduction on noncooperative dynamic game theory, Pursuit-evasion (zero-sum) dynamic games, Numerical solution techniques for zero-sum dynamic games, Maneuvers of two competing vehicles in atmospheric flight, Maneuvers of two competing vehicles in exoatmospheric flight, Maneuvers of two orbiting spacecraft in close proximity.