# AAE 50800: Optimization in Aerospace Engineering

Description:

Formulation of optimization problems encountered in aerospace engineering. Minima of functions and functionals, necessary conditions, calculus of variations, control formulation, two-point boundary-value problems. Applications to typical problems in aerospace engineering such as optimal launch, minimum time to climb, maximum range, and optimal space trajectories.

Format: 3 hrs lecture per week

Credit hours: 3

Status: Elective, Dynamics and Control and Design

Offered: Spring

Pre-requisite: None

Co-requisite: None

Course Instructor: Professor Longuski

Text: None

Assessment Method: Grading is weighted 20% on homework, 20% on a project, 40% on two in-class exams, and 20% on a final exam

## Course Objective: To introduce students to the theory and numerical calculation of optimal space trajectories

Necessary Background: Students should be senior or graduate standing in engineering, science, or mathematics

Topics (number of Lectures):

1. The Problems of Bolza, Lagrange and Mayer (5 lectures)

Interchangeability of the problems, introductory concepts, Zermelo;s problem, Lotka-Volterra model.

1. Proof of the Euler-Lagrange Theorem (7 lectures)

Calculus of variations, necessary conditions, transversality conditions, TPBVP.

Example of flat Earth launch problem and derivation of linear and bilinear tangent steering laws.

1. Proof of the Weierstrass Necessary Condition (2 lectures)
2. Statement of the Maximum Principle (2 lectures)
3. Examples (6 lectures)

Flight envelopes for subsonic and super-sonic aircraft, minimization of time to climb, maximization of the range of a rocket, optimal launching of a satellite.

1. Proof of the Weirstrass-Erdmann Corner Conditions (3 lectures)
2. Optimal Control Problems with Inequality Constraints (3 lectures)

Bounded control problem, singular subarcs, switching functions, generalized Legendre-Clebsch condition

1. General Theory of Optimal Rocket Trajectories (7 lectures)

Extremal arcs, impulsive, thrust, optimal trajectories in a uniform field, the primer in a inverse square field, orbital transfer maneuvers

1. Computational Techniques (5 lectures)