AAE 50800: Optimization in Aerospace Engineering
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
Course Instructor: Professor Longuski
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):
- The Problems of Bolza, Lagrange and Mayer (5 lectures)
Interchangeability of the problems, introductory concepts, Zermelo;s problem, Lotka-Volterra model.
- 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.
- Proof of the Weierstrass Necessary Condition (2 lectures)
- Statement of the Maximum Principle (2 lectures)
- 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.
- Proof of the Weirstrass-Erdmann Corner Conditions (3 lectures)
- Optimal Control Problems with Inequality Constraints (3 lectures)
Bounded control problem, singular subarcs, switching functions, generalized Legendre-Clebsch condition
- 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
- Computational Techniques (5 lectures)
Newton shooting method, gradient method, conjugate-gradient, trajectory design, trajectory optimization
Relationship of course to program objectives:
This course provides the essential technical components (1) of space trajectory design and space trajectory optimization. Students develop basic engineering skills in formulating and solving open-ended problems (2a) and in writing a project report (2c). Some students have turned their projects into directed studies (AAE 590), conference papers, and journal submissions (3).
Prepared by: James M. Longuski
Date: March 19, 2001