Astrodynamics Graduate Courses

(August 2004)


Bruce A. Conway, Professor

Victoria L. Coverstone, Associate Professor

Natasha Neogi, Assistant Professor

John E. Prussing, Professor


AE 402. Orbital Mechanics Analysis of orbits in an inverse-square gravitational field; elementary rocket dynamics, impulsive orbit transfer and rendezvous, and Lambert's Theorem with applications; patched-conic trajectories, planetary gravity-assist maneuvers, and linearized orbit theory with application to simplified analytical models; perturbations. Prerequisite: AE 302 or consent of instructor. 3 undergraduate hours. 3 or 4 graduate hours.

AE 403. Spacecraft Attitude Control Theory and applications of spacecraft attitude dynamics and control; Euler angles, direction cosines, quaternions, and Gibbs-Rodrigues parameters; attitude sensors and control actuators; spin, three-axis active, reaction wheel, control moment gyro, and gravity gradient control systems; environmental effects. Prerequisite: AE 352 and AE 353 or equivalent. 3 undergraduate hours. 3 or 4 graduate hours.

AE 407. Passive Spacecraft Control Free molecule aerodynamics; gravity gradient and solar radiation torques on satellites; interaction of on-board magnetic dipoles with the earth's magnetic field solar wind; cosmic dust and micrometeroid torques; lifetime problem and density determination; and utilization of these various environmental effects in satellite attitude control. Prerequisite: AE 302 and 311. 3 undergraduate hours. 3 or 4 graduate hours.

AE 434. Rocket Propulsion Basic principles of rocket propulsion and rocketry, propel lants and their influence on design of rockets, internal and external ballistics, combustion processes, design of components, flight performance, and rocket testing. Prererequisite: AE 312 or equivalent and AE 433. 3 undergraduate hours. 3 or 4 graduate hours.

AE 435. Electric Propulsion Elements of propulsion as applied to deep-space missions; physics of ionized gases; plasmadynamics; electrothermal, electromagnetic, and electrostatic acceleration of gases to high velocity; high-impulse thruster design and performance; and the resistojet, arcjet, ion engine, MPD arc, and plasma gun. Prerequisite: AE 433. 3 undergraduate hours. 3 or 4 graduate hours.

AE 468. Optical Remote Sensing Introduction to Optical Remote Sensing. Optical sensors including single element and area arrays (CCDs). Systems including imager, spectrometer, interferometer and lidar optical principles and light gathering power. Electromagnetics of atomic and molecular emission and scattering with applications to the atmosphere as an example. Applications include ground and spacecraft platforms. Four laboratory sessions (4.5 hours each) will be arranged during the semester in lieu of four lectures. Course information: Same as ATMS 468 and ECE 468. This course may not be repeated for credit. Prerequisite: PHYS 214; and ECE 329 and ECE 210 and a course in probability or statistics, or consent of instructor. 3 hours.

AE 498 D. Introduction to System Dynamics and Control This course covers what constitutes the common core of Dynamics and Control Theory. The mathematical background that is required of the students is a working knowledge of linear algebra and differential equations. The dynamics and control concepts from AE 352 and 353 or equivalent courses are required. For graduate students intending to pursue a graduate degree in in general area of Dynamics and Control Theory. This course will form the basis for more advanced courses. 3 undergraduate hours. 3 or 4 graduate hours.

AAE 498 E. Nuclear Rocket Propulsion

AE 498 ID, ID1, ID2. Interdisciplinary Design I, II (Cubesat)

AE 498 RLV. Reusable Launch Vehicles

AE 498 S. Access to Space

AE 498 SSS. Software and Systems Safety

AE 502. Advanced Orbital Mechanics Circular restricted three-body problem; surfaces of zero velocity, libration points, halo orbits, perturbed two-body motion; Gauss and Lagrange planetary equations, Hamilton's principle, canonical equations and the Delaunay variables, application to artificial earth satellites; orbit determination. Prerequisite: AE 402 or consent of instructor. 4 hours.

AE 504. Optimal Aerospace Systems Formulation of parameter and functional optimization problems for dynamic systems; applications of optimization principles to the control and performance of aerospace vehicles, including optimal flight paths, trajectories, and feedback control. Prerequisite: AE 352 or equivalent. 4 hours.

AE 508. Optimal Space Trajectories Optimal rocket trajectories in inverse-square and linearized gravitational fields; orbital transfer, intercept, and rendezvous; high-thrust (impulsive) and low-thrust (continuous) trajectories; primer vector theory and applications; cooperative rendezvous. Prerequisite: Credit or concurrent registration in AE 504 or equivalent, or consent of instructor. 4 hours.

AE 554. Dynamical Systems and Bifurcation Theory Fundamental concepts of nonlinear oscillations, structural stability, local and global bifurcations in the context of ordinary and partial differential equations; introduction to dynamical systems, structural stability and Lyapunov-Schmidt Reduction, bifurcations of equilibrium points, limit cycles and tori, the center manifold and Poincare normal forms, co-dimension two and higher order bifurcations, bifurcation theory of maps, the Birkhoff-Smale homoclinic theorem and horseshoes, Melnikov's method and Silnikov phenomena, period doubling and other routes to chaos. Applications to many engineering problems, such as aircraft at high angles of attack, pipes conveying fluid and panel flutter will be demonstrated. Prerequisite: AE 352 or TAM 412. 4 hours.

AE 555. Multivariable Control Design Same as General Engineering 521. Frequency response design specifications; algebraic and analytical constraints in scalar systems; uncertainty representation; Nyquist stability theory, small gain condition, multi-input multi-output systems; singular value decomposition; robustness and mu-function; linear quadratic regulator based design; recovery of LQ design properties; Kalman filter; Riccati equations; H-infinity-based design; mu-synthesis; order reduction; balanced truncation; Hankel singular values; coprime factor reduction; loop shaping. Prerequisite: ECE 515. 4 hours.

AE 556. Robust Control Signal and system spaces; stability, robustness, and the small gain theorem; factorization and parametrization of all stabilizing controllers; performance and achievable closed loop maps; model matching; design of optimal single-input single-output systems in H-infinity, H-2, L-1 senses; extensions to multi-output systems; structured and unstructured uncertainty; robust performance analysis and synthesis; multi-objective control. Prerequisite: ECE 515, MATH 446. 4 hours.


CS 450. Intro to Numerical Analysis Same as CSE 401, ECE 491 and MATH 450. Introduction to numerical analysis that includes linear system solvers, optimization techniques, interpolation and approximations of functions, solving systems of nonlinear equations, eigenvalue problems, least squares, quadrature, as well as numerical handling of ordinary and partial differential equations. Prerequisite: CS 101 or 125; CS 257 or MATH 415; MATH 385, 386 or 441; or consent of instructor. 3 undergraduate hours. 3 or 4 graduate hours.

CS 459. Numerical Approx and ODEs Same as CSE 413 and MATH 459. Polynomial and spline inter polation; least squares and uniform approximation; numerical differentiation and integration; initial-value and boundary-value problems in ordinary differential equations. Prerequisite: CS 257 and MATH 385 or 441, or consent of instructor. 3 undergraduate hours. 3 or 4 graduate hours.

CS 554. Parallel Numerical Algorithms Same as CSE 512. Introduction of numerical algorithms for vector and parallel computers; parallel algo rithms in numerical linear algebra (dense and sparse solvers for linear systems and the algebraic eigenvalue problem), numerical handling for ordinary and partial differential equations, and numerical optimization techniques. Prerequisite: At least one of CS 450, 455, 458, or 459, or consent of instructor. 4 hours.

ECE 470. Introduction to Robotics Same as GE 370 and CS 343. Fundamentals of robotics, rigid motions, homogeneous transformations, forward kinematics, inverse kinematics, velocity kinematics, dynamics, and laboratory projects. Prerequisites: MATH 415 or 418, or consent of instructor. 4 hours.

ECE 486. Control Systems, I Analysis and design of control systems with emphasis on modeling, state variable representation, computer solutions, modern design principles, and laboratory techniques. Prerequisite: ECE 210 or consent of instructor. 4 hours.

ECE 490. Introduction to Optimization Same as CSE 441. Basic theory and methods for the solution of optimization problems; iterative techniques for uncon strained minimization; and introductory presentation of linear and nonlinear programming with engineering applications. Prerequisite: CS 101 or 125, and MATH 380, or consent of instructor. 3 undergraduate hours, 3 or 4 graduate hours.

ECE 515. Control Syst Theory & Design Synthesis of feedback control systems to meet design specifications, including sensitivity; multivariable systems; introduction to systems with random inputs; state variable techniques; and nonlinear systems. Prerequisites: ECE 486 or equivalent, or consent of instructor. 4 hours.

ECE 517. Nonlinear & Adaptive Control Studies design of nonlinear control systems based on stability considerations; examines Lyapunov and hyperstability approaches to analysis and design of model reference adaptive systems; identifiers, observers, and controllers for unknown plants. Prerequisite: ECE 515. 4 hours.

ECE 528. Analysis of Nonlinear Systems Same as GE 520 and ME 546. Treatment of singular points and stability considerations; consideration of graphical and analytical methods, including the perturbation method, variation of parameters, Galerkin's method, and the Ritz method for solving nonlinear differential equations. Prerequisite: ECE 515 or equivalent; MATH 385. 386, or 441 or consent of instructor. 4 hours.

ECE 534. Random Processes Basic concepts of random processes; linear systems with ran dom inputs; Markov processes; spectral analysis, Wiener and Kalman filtering and applications to systems engineering. Prerequisite: ECE 413 or MATH 461 or STAT 400, or consent of instructor. 4 hours.

ECE 553. Optimum Control Systems Formulation of the optimization problem; controllability, observability; stability; Lyapunov's second method; application of variational calculus, maximum principle, and principle of optimality to control problems; stochastic control and adaptive control. Prerequisite: ECE 515. 4 hours.

ECE 554. Sample-Data Control Systems Analysis and design of feedback control systems with digital and sampled data. Prerequisite: ECE 515. 4 hours.

ECE 555. Control of Stochastic Systems Stochastic control models; development of control laws by dynamic programming; separation of estimation and control; Kalman filtering; self-tuning regulators; dual controllers; decentralized control. Prerequisite: ECE 515 and 534. 4 hours.

GE 420. Digital Control of Dynam Systems Examines theory and techniques for control of dynamic processes by digital computers; linear discrete systems, digital filters, sampling signal reconstruction, digital design, state space methods, computers, state estimator, laboratory techniques. Prerequisite: GE 320 or equivalent. 4 hours.

GE 422. Robot Dynamics and Control Same as ECE 489 and ME 446. Dynamics and control of robotic manipulators. Emphasis on fundamental concepts and analytical methods for analysis and design of robotic systems. Laboratory experiments complement the theoretical development. Prerequisite: GE 320 or equivalent or consent of instructors. Recommended: ECE 470. 4 hours.

GE 424. State Space Design Meth in Cntrl Design methods; time domain modeling; trajectories and phase plane analysis; similarity transforms; controllability and observability; pole placement and observers; linear quadratic optimal control; Lyapunov stability and describing functions; simulation. Prerequisite: GE 320 and MATH 225. 3 hours.

GE 531. Genetic Algorithm Methods Genetic algorithm search -- procedures based on the mechanics of natural genetics and natural selection -- are finding increased application to the difficult problems of engineer ing, science, and commerce. This course surveys what genetic algorithms are, where they come from, how they work, and how and where they have been applied. Prerequisites: MATH 242 and CS 101. 4 hours.

ME 462. Modern Control Theory The concept of state; state-space representation of systems; transfer function decomposition and state-variable diagrams; state response of continuous and discrete-data systems; determination of the transition matrix; diagonalization; state response of time-varying systems; controllability and observability; stability and Lyapunov's method; and intro duction to optimization and design. Prerequisite: ME 340. 4 hours.

PHYS 500. Advanced Mechanics Fundamentals of classical Lagrangian and Hamiltonian mechanics, with emphasis on the relation between dynamical symmetries and constants of the motion; use of conservation laws to derive basic equations of fluid dynamics; discussion of some applications. Prerequisite: Mechanics course at the level of PHYS 326 or consent of instructor. 2 hours.

PHYS 510. Nonlinear Dynamics Broad introduction to nonlinear dynamics of physical systems with varying degrees of complexity; surveys a variety of concept associated with bifurcation phenomena, mappings, nonlinear oscillations, chaotic behavior, strange attractors, solitons, and topics of current interest. Prerequisite: MATH 380 or 385 or equivalent; PHYS 326 or equivalent; or consent of instructor. 4 hours.

PHYS 511. Advanced Nonlinear Dynamics Analysis of the dynamics of spatially extended and other complex physical systems using analytical, experimental, computational, topological, and symbolic methods; examples may involve mechanical, electrical, optical, solid state, fluid, chemical, biological, and network systems. Prerequisite: PHYS 510. 4 hours.

TAM 541. Mathematical Methods I Vector and tensor algebra, introduction to complex-variable methods; ordinary differential equations, qualitative questions of existence and uniqueness; analytical solution methods, numerical methods, power-series solution and special functions; eigenvalue problems, Green's functions, Laplace transforms, stability of solutions; engineering applications drawn from mechanics. Prerequisite: MATH 380 and MATH 385. 386, or 441. 4 hours.

TAM 542. Mathematical Methods II Continuation of TAM 541. Modeling, inequalities, elements of functional analysis; partial differential equations, existence and uniqueness, second-order equations; hyperbolic conservation laws; numerical methods, eigenfunction expansions, integral transforms, fundamental solutions; engineering applications drawn from mechanics. Prerequisite: TAM 541. 4 hours.


ASTR 404. Stellar Astrophysics Introduction to astrophysical problems, with emphasis on underlying physical principles; includes the nature of stars, equations of state, stellar energy generation, stellar structure and evolution, astrophysical neutrinos, binary stars, white dwarfs, neutron stars and pulsars, and novae and supernovae. Prerequisites: PHYS 213 or 214, or consent of instructor. 3 undergraduate hours.

ASTR 405. The Solar Sys and IS Medium Physical processes in the solar system; dynamics of the solar system; physics of planetary atmospheres; individual planets; comets, asteroids, and other constituents of the solar system; extra-solar planets; formation of the solar system, stars, and planets; components of the interstellar medium; ionization and recombination; heating and cooling processes; comparison of theory with observations; composition and characteristics of interstellar dust; dynamics of the interstellar medium; interaction of stars with the interstellar medium: H II regions, planetary nebulae, and supernova remnants. Prerequisite: PHYCS 213 or 214. 3 hours.

ASTR 414. Astronomical Techniques Introduction to techniques used in modern optical and radio astronomy with emphasis on the physical and mathematical understanding of the detection of electromagnetic radiation; includes such topics as fundamental properties of radio and optical telescopes and detectors that are used with telescopes. Lectures and laboratory. Prerequisite: MATH 242; PHYCS 213 or 214; or consent of instructor. ASTR 210 is recommended. 4 undergraduate hours.


MATH 446. Applied Complex Variables For students who desire a working knowledge of complex variables; covers the standard topics and gives an introduction to integration by residues, the argument principle, conformal maps, and potential fields. Students desiring a systematic development of the foundations of the subject should take MATH 448. Prerequisite: MATH 243 or MATH 380 or consent of instructor. 3 undergraduate hours. 3 or 4 graduate hours. 4 hours of credit requires approval of the instructor and completion of additional work of substance. Credit is not given for both MATH 446 and 448. MATH 447. Real Variables Careful development of elementary real analysis including such topics as completeness property of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration. Prerequisite: MATH 242 or 243 or 245, or equivalent, and junior standing; or consent of instructor. 3 hours or 3/4 or 1 unit. One unit credit requires approval of the instructor and completion of additional work of substance. Credit is not given for both MATH 447 and 444. 3 undergraduate hours. 3 or 4 graduate hours.

MATH 461. Probability Theory I Same as STAT 451. Introduction to mathematical probability; includes the calculus of probability, combinatorial analysis, random variables, expectation, distribution functions, moment-generating functions, and central limit theorem. Prepares student for MATH 466. Prerequisite: MATH 242 or 243 or equivalent. 3 undergraduate hours. 3 or 4 graduate hours. Four hours of credit requires approval of the instructor and completion of additional work of substance.

MATH 466. Probability Theory II Continuation of MATH 461. Same as STAT 456. Includes random walks, discrete and continuous Markov chains, and special topics from weak stationarity, multivariate central limit theorem, probability model building, stochastic equations, martingale theory, and renewal theory. Prerequisite: MATH 461 or STAT 410. 3 undergraduate hours. 3 or 4 graduate hours. Four hours of credit requires approval of the instructor and completion of additional work of substance.

MATH 482. Linear Programming Rigorous introduction to a wide range of topics in optimization, including a thorough treatment of basic ideas of linear programming, with additional topics drawn from numerical considerations, linear complementarity, integer programming and networks, polyhedral methods. Prerequisite: MATH 415. 3 undergraduate hours. 3 or 4 graduate hours. Four hours of credit requires approval of the instructor and completion of additional work of substance.

MATH 484. Nonlinear Programming Iterative and analytical solutions of constrained and unconstrained problems of optimization; gradient and conjugate gradient solution methods; Newton's method, Lagrange multipliers, and duality and the Kuhn-Tucker theorem; and quadratic, convex, and geometric programming. Prerequisite: MATH 242 or 243, MATH 347 or 348 or equivalent; and a knowledge of linear algebra equivalent to MATH 415, or consent of instructor. 3 undergraduate hours. 3 or 4 graduate hours. Four hours of credit requires approval of the instructor and completion of additional work of substance.

MATH 489. Differential Equations II Continuation of MATH 385. Linear systems of differential equations, including a self-contained development of the necessary matrix theory; the Laplace transform; and nonlinear differential equations. Prerequisite MATH 385 or 441. 3 undergraduate hours, 3 or 4 graduate hours. Four hours of credit requires approval of the instructor and completion of additional work of substance.

MATH 498. Mathematical Methods in Engineering Matrices, determinants, bounds and approximations to eigenvalues, introduction to linear operator theory and inner product spaces, orthogonal expansions, and Fourier transforms. Prerequisite: MATH 380 or equivalent. 3 undergraduate hours. 3 or 4 graduate hours. Four hours of credit requires approval of the instructor and completion of additional work of substance.

MATH 550. Ordinary Diff Equations Existence, uniqueness, and continuation of solutions; topics selected from the following: the theory of linear differential operators, Sturm-Liouville theory, stability theory, and qualitative theory of differential equations. Prerequisite: MATH 447; a first course in ordinary differential equations. 4 hours.

MATH 551. Dynamical Systems Theory Course is an introduction to the study of dynamical systems. Students who intend to do research in nonlinear dynamics are encouraged to take this course. Specific topics will be chosed to illustrate the theory and use of techniques from global analysis and nonlinear dynamics such as (1) discrete dynamical systems, (2) global theory of ordinary differential equations, (3) Hamiltonian systems, (4) KAM theory, (5) bifurcation and stability, (6) Hopf index theory of vector fields, (7) Morse theory of gradient vector fields, (8) Lyapunov theory, (9) infinite dimensional dynamics systems, (10) structural stability. Prerequisite: Consent of instructor. 4 hours.

MATH 553. Partial Differential Equations Basic introduction to the study of partial differential equations; topics include the Cauchy problem, power-series methods, characteristics, classification, canonical forms, well-posed problems, Riemann's method for hyperbolic equations, the Goursat problem, the wave equation, Sturm Liouville problems and separation of variables, Fourier series, the heat equation, integral transforms, Laplace's equation, harmonic functions, potential theory, the Dirichlet and Neumann problems, and Green's functions. Prere quisite: Consent of instructor. 4 hours.

MATH 556. Methods of Math Physics I Course covers several basic mathematical methods of wide use in physics and engineering. Topics will be selected from the following: calculus of variations, Sturm-Liouville theory and eigenvalue problems, Green's functions and generalized functions, Hilbert space techniques. Prerequisite: Advanced calculus. 4 hours.

MATH 557. Methods of Math Physics II Course covers several basic mathematical methods of wide use in physics and engineering. Topics will be selected from the following: integral equations, spectral theory and Hilbert spaces, inverse spectral theory, soliton and waterwave theory, asymptotic methods. Prerequisite: MATH 556 or con sent of instructor. 4 hours.

MATH 587. Optimization by Vector Space Methods Same as ECE 580. Introduction to normed, Banach, and Hilbert spaces; applications of the projection theorem and the Hahn-Banach Theorem to problems of minimum norm, least squares estimation, mathematical programming, and optimal control; the Kuhn-Tucker Theorem and Pontryagin's maximum principle; and introduction to iterative methods. Prerequisite: MATH 415 or 482, and MATH 447 or consent of instructor. 4 hours.