**Astrodynamics Graduate Courses **

(August 2004)

**FACULTY **

**Bruce A. Conway**, Professor

**Victoria L. Coverstone**, Associate Professor

**Natasha Neogi**, Assistant Professor

**John E. Prussing**, Professor

AE COURSES

**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.

**ENGINEERING COLLEGE COURSES**

**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.

**ASTRONOMY COURSES**

**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.

**MATHEMATICS COURSES**

**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.