Mathematical and Information Sciences for Industry:

Mathematical Control and Information (MCI) Track

*(Tentative Proposal)*

- Robert Grossman
- Floyd B. Hanson
- T.E.S. Raghavan
- Stephen S. Yau

We are currently developing an innovative, interdisciplinary program in
control, information, and optimization. We already have a good number of
students doing thesis research in these areas. In fact, at least ten students
have earned Ph.D.s in Control Theory, Filtering Theory, and Game Theory,
supervised by one of us. In addition, we have a well-supported, highly
innovative research program in these areas already within the department.
Although we have had a number of students crossing areas to take control,
optimization, and information courses, we need to give these areas more
structure and cohesion in order to optimize the benefit of our training. Our
informal program has reached a critical mass for going to a more formal and
vital structure. Thus, we need a formal program in *Mathematical Control and
Information (MCI)*.

A formal program in mathematical control, information, and optimization would allow us to develop a strong, integrated graduate program in these areas. In addition, a formal program would help graduates of the program obtain jobs more easily in industry, government, and academia. Control, information, and optimization enables many products to become technically advanced in spite of the great complexity and uncertainty of the tasks that these products can accomplish. Unfortunately, many of the applications of control, information, and optimization are foreign, rather that domestic, implementations. More effort needs to be made to bring mathematical and computational advances rapidly into our graduate programs, justified by current implementation, national trends, and the great potential that these advances offer.

Much control, information, and optimization theory and practice is carried out in engineering departments. Having a separate program identity would make it easier to collaborate on collateral courses, which our program would depend upon. In addition, there should be a rich exchange of students between our program and the ones in engineering. There are also faculty in other departments with whom there could be fruitful intersections. Our courses could, in general, deal more in depth with the mathematical or computational aspects of control, information, and optimization. Thus, our program would complement those in engineering and related departments. Professor D. Graupe of EECS has said that our program has his and his department head's (Professor W. K. Chen) support. Dr. Graupe also gave us some helpful advice to improve our program, suggesting that we form a center for control.

Finally, having an official program in *Mathematical Control and
Information* would supply a strong argument for getting research funds,
since many granting or contracting agencies either expect explicitly or
implicitly expect a serious graduate program in the areas as a condition for
awarding funds.

The core consists of 6 courses:

**Control**Choose two:

- Math 574 (MCI 501): Applied Optimal Control
- MCI 503: Linear and Nonlinear Control I
- MCI 525: Linear and Nonlinear Filtering I

- Math 574 (MCI 501): Applied Optimal Control
**Information Theory**- MCI 550: Information Theory
- MCI 527: Wavelet Signal Processing

- MCI 550: Information Theory
**Optimization**- Stat 471 (MCI 471): Linear and Nonlinear Programming
- Stat 473 (MCI 473): Game Theory

- Stat 471 (MCI 471): Linear and Nonlinear Programming

- Charles Tier, MSCS; Telecommunications and Computer Performance
Evaluation.
- Charles Lin, MSCS; Wavelet Analysis.
- David Boyce, Urban Transportation Center; Traffic Control Models,
Computational Control.
- Daniel Graupe, EECS; Signal Processing, Adaptive and Stochastic
Control, Identification and Adaptive Filtering.
- William D. O'Neill, EECS; Information Theory Models.
- C.K. Sanathanan, EECS; Automatic Control Theory, Modeling, Controller
Design and Simulation of Large-Scale Systems.
- Stanley Pliska, Finance/IDS; Stochastic Financial Models.
- Ahmed A. Shabana, ME; Multi-Body Dymanics, Computational Control.

- Siu Leung Chung,
*Supercomputing Optimization for Stochastic Dynamic Programming*, From Fall 1991, on faculty of University of Toledo. From Summer 1992, a Lecturer (British equivalent of Assistant Professor) in the Department of Information Systems and Computer Science, National University of Singapore. - Wen-Lin Chiou,
*Some Results on Nonlinear Filtering Theory*, Ph.D. Thesis (Chair: S.S. Yau), MSCS, 25 June 1991. - Hsiao-Chih Ru,
*Shapley Value for Multichoice Games*, Ph.D. Thesis (Chair: T.E.S. Raghavan), MSCS, June 1991. Presently, Associate Professor, Soochow University, Taiwan. - Huihuang Xu,
*Data Parallel Methods in Large Scale Scientific Computation*, Ph.D. Thesis (Chair: F.B. Hanson; Co-Advisor: C. Tier), MSCS, 4 June 1992. From Fall 1992, in Networking Research at US Robotics. - Dennis J. Jarvis,
*Performance and Applications of Multiprocessor Systems*, Ph.D. Thesis (Chair: F.B. Hanson; Co-Advisor: C. Tier), MSCS, 17 June 1992. From Fall 1992, on postdoctoral appointment in Computational Control, Applied Mathematics Division at Brown University. - Kumarss Naimipour,
*Numerical Convergence for the Bellman Equation of Stochastic Optimal Control with Quadratic Costs and Constraints*, Ph.D. Thesis (Chair: F.B. Hanson), MSCS 19 June 1992. Presently, Tenured Associate Professor at Northeastern Illinois University. - Christopher J. Pratico,
*I/O View: A Multi-State Visualization Tool Using Inner and Outer Worlds*, Masters Project (Advisor: T.A. Defanti; Secondary Advisor: F.B. Hanson), EECS, 21 June 1992. Presently, on staff of Sarnoff Research Center in NJ. - Chi-Wah Leung,
*Classification of Finite Dimensional Maximal Rank Estimation Algebras with State Space Dimension 3*, Ph.D. Thesis (Chair: S.S. Yau), MSCS, 10 December 1992. Presently, Associate Professor in the Department of Mathematics at National Central University, Chung-Li, Taiwan, ROC. - Tamas Solymosi,
*Algorithm for Computing the Nucleolus of Cooperative Games*, Ph.S. Thesis (Chair: T.E.S. Raghavan), MSCS, June 1993. - Jie Chen,
*The Classification of Estimation Algebras for Nonlinear Filtering Problems*, Ph.D. Thesis (Chair: S.S.T. Yau), MSCS, 4 Match 1994. - Raghib Abu-Saris,
*Filtering of a Wiener-Poisson Driven Stochastic Process*, Ph.D. Thesis (Chair: F.B. Hanson), MSCS, 20 June 1994. - Lixing Jia,
*Modified Sequential Unconstrained Minimization Technique Methods for Constrained Optimization*.## Abbreviated Graduate Handbook

## for Mathematical Control and Information Track

### The Master's Degree Programs

**Mathematical Control and Information.**The M.S. degree with a concentration in Mathematical Control and Information is designed for students who have a bachelor's degree in mathematics, computer science, engineering, or in the physical sciences and have a good background in undergraduate mathematics.The MCI core course requirement consists of the following, selecting 6 or 7 courses (24 hours):

- Choose two of the following three courses:
- MCI 525 (EECS 550) Linear and Nonlinear Filtering I
- MCI 501 (Math 574) Applied Optimal Control
- MCI 503 Linear and Nonlinear Control I

- MCI 525 (EECS 550) Linear and Nonlinear Filtering I
- MCI 527 Wavelet Signal Processing
- MCI 550 Information Theory
- MCI 471 (Stat 471) Linear and Nonlinear Programming
- MCI 473 (Stat 473) Game Theory

In order to get the M.S. degree in Industrial Mathematics, the student has either to write a Master Thesis or take a written Master's Examination.

The written Master's Examination has 14 questions and is based on the following courses:

- MCI 471 (Stat 471) Linear and Nonlinear Programming 2 questions
- MCI 473 (Stat 473) Game Theory 2 questions
- MCI 501 (Math 574) Applied Optimal Control 2 questions
- MCI 503 Linear and Nonlinear Control I 2 questions
- MCI 527 Wavelet Signal Processing 2 questions
- MCI 550 Information Theory 2 questions

### The Doctoral Degree Programs

**The Doctoral preliminary examination.**The Doctoral preliminary examination consists of two written prelims, each of which is in a major subject area. These subject must be in different clusters (subject areas are grouped in clusters). The grades of the written part of the preliminary examination are the grades for these two written prelims. There are four prelim clusters. Students may prepare for the written prelims in the Mathematical Control, Information and Optimization Program by taking the indicated courses.- Linear and Nonlinear Control Theory
- MCI 503 Linear and Nonlinear Control I
- MCI 504 Linear and Nonlinear Control II
- MCI 525 Linear and Nonlinear Filtering I
- MCI 526 Linear and Nonlinear Filtering II

- MCI 503 Linear and Nonlinear Control I
- Applied Control Methods
- MCI 501 (Math 574) Applied Optimal Control
- MCI 521 Applied Stochastic Control
- MCI 585 Computational Control

- MCI 501 (Math 574) Applied Optimal Control
- Information Theory
- MCI 537 Wavelet Signal Processing
- MCI 563 Information Theory
- MCS 531 Error-Correcting-Codes

- MCI 537 Wavelet Signal Processing
- Optimization
- MCI 571 (Stat 571) Non-Cooperative Games
- MCI 572 (Stat 572) Cooperative Game Theory
- MCI 575 (Stat 575) Optimization Methods in Matrices

## Course Descriptions:

### MCS 501Applied Optimal Control

**Catalog Description**Same as Math 574, Applied Optimal Control. Introduction to optimal control theory, calculus of vatiations, maximum principle, dynamic programming, feedback control, linear systems with quadratic criteria, singular control, optimal filtering, stochastic control.**Prerequisites**Graduate standing and Math 411, Advanced Calculus II, or Math 427, Analysis in Several Variables, or consent of the instructor.**List of Topics**- Introduction to applied optimal control theory.
- Calculus of variations.
- The maximum principle.
- Optimal feedback control.
- Linear systems with quadratic criteria.
- Dynamic programming.
- Singular control.
- Optimal filtering.
- Stochastic control.

- Introduction to applied optimal control theory.
**Texts**- A.E. Bryson, Jr., and Y.-C. Ho, Applied Optimal Control, Wiley, 1975
- D.E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall, 1970

- A.E. Bryson, Jr., and Y.-C. Ho, Applied Optimal Control, Wiley, 1975

### MCI 503 Linear Control I 4 Hours

**Catalog Description**Mathematics of dynamic processes, characterization of systems, stability analysis, controllability, observability, canonical forms, realization, estimation, and design.**Prerequisite**Graduate standing and Math 325, Linear Algebra II, and ordinary differential equation, or consent of instructor.**List of Topics**- Mathematics of dynamic processes: solution of ordinary differential
equations, solution of difference equations.
- Characterization of systems: The concept of dynamic systems, equilibrium,
and linearization, continuous linear systems, Discrete systems, Applications.
- Stability analysis: The elements of the Lyapunov stability theory, the
stability of time-invariant linear systems, BIBO stability, Applications.
- Controllability: Continuous systems, Discrete system, Applications.
- Observability: Continuous systems, Discrete system, Duality,
Applications.
- Canonical forms: Controllability canonical forms, Observability canonical
forms, Applications.
- Realization: Realizability of weighting patterns, Realizability of transfer
functions, Applications.
- Estimation and design: The eigenvalue placement theorem, observers,
reduced-order observers, The eigenvalue separation theorem, Applications.

- Mathematics of dynamic processes: solution of ordinary differential
equations, solution of difference equations.

### MCI 504 Linear and Nonlinear Control II 4 Hours

**Catalog Description**Local decomposition of control systems, global decompositions of control systems, input-output maps, realization theory, elementary theory of nonlinear feedback for multi-input multi-output systems, geometric theory of state feedback.**Prerequisite**Graduate standing and MCI 503, Linear and Nonlinear Control I, or EECS 550, Linear Systems Theory and Designs, or consent of instructor.**List of Topics**- Local Decompositions of Control Systems: Distributions, Frobenius
theorem, differential geometric point of view, local reachability, local
observability.
- Global Decompositions of control systems: Sussmann's theorem, control Lie
algebra, observation space.
- Input-Output Maps and Realization Theory: Fliess functional expansions,
Volterra series expansions, output invariance, realization theory.
- Elementary Theory of Nonlinear Feedback for Multi-Input Multi-Output
Systems: Exact linearization via feedback, noninteracting control, exact
linearization of the input-output response.
- Geometric Theory of State Feedback: Zero dynamics, controlled invariant
distributions, controllability distribution, asymptotic stabilization via state
feedback.

- Local Decompositions of Control Systems: Distributions, Frobenius
theorem, differential geometric point of view, local reachability, local
observability.
**Text**- A. Isidori, Nonlinear control systems, Springer-Verlag, New York.

- A. Isidori, Nonlinear control systems, Springer-Verlag, New York.

### MCI 525 Linear and Non-linear Filtering I 4 Hours

**Catalog Description**Probability Theory and Random Variables, Stochastic Processes, Stochastic Differential Equations, Introduction to Filtering Theory, Nonlinear Filtering Theory.**Prerequisite**Graduate standing and Math 325, Linear Algebra II, and Math 411, Advanced Calculus II, and Ordinary differential equation, or consent of instructor.**List of Topics**- Probability Theory and Random Variables: Probability axioms, Random
variables, Jointly distributed random variables, Conditioned probabilities and
expectations, Properties of Gaussian random variables.
- Stochastic Processes: Probability law of a stochastic process, convergence
of random sequences, mean square calculus, Independence, Conditioning, the
Brownian motion processes, Gaussian processes, Markov processes, White Noise.
- Stochastic Differential Equations: Itô Stochastic Integral,
Stochastic Differential Equations, Itô stochastic calculus, Stochastic
integral of Stratonovich, Kolmogorov's equation.
- Introduction to filtering theory: Probabilistic approach, statistical
method.
- Nonlinear Filtering Theory: Discrete Filtering, Continuous Filtering,
Evolution of the conditional density, Evolution of moments.

- Probability Theory and Random Variables: Probability axioms, Random
variables, Jointly distributed random variables, Conditioned probabilities and
expectations, Properties of Gaussian random variables.
**Text**- A. Jazwinski, Stochastic Processes and Filtering Theory, 1970, Academic
Press

- A. Jazwinski, Stochastic Processes and Filtering Theory, 1970, Academic
Press

### MCI 526 Linear and Non-linear Filtering II 4 Hours

**Catalog Description**Duncan-Mortensen-Zakai equation, Finte dimensional filter, Linear filtering, Nonlinear filtering, Numerical method, Applications**Prerequisites**Graduate standing and Math 325, Linear Algebra II, and Math 411, Advanced Calculus II, and Ordinary differential equation, or consent of instructor.**List of Topics**- Duncan-Mortensen-Zakai equation: Giranov's Theorem, The Innovation
Process, The Duncan-Mortensen-Zakai equation.
- Finite Dimensional Filters: Estimation algebra, Vector fields, Homomorphism
principle, Wei-Norman approach.
- Linear filtering: Kalman-Bucy filter with Gaussian initial condition,
Linear filtering with Non-Gaussian initial condition.
- Non-linear filtering: Benes filter, Yau filter, Classification of finite
dimensional estimation algebras, Direct method.
- Numerical method: Kolmogorov equation, system of ordinary differential
equations.
- Applications: Applications of filtering theory and modeling techniques,
Free flight and powered flight navigation, error analysis and sub-optimal
modeling.

- Duncan-Mortensen-Zakai equation: Giranov's Theorem, The Innovation
Process, The Duncan-Mortensen-Zakai equation.
**Texts**- K. Bucy and P. Joseph, Filtering for Stochastic Processes with Applications to Guidance, Chelsea Publishing Company, New York, NY, 1987
- lecture notes.

### MCI 511 Applied Stochastic Control 4 Hours

**Catalog Description**Introduction to stochastic optimal control in continuous time. Discounted performance models. The applied derivation of the Bellman partial differential equation of stochastic dynamic programming. Applications. Numerical considerations and the curse of dimensionality.**Prerequisites**Graduate standing and MCI 525, Linear and Non-Linear Filtering I, or consent of the instructor.**List of Topics**- Introduction to Stochastic Optimal Control.
- Markov Assumptions for Gaussian and Poisson Noise.
- Discounted Performance Models.
- Derivation of Bellman Equation.
- Stochastic Control Considerations.

- Introduction to Stochastic Optimal Control.
**Texts**- R.F. Stengel, Stochastic Optimal Control: Thoery and Application, Wiley, 1986
- lecture notes.

### MCI 530 Computational Control 4 Hours

**Catalog Description**Numerical techniques for solving control problems. Deterministic control, stochastic control. Differential equation solution method, probabilistic methods, and simulations. Treatment of boundary conditions.**Prerequisites**Graduate standing and MCS 481, Numerical Analysis, and MCI 511, Applied Stochastic Control, or consent of the instructor.**List of Topics**- Linear Algebra Computations.
- Feedback Stabilization Problems.
- Pole Assignment Problems.
- Lyapunov Equations.
- Riccati Equations.
- Signal Processing Applications.
- Methods for General Linear Control Systems.
- Methods for Nonlinear Control Systems.

- Linear Algebra Computations.
**Text**- Lecture Notes

### MCI 550 Information Theory 4 Hours

**Catalog Description**Discrete and continuous sources, Discrete and continuous channels, data translation, compaction, transmission and compression codes, estimation, multi-terminal information networks, performance bounds, and Shannon's Thoery.**Prerequisites**Graduate standing and Math 411, Advanced Calculus II, or consent of instructor.**List of Topics**- Discrete Sources and Channels.
- Continuous Sources and Channels.
- Data Translation.
- Data Compression.
- Estimation.
- Multi-Terminal Information Networks.
- Performance Bounds.
- Shannon's Information Theory.

- Discrete Sources and Channels.
**Text**- Richard Blahut, Principles and Practice of Information Theory,
Addison-Wesley

- Richard Blahut, Principles and Practice of Information Theory,
Addison-Wesley

### MCI 527 Wavelet Signal Processing 4 Hours

**Catalog Description**Fourier analysis, continuous wavelet transform, discrete wavelet transforms, copmactly supported wavelets, the z-transform and its application in signal processing, correlation and convolution, A framework for digital filter design, Finite impulse response (FIR) filter design, Design of infinite impulse response (IIR) digital filters, Multirate digital signal processing.**Prerequisites**Graduate standing and Math 325, Linear Algebra II, or consent of instructor.**List of Topics**- Continuous and Discrete Fourier transforms.
- Continuous and Discrete Wavelet transforms.
- Compactly supported wavelets.
- The z-transform and its application in signal processing.
- Correlation and convolution.
- A framework for digital filter design.
- Finite impulse response (FIR) filter design.
- Design of infinite impulse response (IIR) digital filters.
- Multirate digital signal processing.

- Continuous and Discrete Fourier transforms.
**Text**- E.C. Ifeachor and B.W. Jervis, Digital Signal Processing, A Practical
Approach, 1993, Addison-Wesley.

- E.C. Ifeachor and B.W. Jervis, Digital Signal Processing, A Practical
Approach, 1993, Addison-Wesley.

### MCI 571 Non-Cooperative Games 4 Hours

**Catalog Description**Same as Stat 571, Non-Cooperative Games. Extensive games. Separation and fixed point theorems. General minimax theorems. Nash equilibria. War Duels. Completely mixed games. Games with convex payoff. Stochastic games.**Prerequisites**Graduate standing and Stat 461, Applied Probability Models I, or Mth 411, Advanced Calculus.**List of Topics**- Extensive Form and Information Pattern.
- Perfect Recall and Kuhn's Theorem.
- Convex Sets and Separation Theorem.
- Fixed Point Theorems.
- General Minimax Theorems.
- Structure of Optimal Strategies.
- Games on the Unit Square.
- Bohnenblust, Karlin, Shapley Theorem.
- War Duels.
- Nash Equilibria and Classification.
- Complementary Pivoting and Lemke-Howson Algorithm.
- Stochastic Games and Algorithm.
- Games and Statistical Decisions.
- Exams and leeway.

- Extensive Form and Information Pattern.
**Texts**- G. Owen, Game Theory, Academic Press (2nd Edition), 1985;
- T. Parthasarathy and T.E.S. Raghavan, Some Topics in Two-Person Games,
American Elsevier, 1971 (reference).
- Research papers and journals and class lectures only.

### MCI 572 Cooperative Game Theory 4 Hours

**Catalog Description**Same as Stat 572, Cooperative Game Theory. Utility Theory. Games with side payments, stable sets, core, bargaining sets, Shapley value, Nucleolus. Market games. NTU value. Multilinear extensions, non-atomic games.**Prerequisites**Graduate standing and Stat 571, Non-Cooperative Games, or consent of the instructor.**List of Topics**- Utility and Preferences.
- Characteristic Function for TU Games.
- Core, Bargaining Sets for Special Games.
- Shapley Value and Axiomatic Approaches.
- Multilinear Extensions.
- Nucleolus and Applications to Ricardean Economies.
- Core and the Competitive Equilibrium.
- Axiomatic Bargaining of Nash and Harsanyi.
- Market Games.
- NTU Value.
- Non-Atomic Games and the Core Equivalence Theorem.
- Exams and leeway.

- Utility and Preferences.
**Texts**- G. Owen, Games Theory, Academic Press, 1986.
- R.J. Aumann and L.S. shapley, Non Atomic Games, Princeton Univ. Press, 1974.
- A. Roth, Axiomatic Bargaining, Springer Lecture Notes, 1979.

### MCI 590 Advanced Topics in Control Theory 4 Hours

**Catalog Description**Topics from areas such as: H-infinity Control, Perturbations of Control Systems, Adaptive control, Parallel Computational Control, Optimal Filtering, Hybrid control systems, and cross-disciplinary approaches to control problems using algebraic, applied, computational, database, game-theoretic, geometrical, or statistical analysis.**Prerequisites**Graduate standing. Consent of the instructor.

### MCI 591 Advanced Topics in Information Theory 4 Hours

**Catalog Description**Topics from areas such as Adaptive Signal Processing, Telecommunication, Communication, Algorithmic information theory, multimedia communication, control theory for queues, distributed system algorithms, or probabilistic computation algorithms.**Prerequisites**Graduate standing. Consent of the instructor.

### MCI 592 Advanced Topics in Optimization 4 Hours

**Catalog Description**Topics from areas such as advanced game theory, nonsmooth analysis, variational inequalities, nonlinear optimization, computational problems in optimization and differential inclusions.**Prerequisites**Graduate standing. Consent of the instructor.

### MCI 593 Graduate Student Seminar I 1 Hour

**Catalog Description**Graduate student seminar. For graduate students who wish to receive credit for participating in a learning seminar whose weekly time commitment is not sufficient for a reading course. This seminar must be sponsored by a faculty member.**Prerequisites**Graduate standing**List of Topics**- Graduate Student Seminar.

- Graduate Student Seminar.
**Text**- Current literature.

- Current literature.

### MCI 595 Research Seminar 1 Hour

**Catalog Description**Research seminar. Current developments in research with presentations by faculty, students, and visitors.**Prerequisites**Graduate standing.**List of Topics**- Research seminar.

- Research seminar.
**Text**- Current literature.

- Current literature.

### MCI 598 Master's Thesis Research 0-16 Hours

**Catalog Description**Master's Thesis research. Satisfactory/unsatisfactory grade only. Research work under the supervision of a faculty member leading to the completion of a master's thesis.**Prerequisites**Graduate standing and approval of the department.**List of topics**- Individual research.

- Individual research.
**Text**- Current literature.

- Current literature.

### MCI 599 Thesis Research 0-16 Hours

**Catalog Description**Thesis Research. May be repeated for credit. Students may register for more than one section per term. Satisfactory/unsatisfactory grade only. Required for Ph.D.

- G. Owen, Games Theory, Academic Press, 1986.

- MCI 571 (Stat 571) Non-Cooperative Games

- Choose two of the following three courses: