Suzanne Lenhart
University of Tennessee at Knoxville
1. Monday morning: Introduction to optimal control of ordinary differential equations Monday afternoon: Short presentation about numerical solutions and an activity with a demonstration MATLAB code
2. Tuesday afternoon: Brief presentation about optimal control of systems and an activity with a demonstration MATLAB code
3. Wednesday morning: Illustration of ODE problems which are linear in the control and the beginning of optimal control of parabolic partial differential equations
4. Thursday afternoon: Optimal control of systems of parabolic PDEs
5. Friday morning: Applications of optimal control in fishery models
Reference: S. Lenhart and J. T. Workman, Optimal Control Applied to
Biological Models, Chapman and Hall/ CRC Press, 2007.
Note that the first two chapters of this book will be made available. Several MATLAB codes and their explanations will be provided. Some exercises will be worked during this workshop.
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Introduction to optimal control and Hamilton-Jacobi equations
Khai T. Nguyen
NCSU Mathematics Department, USA
Course Objectives/Goals: The goal of this course is to provide an introduction to optimal control and the basic theory of viscosity solutions for first-order Hamilton-Jacobi Equations.
Several examples will be given in order to motivate and introduce the main problems, to illustrate and explain the critical parts of the proofs.
Specific goals include:
- Elementary introduction to the basic problems in dynamic optimization: both in finite and infinite time horizon
- Necessary and sufficient conditions in optimal control
- Viscosity solutions: definition, existence, stability properties, and comparison principle
Student Learning Outcomes: By the end of this course, the students expect to know standard problems in optimal control, Hop-Lax formula, both Dynamic Programming Principle and Pontryagin maximum principle, and understand the concept, basic ideas and theory of viscosity solutions for first-order Hamilton-Jacobi equations.
Course outline:
I. Introduction
– Ordinary differential equations and control dynamics
– Standard optimal control problems
– Existence of optimal open-loop control
– Hop-Lax formula
II. Necessary and sufficient conditions
– Bang-Bang principle
– The Pontryagin maximum principle
– Dynamic programming principle and HJ equations
– Recovering the optimal control problem from the value function
III. Viscosity solutions
– The method of characteristics
– Generalized differentials
– Stability properties
– Comparison principle
– Perron’s method
Recommended notes or books:
- Introduction to the mathematical theory of control, Alberto Bressan
- Lecture note on viscosity solutions and optimal control problems, Khai T. Nguyen
- Hamilton–Jacobi Equations: Theory and Applications, Hung V. Tran
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Introduction to kinetic equations for waves
Minh-Binh Tran
Texas A&M University
Course Objectives: This course introduces mathematical techniques based on kinetic equations
that have been developed to analyze the asymptotic behavior of waves and particles
propagating in a heterogeneous medium.
Specific goals include:
- Introduction to the kinetic theory of linear waves
- Basic understanding of Wigner transform
- Different types of kinetic limits
Outlines:
I. The Wigner transform.
1. The basic properties of the Wigner transform.
2. The semiclassical Wigner transform
3. The semiclassical operators
II. The evolution of the Wigner transform
1. The Liouville equation and geometric optics
2. Wigner transforms of mixtures of states
III. Kinetic limits for the Liouville equations.
1. The Fokker-Planck limit.
2. Radiative transport regime for the Schrodinger equation.
Recommended references:
[1] Bal, Guillaume, Tomasz Komorowski, and Lenya Ryzhik. "Kinetic limits for waves in a
random medium." Kinet. Relat. Models 3.4 (2010): 529-644.
[2] Amirali Hannani, Minh Nhat Phung, Minh-Binh Tran, Emmanuel Trelat. Controlling the Rates
of a Chain of Harmonic Oscillators with a Point Langevin Thermostat 2024
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