The course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). We will consider optimal control of a dynamical system over both a finite and an infinite number of stages. This includes systems with finite or infinite state spaces, as well as perfectly or imperfectly observed systems. We will also discuss approximation methods for problems involving large state spaces. Applications of dynamic programming in a variety of fields will be covered in recitations.

The course addresses dynamic systems, i.e., systems that evolve with time. Typically these systems have inputs and outputs; it is of interest to understand how the input affects the output (or, vice-versa, what inputs should be given to generate a desired output). In particular, we will concentrate on systems that can be modeled by Ordinary Differential Equations (ODEs), and that satisfy certain linearity and time-invariance conditions. We will analyze the response of these systems to inputs and initial conditions. It is of particular interest to analyze systems obtained as interconnections (e.g., feedback) of two or more other systems. We will learn how to design (control) systems that ensure desirable properties (e.g., stability, performance) of the interconnection with a given dynamic system.

Momentum principles and energy principles. Lagrange's equations, Hamilton's principle. Applications to mechanical systems including gyroscopic effects. Study of steady motions and nature of small deviations therefrom. Natural modes and natural frequencies for continuous and lumped parameter systems. Forced vibrations. Dynamic stability theory. Causes of instability. This course reviews momentum and energy principles, and then covers the following topics: Hamilton's principle and Lagrange's equations; three-dimensional kinematics and dynamics of rigid bodies; steady motions and small deviations therefrom, gyroscopic effects, and causes of instability; free and forced vibrations of lumped-parameter and continuous systems; nonlinear oscillations and the phase plane; nonholonomic systems; and an introduction to wave propagation in continuous systems. This course was originally developed by Professor T. Akylas.

This class is an introduction to the dynamics and vibrations of lumped-parameter models of mechanical systems. Topics include kinematics; force-momentum formulation for systems of particles and rigid bodies in planar motion; work-energy concepts; virtual displacements and virtual work; Lagrange's equations for systems of particles and rigid bodies in planar motion; linearization of equations of motion; linear stability analysis of mechanical systems; free and forced vibration of linear multi-degree of freedom models of mechanical systems; and matrix eigenvalue problems. The class includes an introduction to numerical methods and using MATLABĺ¨ to solve dynamics and vibrations problems.

Upon successful completion of this course, students will be able to: * Create lumped parameter models (expressed as ODEs) of simple dynamic systems in the electrical and mechanical energy domains * Make quantitative estimates of model parameters from experimental measurements * Obtain the time-domain response of linear systems to initial conditions and/or common forcing functions (specifically; impulse, step and ramp input) by both analytical and computational methods * Obtain the frequency-domain response of linear systems to sinusoidal inputs * Compensate the transient response of dynamic systems using feedback techniques * Design, implement and test an active control system to achieve a desired performance measureMastery of these topics will be assessed via homework, quizzes/exams, and lab assignments.

Introduction to dynamics and vibration of lumped-parameter models of mechanical systems. Three-dimensional particle kinematics. Force-momentum formulation for systems of particles and for rigid bodies (direct method). Newton-Euler equations. Work-enery (variational) formulation for systems particles and for rigid bodies (indirect method). Virtual displacements and work. Lagrange's equations for systems of particles and for rigid bodies. Linearization of equations of motion. Linear stability analysis of mechanical systems. Free and forced vibration of linear damped lumped parameter multi-degree of freedom models of mechanical systems. Application to the design of ocean and civil engineering structures such as tension leg platforms.

An introduction to theoretical studies of systems of many interacting components, the individual dynamics of which may be simple, but the collective dynamics of which are often nonlinear and analytically intractable. Topics vary from year to year. Format includes both pedagogical lectures and round-table reviews of current literature. Subjects of interest include: problems in natural science (e.g., geology, ecology, and biology) where quantitative theory is still in development; problems in physics, such as turbulence, that demonstrate powerful concepts such as scaling and universality; and modern computational methods for the simulation and study of such problems. Discussions in context of contemporary experimental or observational data.

An introduction to theoretical studies of systems of many interacting components, the individual dynamics of which may be simple, but the collective dynamics of which are often nonlinear and analytically intractable. Topics vary from year to year. Format includes both pedagogical lectures and round-table reviews of current literature. Subjects of interest include: problems in natural science (e.g., geology, ecology, and biology) where quantitative theory is still in development; problems in physics, such as turbulence, that demonstrate powerful concepts such as scaling and universality; and modern computational methods for the simulation and study of such problems. Discussions in context of contemporary experimental or observational data.

An introduction to theoretical studies of systems of many interacting components, the individual dynamics of which may be simple, but the collective dynamics of which are often nonlinear and analytically intractable. Topics vary from year to year. Format includes both pedagogical lectures and round-table reviews of current literature. Subjects of interest include: problems in natural science (e.g., geology, ecology, and biology) where quantitative theory is still in development; problems in physics, such as turbulence, that demonstrate powerful concepts such as scaling and universality; and modern computational methods for the simulation and study of such problems. Discussions in context of contemporary experimental or observational data.

Introduction to nonlinear deterministic dynamical systems. Nonlinear ordinary differential equations. Planar autonomous systems. Fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma. Stability of equilibria by Lyapunov's first and second methods. Feedback linearization. Application to nonlinear circuits and control systems. Alternate years. Description from course website: This course provides an introduction to nonlinear deterministic dynamical systems. Topics covered include: nonlinear ordinary differential equations; planar autonomous systems; fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma; stability of equilibria by Lyapunov's first and second methods; feedback linearization; and application to nonlinear circuits and control systems.

This course begins with a study of the role of dynamics in the general physics of the atmosphere, the consideration of the differences between modeling and approximation, and the observed large-scale phenomenology of the atmosphere. Only then are the basic equations derived in rigorous manner. The equations are then applied to important problems and methodologies in meteorology and climate, with discussions of the history of the topics where appropriate. Problems include the Hadley circulation and its role in the general circulation, atmospheric waves including gravity and Rossby waves and their interaction with the mean flow, with specific applications to the stratospheric quasi-biennial oscillation, tides, the super-rotation of Venus' atmosphere, the generation of atmospheric turbulence, and stationary waves among other problems. The quasi-geostrophic approximation is derived, and the resulting equations are used to examine the hydrodynamic stability of the circulation with applications ranging from convective adjustment to climate.

Examines the long term effects of information technology on business strategy in the real estate and construction industry. Considerations include: supply chain, allocation of risk, impact on contract obligations and security, trends toward consolidation, and the convergence of information transparency and personal effectiveness. Resources are drawn from the world of dot.com entrepreneurship and "old economy" responses. Taught by case study method and grading is based on class participation and papers.

This course examines European music from the early Middle Ages until the end of the Renaissance. It includes a chronological survey and intensive study of three topics: chant and its development, music in Italy 1340-1420, and music in Elizabethan England. Instruction focuses on methods and pitfalls in studying music of the distant past. Students' papers, problem sets, and presentations explore lives, genres, and works in depth. Works are studied in facsimile of original notation, and from original manuscripts at MIT, where possible.

If you are an entrepreneur, one of your priorities, in addition to building your company, is ensuring you have enough money at the right times. Early Stage Capital will consider a broad range of questions that entrepreneurs deal with on this front, including the following: What should your strategy and your priorities be in raising early stage capital? What are the market norms and standards in structuring VC deals? What are the critical negotiating strategies and tactics? How will your company be valued? How can you obtain the optimal valuation for your new venture? What are the critical elements in the relationship between venture capitalists and entrepreneurs? How is the "venture model" evolving? Is it broken? What is the impact of Super Angels and micro VCs? These are key questions that face all entrepreneurs in 2010, particularly first-time entrepreneurs. This course aims to prepare you for these decisions, as either a potential entrepreneur or venture capitalist. Using live interactions with leading figures in the venture finance community, most of the class sessions will analyze fundamental strategies of the venture-capital investment process and the critical importance of the relationship between entrepreneur and investor. As well, we will have a tactical focus on demystifying the legalities and jargon of the term sheet and the "A round" financing process. Significantly for 2010, we will also frequently consider the rapid and arguably fundamental change in VC today as the "lean startup" model threatens much of the traditional role and value of the venture investor. Disclaimer: The websites for this course and the materials they offer are provided for educational use only. They are not a substitute for the advice of an attorney and no attorney-client relationship is created by using them. All materials are provided "as-is", without any express or implied warranties.

The Early Universe provides an introduction to modern cosmology. The first part of the course deals with the classical cosmology, and later part with modern particle physics and its recent impact on cosmology.

Examines traditional forms of East Asian culture (including literature, art, performance, food, and religion) as well as contemporary forms of popular culture (film, pop music, karaoke, and manga). Covers China, Japan, Korea, Taiwan, and Hong Kong, with an emphasis on China. Attention given to women's culture. The influence and presence of Asian cultural expressions in the US are also considered. Use made of resources in the Boston area, including the MFA, the Children's Museum, and the Sackler collection at Harvard. Taught in English.

This lesson is about the flow of energy in ecosystems. The setting is Plimoth Plantation, a living history museum in Plymouth, Massachusetts, USA, where students will learn about the first Thanksgiving meal in America, celebrated in 1621 by early American settlers and Wampanoag Indians. By examining this meal and comparing it to a modern day Thanksgiving celebration, students will be able to explore the way in which food energy moves and is transformed in an ecosystem. The learning goals focus on the movement of energy from one feeding level to the next within a food web, the way in which energy changes form, and the inefficiency of energy transfer, which in turn affects the availability of food energy for organisms at the highest feeding level. The lesson is directed at high school level biology students. Students should be familiar already with food webs, food chains, and trophic (feeding) levels. They should also be familiar with the general equations for photosynthesis (CO2 + H2O => C6H12O6) and cell respiration (C6H12O6 => CO2 + H2O), and understand the basic purpose of these processes in nature. This lesson can be completed during one long classroom period, or can be divided over two or more class meetings. The duration of the lesson will depend on prior knowledge of the students and on the amount of time allotted for student discussion. There are no supplies required for this lesson other than the downloadable worksheets (accessed on this BLOSSOMS site), paper and some glue or tape.

The major goal of this lesson is to provide students with some of the tools they will need to analyze and solve the many complex problems they will face during their lifetimes. In the lesson, students learn to use Flow Charts and Feedback Diagrams to analyze a very complex problem of ecological sustainability. The lesson looks at a specific case study—from my home town in the Philippines—of the Live Reef Fish Trade now threatening survival of the Coral Reef Triangle of Southeast Asia. Live reef fish have long been traded around Southeast Asia as a luxury food item, but in recent decades trade in fish captured on coral reefs has expanded rapidly. Although the trade has provided communities with additional income, these benefits are unsustainable and have come at considerable cost to the environment. This lesson begins by having students analyze a familiar or personal problem, using Flow Charts and Feedback Diagrams, and then moves on to the application of those tools to a complex environmental problem. The lesson could be completed in a 50-minute class session, but using it over two class sessions would be preferable. Everything needed for the lesson is downloadable from the BLOSSOMS website, including blank Flow Charts and Feedback Diagrams, as well as articles on the Philippines case study from the World Wildlife Fund and the United States Agency for International Development.

Ecologies of Construction examines the resource requirements for the making and maintenance of the contemporary built environment. This course introduces the field of industrial ecology as a primary source of concepts and methods in the mapping of material and energy expenditures dedicated to construction activities.

This course provides a review of physical, chemical, ecological, and economic principles used to examine interactions between humans and the natural environment. Mass balance concepts are applied to ecology, chemical kinetics, hydrology, and transportation; energy balance concepts are applied to building design, ecology, and climate change; and economic and life cycle concepts are applied to resource evaluation and engineering design. Numerical models are used to integrate concepts and to assess environmental impacts of human activities. Problem sets involve development of MATLABĺ¨ models for particular engineering applications. Some experience with computer programming is helpful but not essential.