Foundations of neoclassical growth lecture notes 5. Stability, controllability, observability the mathematical structure most naturally adapted to the description of systems is the state space representation. It states that it is necessary for any optimal control along with the optimal state trajectory to solve the so. Optimal control problems lecture notes on calculus of. Lectures on the calculus of variations and optimal control. Optimal control theory is a modern extension of the classical calculus of variations. We could spend an entire lecture discussing the importance of control theory and tracing its development through the windmill, steam governor, and so on. Starting with calculus of variations and optimal control. The answer will again turn out to be a bangbang control, as we will explain later.
In the theory of mathematical optimization one try to nd maximum or minimum points of functions depending of real variables and of other func tions. Such optimization problems are therefore called optimal control problems. The canadian mathematical congrss has held biennual seminars since 1947, and these have covered a wide range of topics. In this lecture, we only give a brief introduction. Introduction to optimal control one of the real problems that inspired and motivated the study of optimal control problems is the next and so called \moonlanding problem. An introduction to optimal control ugo boscain benetto piccoli the aim of these notes is to give an introduction to the theory of optimal control for nite dimensional systems and in particular to the use of the pontryagin maximum principle towards the constructionof an optimal synthesis. This section provides the lecture notes from the course along with information on lecture topics. One good one is dynamic programming and optimal control, vol. Rough lecture notes from the spring 2018 phd course ieor e8100 on mean field games and interacting diffusion models. In this lecture notes, we consider mainly the following two types. Read the texpoint manual before you delete this box. This lecture provides an overview of optimal control theory. Whereas discretetime optimal control problems can be solved by classical optimization techniques, continuoustime problems involve optimization in in. Lecture notes, and per krusells lecture notes for macroeconomic i.
This is an 11 part course designed to introduce several aspects of mathematical control theory as well as some aspects of control in engineering to mathematically mature students. Evans department of mathematics university of california, berkeley. This book grew out of my lecture notes for a graduate course on optimal control theory which i taught at the university of illinois at urbanachampaign during the period from 2005 to 2010. Course notes anthony bright 41510 1 modern control a. Consider the problem of a spacecraft attempting to make a soft landing on the moon using a minimum amount of fuel. In most applications, a general solution is desired that establishes the optimal input as a function of the systems initial condition. Lyapunov theory is covered in many texts on linear systems, e. Refer to x as the state variable, governed by a vectorvalued di erential equation given behavior of control variables yt. Infinitehorizon optimization and dynamic programming lecture notes 6. Before we start on the calculus of variations and control theory, we. Applications of optimal stochastic control are to be found in science, economics, and engineering.
It is oriented towards those who are beginning their study of optimal control, those who are interested in exploiting optimal control theory to treat specific problems arising in applications, in general, and to practitioners in the area of aerospace engineering, in particular. This course studies basic optimization and the principles of optimal control. I stochastic control theory and optimal filtering i brown and hwang, introduction to. The first of the two volumes of the leading and most uptodate textbook on the farranging algorithmic methododogy of dynamic programming, which can be used for optimal control, markovian decision problems, planning and sequential decision making under uncertainty, and discretecombinatorial. The following lecture notes are made available for students in agec 642 and other interested readers. Principles of optimal control aeronautics and astronautics. Lecture notes by alberto bressan scroll to the bottom of the page tutorial on viscosity solutions of hjb equation by alberto bressan. An introduction to mathematical optimal control theory. Optimal control oc study materials pdf free download. Hailed as a breakthrough software, dido is based on the pseudospectral optimal control theory of ross and fahroo. While optimal control theory was originally derived using the techniques of. Lecture 20 optimal control in linear systems jhu learning theory. An introduction to optimal control ugo boscain benetto piccoli the aim of these notes is to give an introduction to the theory of optimal control for nite dimensional systems and in particular to the use of the pontryagin maximum principle towards the constructionof an optimal. Ece 821 optimal control and variational methods lecture notes.
Control system engineeringii 3 10 modulei 10 hours state variable analysis and design. This rst equation is interpreted as the value function associated with a typical small player. Aug 30, 2012 a video introduction to the lecture 3 notes on the basic principles of optimal control. In these notes, both approaches are discussed for optimal control. Portugal october 15, 2011 abstract the aim of this lecture notes is to provide a selfcontained introduction to the subject of dynamic optimization for the msc course on mathematical economics, part of the msc. Frequency domain analysis chapter 5 output feedback chapter 6 lqgltr.
References from our text books are chapter 10 of dixit 1990, chapter 20 chiang and wainwright 2005, and chapter 12. Finally, chapter 8 discusses optimal stopping and impulse control problems. May 01, 2014 learning theory reza shadmehr, phd optimal feedback control of linear dynamical systems with and without additive noise. Optimal control, oc study materials, engineering class handwritten notes, exam notes, previous year questions, pdf free download. Buy lectures on the calculus of variations and optimal control theory ams chelsea publishing on free shipping on qualified orders. Optimal control with aerospace applications space technology library page. In the second part of the book, the author discusses optimal control problems. Optimal control theory and its applications proceedings of. Agec 642 lectures in dynamic optimization optimal control and numerical dynamic programming richard t. An introduction to the coordinatefree maximum principle, paper by hector sussmann about maximum principle on manifolds. Optimal control is concerned with the design of control systems to achieve a prescribed performance e. This is the lecture notes for the econ607 course that i am currently teaching at university of hawaii. Bertsekas massachusetts institute of technology chapter 6 approximate dynamic programming this is an updated version of the researchoriented chapter 6 on approximate dynamic programming. For a very deep study of optimal control athans and falb is a classic.
Linear quadratic optimal control, kalman filter, discretization, linear quadratic gaussian problem, loop transfer recovery, system identification, adaptive. Optimal control theory is the study of dynamic systems, where an. This course introduces fundamental mathematics of optimal control theory and implementation of optimal controllers for practical applications. Optimal control with aerospace applications space technology.
It considers deterministic and stochastic problems. This is an 11 part course designed to introduce several aspects of mathematical control theory as well as some aspects of control in engineering to. The foundation of optimal control theory has been developed into a subject of its own. While preparingthe lectures, i have accumulated an entire shelf of textbooks on calculus of variations and optimal control systems. This webpage contains a detailed plan of the course as well as links to home work hw assignments and other resources. Leonhard euler the words \control theory are, of course, of recent origin, but the subject itself is much older, since it contains the classical calculus of variations as a special case, and the rst calculus of varia tions problems go back to classical greece. Several students have pointed out typos or errors most of which are due to the teacher. Another two are optimal filtering and optimal control. Pontryagins maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. While optimal control theory was originally derived using the techniques of calculus of variation, most robust control. An introduction to mathematical optimal control theory version 0. Lecture notes for me233 advanced control systems ii xu chen and masayoshi tomizuka spring 2014.
Stochastic calculus, filtering, and stochastic control. Dynamic programming and optimal control 3rd edition, volume ii by dimitri p. The approach di ers from calculus of variations in that it uses control variables to optimize the functional. Once the optimal path or value of the control variables is found, the. We will start by looking at the case in which time is discrete sometimes called. Jan, 2020 combination of the two leads to optimal stochastic control. Its main ingredient is the euler equation1 which was discovered already in 1744. Through the years, the notes have been deeply revised and integrated. Penaltybarrier functions are also often used, but will not be discussed here. Euler and lagrange developed the theory of the calculus of variations in the eighteenth century. The curve of minimal length and the isoperimetric prob lem suppose we are interested to nd the curve of minimal length joining two distinct points in the plane. It considers deterministic and stochastic problems for both discrete and continuous systems. Contributions to the theory of optimal control, 1960 paper by kalman. Instead, the necessary condition is replaced by the pontryagin maximal principle.
Such classic control theory is largely concerned with the question of stability, and there is much of this theory which we ignore, e. Buy lectures on the calculus of variations and optimal control theory. Tomlin may 11, 2005 these notes represent an introduction to the theory of optimal control and dynamic games. The theory is an extension of classical calculus of variations since it does not rely of the smoothness assumptions made so far. Dydoh is a software product for solving generalpurpose optimal control problems. It is heavily based on stokey, lucas and prescott 1989, ljungqvist and sargent 2004, dirk kruegers. Pontryagin maximum principle and modern optimal control theory. These notes on system theory are a revised version of the lecture notes. D sontag, calculus of variations and optimal control by g. Notes from my minicourse at the 2018 ipam graduate summer school on mean field games and applications, titled probabilistic compactification methods for stochastic optimal control and mean field games.
Similarly, the stochastic control portion of these notes concentrates on veri. Chapter 7 develops ltering theory and its connection with control. A video introduction to the lecture 3 notes on the basic principles of optimal control. Survey of applications of pde methods to mongekantorovich mass transfer problems an earlier version of which appeared in current developments in. Uc berkeley lecture notes for me233 advanced control. Lectures on calculus of variations and optimal control by l. Optimal control theory 6 3 the intuition behind optimal control theory since the proof, unlike the calculus of variations, is rather di cult, we will deal with the intuition behind optimal control theory instead.
Optimal control theory is a modern approach to the dynamic optimization without being constrained to interior solutions, nonetheless it still relies on di erentiability. A concise introduction, by daniel liberzon reading materials. Lectures in dynamic optimization optimal control and numerical dynamic programming richard t. Applied optimal control and estimation course engineering. The decision makers goal is then to select an optimal decision among all possible ones in order to achieve the better possible. There exist two main approaches to optimal control and dynamic games. In the control formulation, these constraints are expressed in a. So in terms of our general notation, we have xt wt,qtt and x0 w0,q0t. Lecture 20 optimal control in linear systems youtube. The seminar reported in this publication was concerned with optimal control theory and its applications, a subject chosen for its active rowth and its wide implications for other fields. Emergence of locomotion behaviours in rich environments video. Learning theory reza shadmehr, phd optimal feedback control of linear dynamical systems with and without additive noise.
Mean eld game theory is devoted to the analysis of di erential games with in nitely. Lecture notes for a graduate course entropy and partial differential equations. The state of a system is described at any instant by a set of. These notes are not meant to be a complete or comprehensive survey on stochastic optimal control. Optimal control paul schrimpf october 3, 2019 university of british columbia economics 526 cba1 1. Lecture notes principles of optimal control aeronautics. The introduction is intended for someone acquainted with ordinary.
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