講義コード		SLXVI15
講義科目名		数理科学特別講義XVI
講義題目
授業科目区分		学際科目
開講年度		2015
開講学期		前期
曜日時限		集中
必修選択		選択
単位数		1
担当教員		Bernard Bonnard，世話人 佐伯
対象学部等		数理学府
対象学年		修士1年|修士2年|博士1年|博士2年|博士3年
開講地区		伊都地区

履修条件		Linear Algebra, Ordinary Differential Equations. Recommended book for the pre-requisite: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third Edition Hardcover March 26, 2012 by Morris W. Hirsch, Stephen Smale. Highly recommended: Geometric Optimal Control with Applications I, graduate course by Prof. Chyba (June 2015).
授業概要		In the firrst part of the class, various applications of the maximum principle will be explored, in particular the time minimal problem and the linear quadratic optimal control case. Higher order conditions with the concept of conjugate point will be introduced, and an introduction to numerical methods will be given. Then, optimal control theory will then be used to answer purely geometric questions, in particular to study extremals in sub-Riemannian geometry and the role of the so-called abnormal geodesics. The second part of the course will be devoted to analyzing two specific applications. First, a new approach to the contrast imaging problem using tools from geometric optimal control will be explored. The problem is to bring one spin the origin of the Bloch Ball while maximizing the square of the norm of the magnetization vector for the second spin. The optimal solution can be found as an extremal, solution of the Maximum Principle and analyzed with the techniques of geometric control. This leads to a numerical investigation based on so-called indirect methods. Second, we will focus on geometric and numerical techniques to study the orbit transfer between Keplerian elliptic orbits in the two-body problem or between quasi-Keplerian orbits in the Earth-Moon transfer when low propulsion is used.
全体の教育目標		This course is mainly based on geometric methods and is oriented towards applications. The major goal is to equip the students with the necessary framework to study optimal control problems and to illustrate these tool son a couple of specific applications, namely the contract problem in medical imaging and orbital transfer for low thrust engine. This course combined with part I by Prof. Chyba should provide the students with an in-depth overview of geometric optimal control.
個別の教育目標		The main objective for the students is to learn techniques from geometric optimal control and apply them to various concrete problems. In particular the student will learn to analyze singular trajectories for different applications. They play a very important role in optimal control, they are in fact universal objects in various optimal control problems and classifications problem similarly to singularity theory when analyzing functions. At the end of the course, a student should be able to analyze an optimal control problem with the tools presented here.
授業計画		Lecture 1: Applications of the maximum principle The time minimal case and the linear quadratic (LQ) optimal control case. Lecture 2: Higher order conditions and numerical methods Second order conditions, Concept of conjugate point. Direct and Indirect numerical methods. Lecture 3: Geometric Applications Riemannian and SR-geometry: Sphere, conjugate and cut points (normal and abnormal case). Lecture 4: Applications I Nuclear magnetic Resonance and Magnetic Resonance Imaging, the saturation and the contrast problem. Lecture 5: Applications II Nuclear magnetic Resonance and Magnetic Resonance Imaging, the saturation and the contrast problem. Continuation. Lecture 6: Applications III Orbital transfer between Keplerian elliptic orbits, time and energy minimal problem with low propulsion.
キーワード		Geometric Optimal Control, Singular Trajectories, Sub-Riemannian Geometry, Conjugate Points, Nuclear magnetic Resonance, Orbital transfer
授業の進め方		By lectures and homeworks.
テキスト		None in particular.
参考書		・Calculus of Variations. I. M. Gelfand, Izrail Moiseevitch Gelfand, Serge Vasilevich Fomin, Richard A. Silverman Courier Corporation, 2000 - Mathematics - 232 pages. ・Singular Trajectories and Their Role in Control Theory. Bernard Bonnard, Monique Chyba. Springer Science & Business Media, May 12, 2003 - Language Arts & Disciplines - 357 pages. ・Control Theory from the Geometric Viewpoint. A. Agrachev and Y. Sachkov. Springer, Berlin, 2004. ・Optimal Control with Applications in Space and Quantum Dynamics. Bernard Bonnard and Dominique Sugny, AIMS Series on Applied Mathematics.
学習相談		Prof. Bonnard will be available in between lectures to answer questions to the students. Appointment can be made in class or by e-mail. Instructor: Prof. Bernard Bonnard Home Institution: Department of Mathematics, University of Burgundy, France Expertise: Geometric Optimal Control E-mail: Bernard.Bonnard@u-bourgogne.fr
試験／成績評価の方法等		The grade for the course will be determined based on class participation, homework and a final projet. Homework will be distributed at the end of each lecture and collected at the beginning of the next lecture. The final project will be discussed in class and needs to be approved by the instructor.
その他		Attendance is expected and is obviously in a student's best interest. Students are responsible for all information provided in class and on the course web page. Electronic devices such as cell phones, pagers, watch alarms, etc. must be turned off during class.
更新日付		2015-04-12 09:22:41.0