Optimal control and geometry: integrable systems (Q2798030)

This book is a natural continuation of the author's classic text on geometric control theory [\textit{V. Jurdjevic}, Geometric control theory. Cambridge: Cambridge Stud. Adv. Math., 52, Cambridge Univ. Press (1997; Zbl 0940.93005)]. The book aims to point out how the subject traditionally concerning the domain of applied mathematics connected with engineering problems made important contributions to mathematics beyond its original intent. The book aims to highlight the important contributions made to mathematics by the traditional domain of applied mathematics connected with engineering problems, beyond its original intent. In particular, the book explains how control-theoretical tools, such as ''space control'' and ``optimal path'', remarkable for the subject within they were exploited, enriched the calculus of variations with new and fresh insights. The new subject, a synthesis of the calculus of variations, modern symplectic geometry and control theory, provides a rich foundation crucial for problems of applied mathematics. By focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the maximum principle of optimality, shed new light on the theory of integrable systems. The book also explains hidden features concerning the most enigmatic problems such as Kepler and Jacobi's problems and Kowalevski's top.NEWLINENEWLINEThis monograph is a very useful text for graduates and researchers in many fields, ranging from mathematical physics to control theory, as well as for scientists involved in applied mathematics.NEWLINENEWLINEThe subject is introduced in a very self-contained way, through the basics of differential geometry, manifolds, vector fields, differential forms, and Lie brackets. Chapters 1 and 2 deal with the accessibility theory based on Lie theoretic methods. Chapter 3, 4 and 5 concern Lie groups, symplectic and Poisson manifolds. Chapter 6 introduces the maximum principle, explaining the role of optimal control for problems of the calculus of variations and provides a natural transition to the second part of the book on integrable systems. These six chapters form the theoretic background of the second part of the book that mostly deals with specific problems. This part of the book is a presentation of the non-Euclidean geometry of Chapter 7. A first class of problems is treated in Chapter 8 through the study of the geodesic problem in the sub-Riemannian framework and its following application to symmetric spaces. A second class of problems, called affine-quadratic, is presented in Chapter 9, that deals with their controllability and the study of the induced Hamiltonians. Chapter 10 studies the cotangent bundles of space forms as well as the cotangent bundles of oriented Stiefel and oriented Grasmannian manifolds. By means of this analysis, an explanation is provided for the enigmatic discovery of \textit{V.A. Fock} [The hydrogen atom and the non-Euclidean geometry, Izv. Akad. Nauk SSSR, Ser. Fizika 8 (1935)] on the connection between solutions of Kepler's problems and the geodesics of the space forms. Chapter 11 focuses on the elliptic geodesic problem on the sphere, by stressing another interesting connection with Jacobi's geodesic problem on the ellipsoid. Chapters 12, 13, 14 and 15 investigate the rigid body. In particular, the study of the integrability of Hamiltonians associated to the affine quadratic problems on different linear groups points out remarkable connections between Kirchhoff's elastica problem, and the dynamics of the heavy top and rigid body, and their generalization to higher dimensions. As well, the complexification of the Hamiltonian associated with Kirchhoff's problem provides a better understanding of the use of complex variable in the famous paper of \textit{S. Kowalevski} [Acta Math. XII. 177-232 (1889; JFM 21.0935.01)]. Chapter 16 deals with the curvature problem for curves, that is, the problem to find out curves which minimize, among a suitable class of curves, the \(L^2\)-norm of curves' curvature. Such curves are historically called \textit{elastic curves}. The chapter also studies the parallel Dubins-Delauney problem for curves. The book ends with Chapter 17, that deals with infinite-dimensional Hamiltonian systems and their relevance for the solutions of the nonlinear Schrödinger equation, the Korteweg-de Vries equations and Heisenberg's magnetic equation. The passage to infinite dimensions is based on the geometric notions developed in the previous chapters. The author includes some concluding remarks about possible future research projects where to apply book's geometric techniques.NEWLINENEWLINEThe book's style and the various historical notes throughout it make reading such monograph a very exciting experience; the bibliography is also very rich.