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There is an ever-growing interest in control problems today, con­ nected with the urgent problems of the effective use of natural resources, manpower, materials, and technology. When referring to the most important achievements of science and technology in the 20th Century, one usually mentions the splitting of the atom, the exploration of space, and computer engineering. Achievements in control theory seem less spectacular when viewed against this background, but the applications of control theory are playing an important role in the development of modern civilization, and there is every reason to believe that this role will be even more signifi­ cant in the future. Wherever there is active human participation, the problem arises of finding the best, or optimal, means of control. The demands of economics and technology have given birth to optimization problems which, in turn, have created new branches of mathematics. In the Forties, the investigation of problems of economics gave rise to a new branch of mathematical analysis called linear and convex program­ ming. At that time, problems of controlling flying vehicles and technolog­ ical processes of complex structures became important. A mathematical theory was formulated in the mid-Fifties known as optimal control theory. Here the maximum principle of L. S. Pontryagin played a pivotal role. Op­ timal control theory synthesized the concepts and methods of investigation using the classical methods of the calculus of variations and the methods of contemporary mathematics, for which Soviet mathematicians made valuable contributions.

Intro -- Preface -- Acknowledgement -- Contents -- 1 Basics on Uncertainty Theory -- 1.1 Uncertainty Space -- 1.1.1 Uncertain Variable -- 1.1.2 Independence -- 1.2 Expected Value -- 1.2.1 Distribution of Function of Uncertain Variable -- 1.2.2 Expected Value of Function of Uncertain Variable -- 1.3 Optimistic Value and Pessimistic Value -- 1.4 Uncertain Simulation -- 1.5 Uncertain Process -- 1.5.1 Liu Process -- 1.5.2 Liu Integral -- 1.6 Uncertain Differential Equation -- References -- 2 Uncertain Expected Value Optimal Control -- 2.1 Problem of Uncertain Optimal Control -- 2.2 Principle of Optimality -- 2.3 Equation of Optimality -- 2.4 Equation of Optimality for Multidimension Case -- 2.5 Uncertain Linear Quadratic Model -- 2.6 Optimal Control Problem of the Singular Uncertain System -- References -- 3 Optimistic Value-Based Uncertain Optimal Control -- 3.1 Optimistic Value Model -- 3.2 Equation of Optimality -- 3.3 Uncertain Optimal Control Model with Hurwicz Criterion -- 3.4 Uncertain Linear Quadratic Model Under Optimistic Value Criterion -- 3.5 Optimistic Value Optimal Control for Singular System -- 3.5.1 Example -- References -- 4 Optimal Control for Multistage Uncertain Systems -- 4.1 Recurrence Equation -- 4.2 Linear Quadratic Model -- 4.3 General Case -- 4.3.1 Hybrid Intelligent Algorithm -- 4.3.2 Finite Search Method -- 4.3.3 Optimal Controls for Any Initial State -- 4.4 Example -- 4.5 Indefinite LQ Optimal Control with Equality Constraint -- 4.5.1 Problem Setting -- 4.5.2 An Equivalent Deterministic Optimal Control -- 4.5.3 A Necessary Condition for State Feedback Control -- 4.5.4 Well Posedness of the Uncertain LQ Problem -- 4.5.5 Example -- References -- 5 Bang-Bang Control for Uncertain Systems -- 5.1 Bang-Bang Control for Continuous Uncertain Systems -- 5.1.1 An Uncertain Bang-Bang Model -- 5.1.2 Example.

© 2015 Society for Industrial and Applied Mathematics. We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained. From the Hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the Lagrangian formalism by means of a proper Lagrangian submanifold. Singular optimal control problems and overdetermined ones are also studied in the paper. ; This work has been partially supported by MICINN (Spain) MTM2008-00689, MTM2009-13383,MTM2009-08166-E, MTM2010-21186-C02-01 and MTM2010-21186-C02-02; 2009SGR1338 fromthe Catalan government and SOLSUB200801000238 from the Canary Islands government; theEuropean project IRSES-project GeoMech-246981 and the ICMAT Severo Ochoa project SEV-2011-0087

1. Introduction -- 2. Control Problem -- Notation and definitions, motivation of later results -- 3. Railroad Train Example -- Motivation of a "bang-bang principle," the time optimal and minimum fuel problems -- 4. Controllability -- Restricted and unrestricted control, the controllability matrix -- 5. Controllability Using Special Controls -- Piecewise constant controls and bang-bang controls -- 6. Linear Time Optimal Systems -- Existence, necessary conditions, extremal control, reachable cone, normal systems and bang-bang controls, application to synthesis of railroad train example -- 7. General Control Systems: Existence -- Motivation behind existence results, two existence theorems and extensions, examples -- 8. General Control Systems: Necessary Conditions -- Principle of Optimality, Pontryagin maximum principle, transversality conditions, application to synthesis of railroad train example -- 9. Further Topics -- References -- List of Symbols -- Index for Definitions.