Singular control

Abstract: BASIC OPTIMAL CONTROL PROBLEMS Preliminaries The Basic Problem and Necessary Conditions Pontryagin's Maximum Principle Exercises EXISTENCE AND OTHER SOLUTION PROPERTIES Existence and Uniqueness Results Interpretation of the Adjoint Principle of Optimality The Hamiltonian and Autonomous Problems Exercises STATE CONDITIONS AT THE FINAL TIME Payoff Terms States with Fixed Endpoints Exercises FORWARD-BACKWARD SWEEP METHOD LAB 1: INTRODUCTORY EXAMPLE LAB 2: MOLD AND FUNGICIDE LAB 3: BACTERIA BOUNDED CONTROLS Necessary Conditions Numerical Solutions Exercises LAB 4: BOUNDED CASE LAB 5: CANCER LAB 6: FISH HARVESTING OPTIMAL CONTROL OF SEVERAL VARIABLES Necessary Conditions Linear Quadratic Regulator Problems Higher Order Differential Equations Isoperimetric Constraints Numerical Solutions Exercises LAB 7: EPIDEMIC MODEL LAB 8: HIV TREATMENT LAB 9: BEAR POPULATIONS LAB 10: GLUCOSE MODEL LINEAR DEPENDENCE ON THE CONTROL Bang-Bang Controls Singular Controls Exercises LAB 11: TIMBER HARVESTING LAB 12: BIOREACTOR FREE TERMINAL TIME PROBLEMS Necessary Conditions Time Optimal Control Exercises ADAPTED FORWARD-BACKWARD SWEEP Secant Method One State with Fixed Endpoints Nonlinear Payoff Terms Free Terminal Time Multiple Shots Exercises LAB 13: PREDATOR-PREY MODEL DISCRETE TIME MODELS Necessary Conditions Systems Case Exercises LAB 14: INVASIVE PLANT SPECIES PARTIAL DIFFERENTIAL EQUATION MODELS Existence of an Optimal Control Sensitivities and Necessary Conditions Uniqueness of the Optimal Control Numerical Solutions Harvesting Example Beaver Example Predator-Prey Example Identification Example Controlling Boundary Terms Exercises OTHER APPROACHES AND EXTENSIONS REFERENCES INDEX

Abstract: For the last 30 years the optimization of nonsingular control problems has been an Important part of control engineering, and its mathematical theory is well developed and widely known. On the other hand, singular control problems prove more difficult to analyse and—although necessary conditions for optimality of singular controls have been established over the past decade—It is only recently that sufficient, and necessary and sufficient, conditions have been formulated. The purpose of this book Is to collect together all known results in optimal control theory (as well as appropriate computational methods) which can be applied to the singular problems In optimal control and which up to now have been scattered In numerous journals. Complete and self-contained, the volume begins with an historical survey of singular control problems and leads to the presentation of important, recent results in the field. There are specific real-world applications and the authors discuss those avenues of research which require further Investigation. All those involved In the optimization of dynamical systems will welcome the publication of this book. In addition to advanced students, lecturers and research workers in universities, this will include practising mechanical, chemical and electrical engineers, builders, textile technologists, paper scientists and chemists, and many concerned with non-technical fields such as economics and business management Contents An historical survey of singular control problems Introduction. Singular control in space navigation. Method of Mlele via Green's theorem. Linear systems—quadratic cost Necessary conditions for singular optimal control. Sufficient conditions and necessary and sufficient conditions for optimality. References. Fundamental concepts Introduction. The general optimal control problem. The first variation of J. The second variation of J. A singular control problem. References. Necessary conditions for singular optimal control Introduction. The generalized Legendre-Clebsch condition. Jacobson's necessary condition. References. Sufficient conditions and necessary and sufficient conditions tor non-negativity of nonsingular and singular second variations Introduction. Preliminaries. The nonsingular case. Strong positivlty and the totally singular second variation. A general sufficiency theorem for the second variation. Necessary and sufficient conditions for non-negativity of the totally singular second variation. Necessary conditions for optimality. Other necessary and sufficient conditions. Sufficient conditions for a weak local minimum. Existence conditions for the matrix Rlccati differential equation. Conclusion. References. Computational methods for singular control problems Introduction. Computational application of the sufficiency conditions of theorems in the previous chapter. Computation of optimal singular controls. Joining of optimal singular and non-singular sub-arcs. Conclusion. References. Conclusion The Importance of singular optimal control problems. Necessary conditions. Necessary and sufficient conditions. Computational methods. Switching conditions. Outlook for the future Author index. Sublect index.

Abstract: Aspects of optimization theory and switching theory are discussed, taking into account the necessary conditions for extrema, a solution subject to constraints, the calculus of variations, the Pontryagin maximum principle, the canonical transformation, Contensou's domain of maneuverability, optimal switching, a junction with singular arc, and linearized singular control. Equations of motion are considered along with aerodynamic and propulsive forces, the general properties of optimal trajectories, flight in a horizontal plane, optimal coasting flight, supersonic cruise, the supersonic turn, supersonic maneuvers in a vertical plane, energy state approximation, a modified Chapman's formulation for optimal reentry trajectories, optimal planar reentry trajectories, and an optimal glide of reentry vehicles. Orbital aerodynamic maneuvers are examined, giving attention to aerodynamic capture, a change in the apogee, a change in the eccentricity, a change in the perigee, an orbital maneuver, an aerodynamic maneuver, and a combined maneuver.