Optimal Interception Problem with Delays (OIP)

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Table of contents

Introduction Générale
General Introduction
I Optimal Control Framework and Dynamic alModel
1 Elements of Optimal Control
1.1 Some Tools from Differential Geometry
1.1.1 Notations and Properties of Vector Fields
1.1.2 Standard Results on Hamiltonian Fields
1.2 Classical Optimal Control Problems
1.3 Optimality Conditions and NumericalMethods
1.3.1 TheMaximumPrinciple
1.3.2 Sufficient Optimality Conditions
1.3.3 Classical NumericalMethods in Optimal Control
1.3.4 Numerical HomotopyMethods
1.4 Problems with Control and State Constraints
1.4.1 General Control and State Constraints
1.4.2 Mixed Control-State Constraints
1.4.3 Numerical Difficulties Due to Control and State Constraints
1.5 Problems with Control and State Delays
1.5.1 MaximumPrinciple for Problems with Delays
1.5.2 Numerical Difficulties Due to Control and State Delays
2 Rendezvous Problems
2.1 Physical Problem and DynamicalModel
2.1.1 Fundamental Coordinate Systems
2.1.2 Environmental and DynamicalModeling
2.2 Optimal Control Problems
2.2.1 General Optimal Guidance Problem (GOGP)
2.2.2 Optimal Interception Problem (OIP)
2.2.3 Optimal Interception Problem with Delays (OIP)¿
II Structure of Extremals and Numerical Strategies of Guidance
3 Structure of Extremals for Optimal Guidance Problems
3.1 Local Change of Problems Under Abstract Framework
3.1.1 Reduction to Local Problems with Pure Control Constraints
3.1.2 Sufficient Conditions Under Reduction to Local Problems
3.2 Local Transformations for (GOGP)
3.2.1 Coordinates Under the Trajectory Reference Frame
3.2.2 Additional Local Euler Coordinates
3.2.3 Global and Local Adjoint Formulations for (GOGP)
3.3 Regular and Nonregular Pontryagin Extremals
3.3.1 Regular Pontryagin Extremals
3.3.2 Nonregular Pontryagin Extremals
3.4 Conclusions
4 Numerical Guidance Strategy
4.1 General Numerical Homotopy Procedure for (GOGP)
4.1.1 General Optimal Guidance Problem of Order Zero (GOGP)0
4.1.2 Parametrized Family of Optimal Control Problems (GOGP)
4.2 Optimal Interception Problem of Order Zero (OIP)0
4.2.1 Approximated Local Controllability of (OIP)s0
4.2.2 Froma LOS Analysis to a Suboptimal Guidance Law for (OIP)s0
4.3 Numerical Simulations for (OIP)
4.3.1 Mathematical Design of theMission
4.3.2 Homotopy Scheme and Numerical Results
4.4 Conclusions
5 Numerical Robustness and Interception Software (ONERA)
5.1 Increasing the Robustness: Initialization Grids
5.1.1 Fast Initialization Grids Design
5.1.2 Numerical Time-RobustnessMonte Carlo Experiments
5.2 Software Design: a Template C++ Library (ONERA)
5.2.1 Library Structure (Simplified UML Class Diagram)
5.2.2 Details on Classes and User Script Examples
5.3 Conclusions
III Continuity of Pontryagin Extremals with Respect to Delays
6 Solving Optimal Control Problems with Delays
6.1 Continuity Properties with Respect to Delays
6.2 Homotopy Algorithmand Numerical Simulations
6.2.1 Solving (OCP)¿ by ShootingMethods and Homotopy on Delays
6.2.2 First Numerical Tests
6.3 Numerical Strategy to Solve (OIP)¿
6.3.1 Local Initialization Procedure for (OIP)¿
6.3.2 Numerical Simulations for (OIP)¿
6.4 Conclusions
7 Continuity Properties of Pontryagin Extremals
7.1 Proof of the PMP Using Needle-Like Variations
7.1.1 Preliminary Notations
7.1.2 Needle-Like Variations and Pontryagin Cones
7.1.3 Proof of TheMaximum Principle
7.2 Conic Implicit Function Theorem with Parameters
7.3 Proof of Theorem 6.1
7.3.1 Controllability for (OCP)¿
7.3.2 Existence of Optimal Controls for (OCP)¿
7.3.3 Convergence of Optimal Controls and Trajectories for (OCP)¿ .
7.3.4 Convergence of Optimal Adjoint Vectors for (OCP)¿
7.4 Conclusions
Conclusion
Bibliography