Optimal Trajectories in Atmospheric Flight - 1st Edition - ISBN: 9780444419613, 9780444601452

Optimal Trajectories in Atmospheric Flight

1st Edition

Authors: Nguyen Vinh
eBook ISBN: 9780444601452
Imprint: Elsevier
Published Date: 1st January 1981
Page Count: 420
Sales tax will be calculated at check-out Price includes VAT/GST
15% off
15% off
15% off
Price includes VAT/GST
× DRM-Free

Easy - Download and start reading immediately. There’s no activation process to access eBooks; all eBooks are fully searchable, and enabled for copying, pasting, and printing.

Flexible - Read on multiple operating systems and devices. Easily read eBooks on smart phones, computers, or any eBook readers, including Kindle.

Open - Buy once, receive and download all available eBook formats, including PDF, EPUB, and Mobi (for Kindle).

Institutional Access

Secure Checkout

Personal information is secured with SSL technology.

Free Shipping

Free global shipping
No minimum order.


Optimal Trajectories in Atmospheric Flight deals with the optimization of trajectories in atmospheric flight. The book begins with a simple treatment of functional optimization followed by a discussion of switching theory. It then presents the derivation of the general equations of motion along with the basic knowledge in aerodynamics and propulsion necessary for the analysis of atmospheric flight trajectories. It goes on to the study of optimal trajectories by providing the general properties of the optimal aerodynamic controls and the integrals of motion. This is followed by discussions of high subsonic and supersonic flight, and approximation techniques to reduce the order of the problem for a fast computation of the optimal trajectory. The final chapters present analyses of optimal reentry trajectories and orbital maneuvers.
This book is intended as a reference text for scientists and engineers wanting to get into the subject of optimal trajectories in atmospheric flight. If used for teaching purposes, the book is written in a self-contained way so that a selective use of the material is at the discretion of the lecturer. The first 11 chapters are sufficient for a one-semester course with emphasis on optimal maneuvers of high performance aircraft.

Table of Contents





0.1 Scope

0.2 Plan of the Study


Chapter 1. Optimization Theory

1.1 Necessary Conditions for Extrema

1.2 Solution Subject to Constraints

1.3 Calculus of Variations

1.3.1 Necessary conditions for optimality

1.3.2 Transversality conditions

1.3.3 Canonical equations

1.3.4 First integrals

1.4 Pontryagin Maximum Principle

1.5 Canonical Transformation


Chapter 2. Switching Theory

2.1 Contensou's Domain of Maneuverability

2.2 Optimal Switching

2.3 Junction With Singular Arc

2.4 Linearized Singular Control


Chapter 3. Equations of Motion

3.1 Introduction

3.2 Relative Angular Motion

3.3 Equations for Flight over a Rotating Spherical Earth

3.4 Equations for Flight over a Flat Earth


Chapter 4. Aerodynamic and Propulsive Forces

4.1 The Atmosphere

4.1.1 Temperature 64

4.1.2 Pressure

4.1.3 Density

4.1.4 Viscosity

4.2 Aerodynamic Forces

4.3 The Drag Polar

4.4 Propulsive Force


Chapter 5. General Properties of Optimal Trajectories

5.1 The Optimal Controls

5.2 Integrals of the Motion

5.3 Application to the Case of Flat Earth

5.4 Application to the Case of Spherical Earth


Chapter 6. Flight in a Horizontal Plane

6.1 Dimensionless Equations of Motion

6.2 Minimum Fuel Turn at Constant Altitude

6.2.1 Optimal controls

6.2.2 Optimal turn to a heading

6.2.3 Optimal turn to a line

6.2.4 Optimal turn to a point

6.3 Optimal Coasting Flight

6.3.1 Optimal lift control

6.3.2 Maximum ranges

6.3.3 The footprint

6.3.4 Some optimal maneuvers


Chapter 7. Optimal Coasting Flight

7.1 Dimensionless Equations of Motion

7.2 Optimal Aerodynamic Control

7.3 Optimal Skip Trajectory

7.3.1 Free final heading

7.3.2 Free final flight path angle

7.3.3 Maximum final altitude

7.3.4 Maximum final speed

7.3.5 Maximum final heading

7.3.6 Maneuvers at small angles

7.4 Optimal Maneuvers in the Vertical Plane

7.4.1 Optimal pull-up maneuver

7.4.2 Solution with bounded lift control

7.4.3 Optimal glide for maximum range

7.4.4 Linearized singular lift control for maximum range


Chapter 8. Supersonic Cruise

8.1 Introduction

8.2 Dimensionless Equations of Motion

8.3 Flight Envelope

8.4 General Analysis

8.5 Maximum Range

8.5.1 Exact analysis

8.5.2 Steady-state cruise

8.6 Maximum Endurance

8.7 Supersonic Cruise With Time Constraint

8.8 Optimality of the Solution


Chapter 9. Supersonic Turn

9.1 Introduction

9.2 Flight Envelope

9.3 Minimum Time Turn

9.3.1 Optimal controls

9.3.2 Turn to a heading

9.3.3 Turn to a point

9. 3.4 Turn to a line

9.4 Minimum Radius Turn

9.5 Minimum Fuel Turn

9.5.1 Optimal controls

9.5.2 Turn to a heading

9.5.3 Turn to a line


Chapter 10. Supersonic Maneuvers in a Vertical Plane

10.1 Dimensionless Equations of Motion

10.2 Glide with Maximum Range

10.3 Cruise by Periodic Control

10.4 Minimum-Time Loop Maneuvers

10.5 Other Maneuvers of Fighter Aircraft

10.6 Minimum-Time to Climb


Chapter 11. Energy State Approximation

11.1 Introduction

11.2 Energy-State Approximation

11.2.1 Minimum-time to climb

11.2.2 Minimum-fuel to climb

11.2.3 Maximum range glide

11.2.4 Minimum-fuel with fixed range

11.3 Loop, Immelman and Split-S

11.4 Three-Dimensional Turns


Chapter 12. Modified Chapmans Formulation for Optimal Reentry Trajectories

12.1 Dimensionless Equations of Motion

12.2 Variational Formulation

12.3 Reduction to the Flat Planet Case

12.4 The Vector Integral

12.5 Flight in the Vacuum


Chapter 13. Optimal Planar Reentry Trajectories

13.1 The Variational Equations

13.2 Optimal Pull Up Maneuver

13.3 Geometry of a Skip Trajectory

13.4 Skip Trajectory for Maximum Final Speed

13.5 Skip Trajectory for Maximum Apogee Altitude

13.6 Skip Trajectory for Maximum Apogee Altitude with Prescribed Apogee Speed

13.7 Skip Trajectory for Maximum Apogee Speed with Prescribed Apogee Altitude

13.8 Skip Trajectory for Maximum Range

13.9 Glide with Maximum Range

13.10 The Nonlinear Equation for the Lift Control


Chapter 14. Optimal Glide of Reentry Vehicles

14.1 Maximum Cross Range

14.2 Approximate Solutions for Maximum Cross Range

14.3 The Footprint of Reentry Vehicle

14.4 Some Related Optimal Glide Problems

14.4.1 Trajectory with maximum final speed

14.4.2 Reaching a latitude with maximum final speed

14.4.3 Closest approach to a prescribed position


Chapter 15. Orbital Aerodynamic Maneuvers

15.1 Introduction

15.2 Orbit Modifications

15.2.1 Aerodynamic capture

15.2.2 Change in the apogee

15.2.3 Change in the eccentricity

15.2.4 Change in the perigee

15.2.5 Change in the major axis

15.3 Rotation of Orbital Plane

15.3.1 Orbital maneuver

15.3.2 Aerodynamic maneuver

15.3.3 Combined maneuver


Author Index

Subject Index


No. of pages:
© Elsevier 1981
eBook ISBN:

About the Author

Nguyen Vinh

Ratings and Reviews