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2019 | OriginalPaper | Chapter

2. Analytical Optimal Control

Author : Ashish Tewari

Published in: Optimal Space Flight Navigation

Publisher: Springer International Publishing

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Abstract

Optimization refers to the process of achieving the best possible result (objective), given the circumstances (constraints). When applied to determine a control strategy for fulfilling a desired task, such an optimization is called optimal control .

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Footnotes
1
Since the dynamic equality constraint of Eq. (2.34) must be enforced separately, \(\dot {x}\) is taken to be an independent variable in the minimization of J a.
 
2
A symplectic matrix, A, satisfies A T JA = J, where
$$\displaystyle \begin{aligned} J=\left(\begin{array}{cccc}0 &&& I\\-I &&& 0\end{array}\right) \end{aligned}$$
is analogous to the imaginary number, \(j=\sqrt {-1}\), in complex algebra since J 2 = −I. A symplectic matrix, A, has the following properties:
1.
Its determinant is unity, i.e., det(A) = 1.
 
2.
If λ is an eigenvalue of A, then 1∕λ is also an eigenvalue of A.
 
 
Literature
13.
Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)MATH
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Goh, B.S.: Compact forms of the generalized Legendre conditions and the derivation of the singular extremals. In: Proceedings of 6th Hawaii International Conference on System Sciences, pp. 115–117 (1973)
42.
Jacobson, D.H.: A new necessary condition of optimality for singular control problems. SIAM J. Control 7, 578–595 (1969)MathSciNetCrossRef
46.
Kelley, H.J., Kopp, R.E., Moyer, H.G.: Singular extremals. In: Leitmann, G. (ed.) Topics in Optimization, pp. 63–101. Academic, New York (1967)CrossRef
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Mattheij, R.M.M., Molenaar, J.: Ordinary Differential Equations in Theory and Practice. SIAM Classics in Applied Mathematics, vol. 43. SIAM, Philadelphia (2002)
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Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)
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Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003)CrossRef
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Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)MATH
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Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (2002)CrossRef
Metadata
Title
Analytical Optimal Control
Author
Ashish Tewari
Copyright Year
2019
Book
Optimal Space Flight Navigation

Print ISBN: 978-3-030-03788-8

Electronic ISBN: 978-3-030-03789-5

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-03789-5_2