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 .
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
-
1.
References
Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)
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)
Jacobson, D.H.: A new necessary condition of optimality for singular control problems. SIAM J. Control 7, 578–595 (1969)
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)
Mattheij, R.M.M., Molenaar, J.: Ordinary Differential Equations in Theory and Practice. SIAM Classics in Applied Mathematics, vol. 43. SIAM, Philadelphia (2002)
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)
Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003)
Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (2002)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Tewari, A. (2019). Analytical Optimal Control. In: Optimal Space Flight Navigation. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-03789-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-03789-5_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-03788-8
Online ISBN: 978-3-030-03789-5
eBook Packages: EngineeringEngineering (R0)