Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-24T05:44:13.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Keeping a satellite aloft: two finite fuel stochastic control models

Part of: Stochastic analysis Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

S. D. Jacka
S. D. Jacka*
Affiliation:
University of Warwick
*
∗Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: s.d.jacka@warwick.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider two models for the control of a satellite–in the first, fuel is expended in a linear fashion to move a satellite following a diffusion–where the aim is to keep the satellite above a critical level for as long as possible (or indeed to reach a higher, ‘safe’ level). Under suitable assumptions for the drift and diffusion coefficients, it is shown that the stochastic maximum of the time to fall below the critical level is attained by a policy which imposes a reflecting boundary at the critical level until the fuel is exhausted and jumps the satellite directly to the safe level if this is ever possible. In the second model, there is a nonlinear response to the expenditure of fuel, and no safe level. It is shown that the optimal policy for maximizing the expected discounted time for the satellite to crash is similar, in that equal packets of fuel are used to jump the satellite upwards each time it reaches the critical level.

Keywords

Stochastic controldiffusionstrong maximum principlefinite-fuel problemtotal positivity

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 1 - 20
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bather, J. A., and Chernoff, H. (1967). Sequential decisions in the control of a space ship (finite fuel). J. Appl. Prob. 4, 584–604.Google Scholar
Beneš, V. E., Shepp, L. A., and Witsenhausen, H. S. (1980). Some solvable stochastic control problems. Stochastics & Stochastics Rep. 4, 39–83.Google Scholar
El Karoui, N., and Chaleyat-Maurel, M. (1978). Un problème de réflexion et ses applications au temps local et aux equations différentielles stochastique sur ℝ–cas continu. Temps Locaux. Astérisque 52–53, 117–144.Google Scholar
Friedman, A. (1975). Stochastic Differential Equations and Applications, Vol. I. Academic Press, New York.Google Scholar
Harrison, J. M., and Taksar, M. I. (1983). Instantaneous control of Brownian motion. Math. Operat. Res. 8, 439–453.Google Scholar
Harrison, J. M., and Taylor, A. J. (1977). Optimal control of a Brownian storage system. Stoch. Proc. Appl. 6, 179–194.Google Scholar
Ikeda, N., and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North Holland-Kodansha, Amsterdam.Google Scholar
Jacka, S. D. (1983). A finite fuel stochastic control problem. Stochastics & Stochastics Rep. 10, 103–114.Google Scholar
Rogers, L. C. G., and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. II. Wiley, New York.Google Scholar
Weerasinghe, A. (1992). Finite fuel stochastic control problem on a finite time horizon. SIAM J. Control Optimization 30, 1395–1408.Google Scholar