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Bang-bang controls of point processes

Published online by Cambridge University Press:  01 July 2016

P. Brémaud
P. Brémaud*
Affiliation:
CEREMADE, Université de Paris IX (Dauphine)
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Abstract

In this paper, we consider the problem of controlling the intensity of a point process in order to maximize the probability that the number of points in a fixed interval equals a given integer, under the constraint that the intensity belong to some closed interval of R +.

The problem is stated as a problem of optimization on the set of probabilities over the basic measurable space of point processes, and shown to be equivalent to a problem of deterministic control. Structural results concerning the set of optimal solutions are given. The existence of the latter is proven; the control is shown to be bang-bang and a complete solution can be obtained by application of Pontryagin's Maximum Principle.

Keywords

POINT PROCESSQUEUING PROCESSMARTINGALEPREDICTABLE PROCESSSTOCHASTIC INTEGRALOPTIMAL CONTROLBANG-BANG CONTROL

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 2 , June 1976, pp. 385 - 394
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Boel, R. (1974) Optimal Control of Jump Processes. Ph.D. Thesis, Memo ERL–M448, Department of EECS, University of California, Berkeley.Google Scholar
[2] Brémaud, P. (1972) A Martingale Approach to Point Processes. Ph.D. Thesis, Memo ERL–M 345, Department of EECS, University of California, Berkeley.Google Scholar
[3] Brémaud, P. (1974) The martingale theory of point processes with an intensity. In Proceedings of the IRIA Colloquium on Control Theory, Lect. Notes in Computer Science and Math. Systems 107, Springer-Verlag, 519–542.Google Scholar
[4] Doleans–Dade, C. and Meyer, P. A. (1970) Intégrales stochastiques par rapport aux martingales locales. In Seminaire de Probabilité Université de Strasbourg, IV, Lect. Notes in Mathematics 124, Springer–Verlag, 77–107.Google Scholar
[5] Jacod, J. (1973) On the stochastic intensity of a random point process. Technical Report, Department of Statistics, Princeton University.Google Scholar
[6] Jacod, J. (1975) Multivariate point process: Predictable projection Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
[7] Meyer, P. A. (1965) Probabilités et Potentiel. Hermann, Paris.Google Scholar
[8] Rishel, R. (1974) A minimum principle for controlled jump processes. In Proceedings of the IRIA Colloquium on Control Theory, Lect. Notes in Comp. Science and Math. Systems 107, Springer–Verlag, 499–518.Google Scholar
[9] Lee, E. B. and Markus, M. (1968) Foundations of Optimal Control Theory. Wiley, New York.Google Scholar