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Optimal control of piecewise deterministic Markov process. (English) Zbl 0566.93074
A piecewise deterministic Markov process which is characterized by its behaviour between the jumps, \(x_ t=f(x_ t),\) and by the jump behaviour given by the jump rate \(\lambda\), the jump height q in the interior and the jump height p on the boundary of the state space, is controlled by strategies depending on the initial state of the deterministic motion and on the time spent since the starting point. The aim is to minimize continuous, jump and terminal costs. An optimality condition is given in the form of a limiting Hamilton-Jacobi-Bellman equation, the existence of optimal controls is derived, and by considering the original problem as a convex mathematical programming problem, the value function is characterized as supremum of smooth subsolutions.
Reviewer: M.Kohlmann

MSC:
93E20 Optimal stochastic control
60J25 Continuous-time Markov processes on general state spaces
49K45 Optimality conditions for problems involving randomness
90C39 Dynamic programming
49L20 Dynamic programming in optimal control and differential games
60J75 Jump processes (MSC2010)
90C25 Convex programming
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References:
[1] Blumenthal R. M., Markov Processes and Potential Theory (1968) · Zbl 0169.49204
[2] Davis M. H. A., J. Royal Stat. Soc 46 (3) (1984)
[3] Doob J. L., Stochastic Processes (1953)
[4] van der Duyn Schouten F. A., Markov Decision Drift Processes (1983) · Zbl 0519.90052
[5] Florentin J. J., J. Basic Engineering (1963)
[6] Gnedenko B. V., Introduction to the Theory of Mass Service (1965)
[7] Kushner H. J., IRE Transactions on Automatic Control 7 (1962) · doi:10.1109/TAC.1962.1105490
[8] Krassovskii N. N., Appl. Math. Mech 25 pp 627– (1961) · Zbl 0107.35001 · doi:10.1016/0021-8928(61)90032-6
[9] Lidskii E. A., Appl. Math. Mech 27 pp 33– (1963) · doi:10.1016/0021-8928(63)90094-7
[10] Rishel R. W., SI AM. J. Control 13 pp 338– (1957) · Zbl 0304.93025 · doi:10.1137/0313020
[11] Rockafellar R. T., Duke Math. J 33 pp 81– (1966) · Zbl 0138.09301 · doi:10.1215/S0012-7094-66-03312-6
[12] Sworder D. D., IEEE Trans. Automatic Control 14 pp 9– (1969) · doi:10.1109/TAC.1969.1099088
[13] Vermes D., Professional Paper No. 80-15, International Institute of Applied Systems Analysis (1980)
[14] Davis M. H. A., Adv. Appl. Probability (1984)
[15] Vermes D., Parameter-free approach to Young’s generalized flows and optimal processes including continuous extinction
[16] Vinter R. B., SI AM J. Control and Optimization 16 pp 546– (1978) · Zbl 0396.49011 · doi:10.1137/0316037
[17] Vinter R. B., SI AM J. Control and Optimization 16 pp 571– (1978) · Zbl 0392.49011 · doi:10.1137/0316038
[18] Warga J., Optimal Control of Differential and Functional Equations (1972) · Zbl 0253.49001
[19] Wonham, W. M. 1970.Probabilistic Methods in Applied Mathematics, Edited by: Bharucha-Reid, A. T. 191–199. New York: Academic Press.
[20] Young L. C., Lectures on the Calculus of Variations and Optimal Control Theory (1969) · Zbl 0177.37801
[21] Yushkevich A. A., Stochastics 10 pp 235– (1983) · Zbl 0498.90081 · doi:10.1080/17442508308833256
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