×

Switching control of piecewise-deterministic processes. (English) Zbl 0631.93079

A finite collection of piecewise-deterministic processes is controlled in order to minimize the expected value of a performance functional with continuous operating cost and discrete switching control costs. The solution of the associated dynamic programming equation is obtained by an iterative approximation using optimal stopping time problems.

MSC:

93E20 Optimal stochastic control
49L20 Dynamic programming in optimal control and differential games
60G40 Stopping times; optimal stopping problems; gambling theory
45K05 Integro-partial differential equations
90C39 Dynamic programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Davis, M. H. A.,Piecewise-Deterministic Markov Processes: A General Class of Nondiffusion Models, Journal of the Royal Statistical Society, Series B, Vol. 46, pp. 353-388, 1984. · Zbl 0565.60070
[2] Davis, M. H. A.,Control of Piecewise-Deterministic Processes via Discrete-Time Dynamic Programming, Proceedings of the 3rd Bad Hannef Symposium on Stochastic Differential Systems, Bad Hannef, Germany, 1985.
[3] Vermes, D.,Optimal Control of Piecewise-Deterministic Markov Processes, Stochastics, Vol. 14, pp. 165-208, 1985. · Zbl 0566.93074
[4] Soner, M.,Optimal Control with State-Space Constraint, II, SIAM Journal on Control and Optimization, Vol. 24, pp. 1110-1122, 1986. · Zbl 0619.49013 · doi:10.1137/0324067
[5] Gugerli, U. S.,Optimal Stopping of a Piecewise-Deterministic Markov Process, Department of Electrical Engineering, Imperial College, London, England, Preprint, 1986. · Zbl 0611.60039
[6] Lenhart, S. M., andLiao, Y. C.,Integro-Differential Equations Associated with Optimal Stopping Time of a Piecewise-Deterministic Process, Stochastics, Vol. 15, pp. 183-207, 1985. · Zbl 0582.60053
[7] Capuzzo-Dolcetta, I., andEvans, L. C.,Optimal Switching for Ordinary Differential Equations, SIAM Journal on Control and Optimization, Vol. 22, pp. 143-161, 1984. · Zbl 0641.49017 · doi:10.1137/0322011
[8] Robin, M.,Impulsive Control of Markov Processes, PhD Thesis, Paris University IX, 1978. · Zbl 0445.93043
[9] Lenhart, S. M., andBelbas, S. A.,A System for Nonlinear Partial Differential Equations Arising in the Optimal Control of Stochastic Systems with Switching Costs, SIAM Journal on Applied Mathematics, Vol. 43, pp. 465-475, 1983. · Zbl 0511.93077 · doi:10.1137/0143030
[10] Lenhart, S. M., andLiao, Y. C.,Integro-Differential Equations Associated with Piecewise-Deterministic Processes, Proceedings of the International Conference on Differential Equations and Mathematical Physics, University of Alabama-Birmingham (to appear). · Zbl 0639.60063
[11] Soner, H. M.,Optimal Control of a One-Dimensional Storage Process, Applied Mathematics and Optimization, Vol. 13, pp. 175-191, 1985. · Zbl 0572.90021 · doi:10.1007/BF01442206
[12] Oleinik, O. A., andRadevic, E. V.,Second-Order Equations with Nonnegative Characteristic Form, American Mathematical Society Translations, Providence, Rhode Island, 1973.
[13] Evans, L. C., andMenaldi, J. L.,Gradient Bounds for Solutions of Variational Inequalities, Applied Mathematics and Optimization, Vol. 7, pp. 247-252, 1981. · Zbl 0477.49009 · doi:10.1007/BF01442119
[14] Perthame, B.,Quasi-Variational Inequalities and Hamilton-Jacobi-Bellman Equations in a Bounded Region, Communications in Partial Differential Equations, Vol. 9, pp. 561-595, 1984. · doi:10.1080/03605308408820342
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.