Abstract
We consider the problem of optimal stochastic scheduling for nonlinear systems with Poisson noise disturbances and a performance index including both operating costs and costs for scheduling changes. In general, the value functions of the dynamic programming, quasi-variational inequalities which define the optimality conditions for such problems are not differentiable. However, we can treat them as “viscosity solutions” as introduced by Crandall and Lions. Existence and uniqueness questions are studied from this point of view.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Crandall MG, Lions PL (1983) Viscosity solution of Hamilton-Jacobi equation. Trans Amer Math Soc 277:1–42
Lions PL (1982) Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London
Crandall MG, Evans LC, Lions PL (1984) Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc 282:487–502
Capuzzo Dolcetta I, Evans LC (1984) Optimal switching for ordinary differential equations. SIAM J Control Optim 22:143–161
Evans LC, Friedman A (1979) Optimal stochastic switching and the Dirichlet problem for the Bellman equation. Trans Amer Math Soc 253:365–389
Lenhart SM, Belbas SA (1983) A system of nonlinear partial differential equations arising in the optimal control of stochastic systems with switching costs. SIAM J Appl Math 43:465–475
Bensoussan A (1978) On the Hamilton-Jacobi approach for the optimal control of diffusion processes with jumps. In: Friedman A, Pinsky M (eds) Stochastic Analysis. Academic Press, Dordrecht, pp 25–55
Evans LC (1983) Nonlinear systems in optimal control theory and related topics. In: Ball JM (eds) Systems of Nonlinear Partial Differential Equations. Reidel, Dordrecht, pp. 95–113
Evans LC (1980) On solving certain nonlinear partial differential equations by accretive operator methods. Israel J Math 36:225–247
Menaldi JL (1980) On the optimal stopping time problem for degenerate diffusions. SIAM J Control Optim 18:697–721
Menaldi JL (1980) On the optimal impulse control problem for degenerate diffusions. SIAM J Control Optim 18:722–739
Blankenship GL, Menaldi JL (1982) Optimal stochastic scheduling of power generation systems with scheduling delays and large cost differentials. SIAM J Control Optim 22:121–132
Lions PL (1983) Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, 1. Comm Partial Differential Equations 8:1101–1174
Lions PL (1983) Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, 2. Comm Partial Differential Equations 8:1229–1276
Li CW, Blenkenship GL Optimal stochastic control of linear systems with Poisson noise disturbances (submitted for publication)
Li CW, Blankenship GL (1986) Almost sure stability of linear stochastic systems with poisson process coefficients. SIAM J Appl Math 46:875–911
Li CW (1984) Almost Sure Stability, Optimal Control and Scheduling of Stochastic Systems with Point Process Coefficients. Ph.D. Dissertation, Applied Mathematics Program, University of Maryland, College Park
Dynkin EB (1965) Markov Processes, vol. 1. Springer-Verlag, Berlin
Author information
Authors and Affiliations
Additional information
This author's research was conducted in the Applied Mathematics Program at the University of Maryland, College Park. It was supported in part by the Department of Energy under contract DE-AC01-79ET-29244.
This author's research was supported in part by the Army Research Office under contract DAAG29-83-C-0028 with SEI, Greenbelt, MD. Address correspondence concerning the paper to this author.
Rights and permissions
About this article
Cite this article
Li, C.W., Blankenship, G.L. Optimal stochastic scheduling of systems with Poisson noises. Appl Math Optim 15, 187–221 (1987). https://doi.org/10.1007/BF01442652
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01442652