Discrete-Time Hybrid Control Processes with Unbounded Costs

• Published in: Applied mathematics & optimization Vol. 90; no. 3; p. 51
• Main Authors: Jasso-Fuentes, Héctor, Salgado-Suárez, Gladys D.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2024; Springer Nature B.V
• Subjects: Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Control; Costs; Dynamic programming; Hybrid control; Inventory control; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Optimal control; Optimization; Simulation; Stochastic models; Systems Theory; Theoretical; 93E20; 34A38; 60J05; Dynamic programming; Markov decision processes; Hybrid control
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-024-10192-9

This paper extends the results provided in Jasso-Fuentes et al. (Appl Math Optim 81(2):409–441, 2020b) and Jasso-Fuentes et al. (Pure Appl Funct Anal 9(3):675–704, 2024) regarding the study of discrete-time hybrid stochastic models with general spaces and total discounted payoffs. This extension incorporates the handling of negative and/or unbounded costs per stage. In particular, it encompasses interesting applications, such as scenarios where the controller optimizes net costs, social welfare costs, or distances between points. These situations arise when assumptions of both non-negativeness and boundedness on the cost per stage do not apply. Our proposal relies on Lyapunov-like conditions, enabling, among other aspects, the finiteness of the value function and the existence of solutions to the associated dynamic programming equation. This equation is crucial for deriving optimal control policies. To illustrate our theory, we include an example in inventory-manufacturing management, highlighting its evident hybrid nature.