The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem with SIR dynamics main feature of our study is the presence of state constraints (related to ...intensive care units ICU capacity) and strict target objectives (related to the immunity threshold). The first class of results provides a comprehensive description of different zones of interest using viability tools. The second achievement is a thorough mathematical analysis of Pontryagin extremals for the aforementioned problem allowing to obtain an explicit closed-loop feedback optimal control. All our theoretical results are numerically illustrated for a further understanding of the geometrical features and scenarios.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean–Vlasov type. Motivated by the recent interest in ...mean-field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic controlled systems with mean-field interactions when subject to a common policy.
Full text
Available for:
BFBNIB, INZLJ, NMLJ, NUK, PNG, SAZU, UL, UM, UPUK, ZRSKP
Using a recently introduced representation of the second order adjoint state as the solution of a function-valued backward stochastic partial differential equation (SPDE), we calculate the viscosity ...super- and subdifferential of the value function evaluated along an optimal trajectory for controlled semilinear SPDEs. This establishes the well-known connection between Pontryagin's maximum principle and dynamic programming within the framework of viscosity solutions. As a corollary, we derive that the correction term in the stochastic Hamiltonian arising in nonsmooth stochastic control problems is nonpositive. These results directly lead us to a stochastic verification theorem for fully nonlinear Hamilton–Jacobi–Bellman equations in the framework of viscosity solutions.
The generally, systems control the saturation results in their dynamics, either in their control signal, in their states or in both. This paper presents the theory and wording of the Pontryagin ...principle, which has been applied as this type of continuous time problems, to obtain control laws that optimize the index in question. Likewise, a problem is presented and solved with which the application of the Pontryagin principle is exposed and finally the conclusions are presented.
In this paper, we determine the optimal control of plant disease model with roguing, replanting, curative, and preventive treatment using the Maximum Pontryagin principle. Numerical simulation ...results show that the procedure can reduce the population of infected plants. Therefore, controlling by roguing, replanting, curative, and preventive treatment is highly recommended to increase the number of susceptible, removed, and protected plants.
We study deterministic nonstationary discrete-time optimal control problems in both finite and infinite horizon. With the aid of Gâteaux differentials, we prove a discrete-time maximum principle in ...analogy with the well-known continuous-time maximum principle. We show that this maximum principle, together a transversality condition, is a necessary condition for optimality; we also show that it is sufficient under additional hypothesis. We use Gâteaux differentials as a natural setting to derive first-order conditions. Additionally, we use the discrete-time maximum principle to derive the discrete-time Euler equation and to characterize Nash equilibria for discrete-time dynamic games.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In the mid‐1950s, Pontryagin et al. published a principle that became a fundamental concept in optimal control (OC) theory. The principle provides theoretical and practical methods to find the ...solution of OC problems, in particular, open‐loop control problems. In chemical engineering, the principle has played an important role as a decision making framework for more than 60 years. This study gathers the main contributions on the application of the Pontryagin's principle to the dynamic optimization of chemical processes. A concise overview of the optimality conditions for a wide class of constrained OC problems is provided. Numerical methods to solve the necessary conditions and strategies to address inequality constraints are summarized. The information and illustrative case study presented in this work can be used as a guide to implement the principle in different settings. Opportunities for further application of the principle in relevant chemical engineering problems are also discussed.
Full text
Available for:
BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
A proposed two-dimensional modified Lotka-Volterra fishery model in terms of predator-prey aims to explore the effect of non-selective harvesting on the predator and prey populations. The study ...delves into various essential aspects of the dynamical system, including positivity, uniform boundedness, and persistence. Points of equilibrium are identified. The system's local and global stability are thoroughly examined and discussed. Moreover, the research explores the concept of bionomic equilibrium, a scenario where economic rent is entirely dissipated. Extending the bioeconomic model, the study investigates a linear optimal control problem to determine the most effective harvesting strategy. Utilising Pontryagin's maximum principle, the optimal control is characterised. The findings indicate that maximum allowable effort should be exerted whenever the net revenue per unit effort surpasses the total net marginal revenue of predator and prey stocks. Numerical simulations, using data on the marine artisanal fishery in Ghana, are conducted to validate the theoretical discoveries. The outcomes reveal that the fishery can support sustainable harvesting of both predator (tuna) and prey (sardinella) populations, as long as the optimal harvesting effort is set at 100,000 fishing trips annually.
We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the ...major player faces an optimization over conditional McKean-Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibriums for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is of independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the Appendix), we reduce the solution of the mean field game to a forward-backward system of stochastic differential equations of the conditional McKean-Vlasov type, which we solve in the linear quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those originally found in the literature.
PDF
