Journal articles: 'Bellman optimality principle'

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Garaev, K. G. "A remark on the Bellman principle of optimality." Journal of the Franklin Institute 335, no. 2 (1998): 395–400. http://dx.doi.org/10.1016/s0016-0032(96)00125-1.

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Chirima, J., F. R. Matenda, E. Chikodza, and M. Sibanda. "Dynamic programming principle for optimal control of uncertain random differential equations and its application to optimal portfolio selection." Review of Business and Economics Studies 12, no. 3 (2024): 74–85. http://dx.doi.org/10.26794/2308-944x-2024-12-3-74-85.

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This study aimed to examine an uncertain stochastic optimal control problem premised on an uncertain stochastic process. The proposed approach is used to solve an optimal portfolio selection problem. This paper’s research is relevant because it outlines the procedure for solving optimal control problems in uncertain random environments. We implement Bellman’s principle of optimality method in dynamic programming to derive the principle of optimality. Then the resulting Hamilton-Jacobi-Bellman equation (the equation of optimality in uncertain stochastic optimal control) is used to solve a proposed portfolio selection problem. The results of this study show that the dynamic programming principle for optimal control of uncertain stochastic differential equations can be applied in optimal portfolio selection. Also, the study results indicate that the optimal fraction of investment is independent of wealth. The main conclusion of this study is that, in Itô-Liu financial markets, the dynamic programming principle for optimal control of uncertain stochastic differential equations can be applied in solving the optimal portfolio selection problem.

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Zhu, Yingjun, and Guangyan Jia. "Dynamic Programming and Hamilton–Jacobi–Bellman Equations on Time Scales." Complexity 2020 (November 19, 2020): 1–11. http://dx.doi.org/10.1155/2020/7683082.

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Bellman optimality principle for the stochastic dynamic system on time scales is derived, which includes the continuous time and discrete time as special cases. At the same time, the Hamilton–Jacobi–Bellman (HJB) equation on time scales is obtained. Finally, an example is employed to illustrate our main results.

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Tarasenko, A., and I. Egorova. "THE OPTIMAL PRINCIPLE OF BELLMAN IN THE PROBLEM OF OPTIMAL MEANS DISTRIBUTION BETWEEN ENTERPRISES FOR THE EXPANSION OF PRODUCTION." Vestnik Universiteta, no. 10 (November 28, 2019): 132–38. http://dx.doi.org/10.26425/1816-4277-2019-10-132-138.

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The method of dynamic programming has been considered, which is used in solving multiple problems in economics, on the example of using Bellman’s optimality principle for solving nonlinear programming problems. On a specific numerical example, the features of the solution have been shown in detail with all the calculations. The problem of optimal distribution of funds among enterprises for the expansion of production has been formulated, which would give the maximum total increase in output. The solution of the task has been presented in the case, when the number of enterprises is 3. It has been shown, that the Bellman optimality principle allows you solve applied problems of cost forecasting with obtaining the optimal solution-maximum profit at minimum costs.

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Gani, Shrishail Ramappa, and Shreedevi Veerabhadrappa Halawar. "Optimal control analysis of deterministic and stochastic epidemic model with media awareness programs." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 9, no. 1 (2018): 24–35. http://dx.doi.org/10.11121/ijocta.01.2019.00423.

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The present study considered the optimal control analysis of both deterministic differential equation modeling and stochastic differential equation modeling of infectious disease by taking effects of media awareness programs and treatment of infectives on the epidemic into account. Optimal media awareness strategy under the quadratic cost functional using Pontrygin's Maximum Principle and Hamiltonian-Jacobi-Bellman equation are derived for both deterministic and stochastic optimal problem respectively. The Hamiltonian-Jacobi-Bellman equation is used to solve stochastic system, which is fully non-linear equation, however it ought to be pointed out that for stochastic optimality system it may be difficult to obtain the numerical results. For the analysis of the stochastic optimality system, the results of deterministic control problem are used to find an approximate numerical solution for the stochastic control problem. Outputs of the simulations shows that media awareness programs place important role in the minimization of infectious population with minimum cost.

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Корнеев, А. М., Т. В. Лаврухина та Т. А. Сметанникова. "Обоснование выбора метода для распределения региональных ресурсов МЧС". ТЕНДЕНЦИИ РАЗВИТИЯ НАУКИ И ОБРАЗОВАНИЯ 70, № 1 (2021): 25–29. http://dx.doi.org/10.18411/lj-02-2021-07.

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The paper describes the choice of a method for the correct and correct allocation of resources within the Ministry of Emergency Situations management. As such a method, it was decided to choose the Bellman algorithm. A mathematical model is introduced that allows applying the chosen optimality principle. The choice of the general type of function that is most suitable for resource allocation is given.

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Gomoyunov, M. I. "On the Relationship Between the Pontryagin Maximum Principle and the Hamilton–Jacobi–Bellman Equation in Optimal Control Problems for Fractional-Order Systems." Дифференциальные уравнения 59, no. 11 (2023): 1515–21. http://dx.doi.org/10.31857/s0374064123110067.

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We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is studied. It is proved that the costate variable in the Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of the optimal result functional calculated along the optimal motion.

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Kuz’michev, Venedikt, Ilia Krupenich, Evgeny Filinov, and Andrey Tkachenko. "Optimization of gas turbine engine control using dynamic programming." MATEC Web of Conferences 220 (2018): 03002. http://dx.doi.org/10.1051/matecconf/201822003002.

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The aim of engine control optimization is to derive the optimal control law for engine operation managing during the aircraft flight. For numerical modeling a continuous flight process defined by a system of differential equations is replaced by a discrete multi-step process. Values of engine control parameters in particular step uniquely identify a system transitions from one state to another. The algorithm is based on the numerical method of dynamic programming and the Bellman optimality principle. The task is represented as a sequence of nested optimization subtasks, so that control optimization at the first step is external to all others. The optimum control function can be determined using the minimax principle of optimality. Aircraft performance calculation is performed by numerical integration of differential equations of aircraft movement.

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Ma, Han, and Qimin Zhang. "Threshold dynamics and optimal control on an age-structured SIRS epidemic model with vaccination." Mathematical Biosciences and Engineering 18, no. 6 (2021): 9474–95. http://dx.doi.org/10.3934/mbe.2021465.

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<abstract><p>We consider a vaccination control into a age-structured susceptible-infective-recovered-susceptible (SIRS) model and study the global stability of the endemic equilibrium by the iterative method. The basic reproduction number $ R_0 $ is obtained. It is shown that if $ R_0 &lt; 1 $, then the disease-free equilibrium is globally asymptotically stable, if $ R_0 &gt; 1 $, then the disease-free and endemic equilibrium coexist simultaneously, and the global asymptotic stability of endemic equilibrium is also shown. Additionally, the Hamilton-Jacobi-Bellman (HJB) equation is given by employing the Bellman's principle of optimality. Through proving the existence of viscosity solution for HJB equation, we obtain the optimal vaccination control strategy. Finally, numerical simulations are performed to illustrate the corresponding analytical results.</p></abstract>

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Shen, Jiayu, Yueqiang Jin, and Bing Liu. "Expected Value Model of an Uncertain Production Inventory Problem with Deteriorating Items." Journal of Advanced Computational Intelligence and Intelligent Informatics 26, no. 5 (2022): 684–90. http://dx.doi.org/10.20965/jaciii.2022.p0684.

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In this study, we present an optimal control model for an uncertain production inventory problem with deteriorating items. The dynamics of the model includes perturbation by an uncertain canonical process. An expected value optimal control model is established based on the uncertainty theory. The aim of this study is to apply the optimal control theory to solve a production inventory problem with deteriorating items and derive an optimal inventory level and production rate that would maximize the expected revenue. The uncertainty theory is used to obtain the equation of optimality. The Hamilton–Jacobi–Bellman (HJB) principle is used to solve the equation of optimality. The results are discussed using numerical experiments for different demand functions.

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Diep, Thang, Nguyen Phung Quang, and Nguyen Duc Huy. "NOVEL CONTROL APPROACH FOR OPTIMAL POWER FLOW IN HYBRID WIND-PHOTOVOLTAIC-DIESEL GENERATION SYSTEMS." Journal of Computer Science and Cybernetics 33, no. 2 (2017): 180–92. http://dx.doi.org/10.15625/1813-9663/33/2/8898.

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The paper deals with a formulation of an analytical control model for optimal power flow in an islanded microgrid (MG). In an MG with load change, wind power fluctuation, sun irradiation power disturbance, that can significant influence the power flow, and hence the power flow control problem in real life system faces some new challenges. In order to maintain the balance of power flow, a diesel engine generator (DEG) needs to be scheduled. The objective of the control problem is to find the DEG output power by minimizing the total cost of energy. Using the Bellman principle, the optimality conditions obtained satisfy the Hamilton-Jacobi-Bellman equation, which depends on time and system states, and ultimately, leads to a feedback control or to the so called energy management to be implemented in a SCADA system

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Li, Ai Zhong, and Ruo En Ren. "Continuous-Time Optimal Portfolio Selection Strategy with Redemption Based on Stochastic Control." Advanced Materials Research 271-273 (July 2011): 592–96. http://dx.doi.org/10.4028/www.scientific.net/amr.271-273.592.

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In this paper a continuous-time portfolio optimization decision with the redemption is made, a typical portfolio selection model is established by use of Bellman principle of optimality and HJB equation, we derive the optimal strategy and efficient frontier with general stochastic control technique. Its research methodologies can be applied in the practical work such as investment funds management and financial risk management to raise the scientificity of decisions. It is of great referential and inspirational value to provide solutions to practical problem in real investment process.

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13

Mohan, Manil T. "Dynamic programming and feedback analysis of the two dimensional tidal dynamics system." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 109. http://dx.doi.org/10.1051/cocv/2020025.

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In this work, we consider the controlled two dimensional tidal dynamics equations in bounded domains. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled 2D tidal dynamics equations. The existence of an optimal control is shown and the dynamic programming method for the optimal control of 2D tidal dynamics system is also described. We show that the feedback control can be obtained from the solution of an infinite dimensional Hamilton-Jacobi equation. The non-differentiability and lack of smoothness of the value function forced us to use the method of viscosity solutions to obtain a solution of the infinite dimensional Hamilton-Jacobi equation. The Bellman principle of optimality for the value function is also obtained. We show that a viscosity solution to the Hamilton-Jacobi equation can be used to derive the Pontryagin maximum principle, which give us the first order necessary conditions of optimality. Finally, we characterize the optimal control using the adjoint variable.

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Kuznetsov, Sergey. "Using Bellman Optimality Principle for the Generative Autoencoder Architecture for the Problems of the Attribute Data Typesetting and Semantic Description in Data Management." Differential Equations and Control Processes, no. 2 (2024): 171–82. https://doi.org/10.21638/11701/spbu35.2024.207.

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The publication presents the problems of identifying data types (typesetting) and semantic description of the attributes when managing structured data and master data (Master Data Management). A formal definition of the generalized attribute typesetting problem is given, which allows generation of the additional data types. This problem allows using the discrete Bellman optimality principle under special criteria of the target function. A unified architecture of the deep generative neural network addressing simultaneously the generalized attribute typesetting and semantic description generation problems is proposed. The architecture is based on the generative adversarial autoencoder architecture (AAE) using the mechanisms of soft-attention, and long-term memory (SCRN). The effectiveness of such implementation, in particular, is achieved through the application of the principles of dynamic programming within each epoch of the network training.

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15

Diep-Thanh, Thang, Quang Nguyen-Phung, and Huy Nguyen-Duc. "Stochastic control for optimal power flow in islanded microgrid." International Journal of Electrical and Computer Engineering (IJECE) 9, no. 2 (2019): 1045. http://dx.doi.org/10.11591/ijece.v9i2.pp1045-1057.

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<p>The problem of optimal power flow (OPF) in an islanded mircrogrid (MG) for hybrid power system is described. Clearly, it deals with a formulation of an analytical control model for OPF. The MG consists of wind turbine generator, photovoltaic generator, and diesel engine generator (DEG), and is in stochastic environment such as load change, wind power fluctuation, and sun irradiation power disturbance. In fact, the DEG fails and is repaired at random times so that the MG can significantly influence the power flow, and the power flow control faces the main difficulty that how to maintain the balance of power flow? The solution is that a DEG needs to be scheduled. The objective of the control problem is to find the DEG output power by minimizing the total cost of energy. Adopting the Rishel’s famework and using the Bellman principle, the optimality conditions obtained satisfy the Hamilton-Jacobi-Bellman equation. Finally, numerical examples and sensitivity analyses are included to illustrate the importance and effectiveness of the proposed model.</p>

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16

Diep-Thanh, Thang, Quang Nguyen-Phung, and Huy Nguyen-Duc. "Stochastic control for optimal power flow in islanded microgrid." International Journal of Electrical and Computer Engineering (IJECE) 9, no. 2 (2019): 1045–57. https://doi.org/10.11591/ijece.v9i2.pp1045-1057.

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The problem of optimal power flow (OPF) in an islanded mircrogrid (MG) for hybrid power system is described. Clearly, it deals with a formulation of an analytical control model for OPF. The MG consists of wind turbine generator, photovoltaic generator, and diesel engine generator (DEG), and is in stochastic environment such as load change, wind power fluctuation, and sun irradiation power disturbance. In fact, the DEG fails and is repaired at random times so that the MG can significantly influence the power flow, and the power flow control faces the main difficulty that how to maintain the balance of power flow? The solution is that a DEG needs to be scheduled. The objective of the control problem is to find the DEG output power by minimizing the total cost of energy. Adopting the Rishel’s famework and using the Bellman principle, the optimality conditions obtained satisfy the Hamilton-Jacobi-Bellman equation. Finally, numerical examples and sensitivity analyses are included to illustrate the importance and effectiveness of the proposed model.

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17

Olha, Kryazhych, Itskovych Victoria, Iushchenko Kateryna, and Kuprin Oleksii. "Features in solving individual tasks to develop service-oriented networks using dynamic programming." Eastern-European Journal of Enterprise Technologies 1, no. 4 (121) (2023): 34–40. https://doi.org/10.15587/1729-4061.2023.274144.

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The object of this study is an approach to solving the problems of designing service-oriented networks that warn about emergencies using dynamic programming. The main issue is the complexity of algorithmization of processes that describe the achievement of an optimal solution in multi-stage nonlinear problems. The possibilities of applying the Bellman optimality principle for solving the set tasks for the purpose of their application in the field of engineering and technology are determined. Based on the Bellman functional equation, a model of the optimal number of sensors in the monitoring system for warning of emergencies was built. A feature of the design is that using the classical Bellman equation, it is proposed to solve problems of various technical directions, provided that the resource determines what exactly makes it possible to optimize work in any way. Important with this approach is the planning of the action as an element of some problem with the augmented state. After that, the proposed structure in formal form extends to other objects. A problem was proposed and considered, which confirmed the mathematical calculations, as a result of which an optimal plan for replacing the sensors of the system was obtained; and the possibilities of significant cost reduction were identified. In the considered example, an optimal plan for replacing the system sensors was compiled and the possibility of reducing costs by 31.9 % was proved. The proposed option was used in the development of information technology for modeling a service-oriented network based on energy-efficient long-range protocols; some of the identified features were further developed in the design of a recommendation system for issuing loans and developing an interactive personnel training system

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18

Bahlali, Dounia, and Farid Chighoub. "A general time-inconsistent stochastic optimal control problem with delay." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 2 (2024): e6922. http://dx.doi.org/10.54021/seesv5n2-115.

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In this paper, we develop a theory addressing a broad class of time-inconsistent stochastic control problems characterized by stochastic differential delayed equations (SDDEs), indicating the absence of a Bellman optimality principle. The approach involves framing these problems within a game theoretic framework and seeking subgame perfect Nash equilibrium strategies. For a general controlled process with delay and a reasonably broad objective functional, we extend the standard Bellman equation into a system of nonlinear equations. This extension facilitates the determination of both the equilibrium strategy and the equilibrium value function. Importantly, to exemplify the theory’s applicability, we delve into specific example such mean-variance portfolio with state dependent risk aversion problem with delay, where we formulate the optimal investment mean-variance problem within a game theoretic framework and then applying a stochastic control theory with delay. Next by solving the extended HJB equations and constructing an exponential martingale process the explicit- form solution of the optimal investment strategy and the corresponding equilibrium value function are derived. This analysis not only extends the theoretical foundations but also provides insights into addressing and resolving time inconsistency in a practical example.

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19

Chan, Terence. "Some diffusion models for the mabinogion sheep problem of williams." Advances in Applied Probability 28, no. 3 (1996): 763–83. http://dx.doi.org/10.2307/1428180.

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The ‘Mabinogion sheep’ problem, originally due to D. Williams, is a nice illustration in discrete time of the martingale optimality principle and the use of local time in stochastic control. The use of singular controls involving local time is even more strikingly highlighted in the context of continuous time. This paper considers a class of diffusion versions of the discrete-time Mabinogion sheep problem. The stochastic version of the Bellman dynamic programming approach leads to a free boundary problem in each case. The most surprising feature in the continuous-time context is the existence of diffusion versions of the original discrete-time problem for which the optimal boundary is different from that in the discrete-time case; even when the optimal boundary is the same, the value functions can be very different.

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Chan, Terence. "Some diffusion models for the mabinogion sheep problem of williams." Advances in Applied Probability 28, no. 03 (1996): 763–83. http://dx.doi.org/10.1017/s0001867800046486.

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The ‘Mabinogion sheep’ problem, originally due to D. Williams, is a nice illustration in discrete time of the martingale optimality principle and the use of local time in stochastic control. The use of singular controls involving local time is even more strikingly highlighted in the context of continuous time. This paper considers a class of diffusion versions of the discrete-time Mabinogion sheep problem. The stochastic version of the Bellman dynamic programming approach leads to a free boundary problem in each case. The most surprising feature in the continuous-time context is the existence of diffusion versions of the original discrete-time problem for which the optimal boundary is different from that in the discrete-time case; even when the optimal boundary is the same, the value functions can be very different.

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21

Panteleev, Andrei V., and Anna A. Kolessa. "Optimal Open-Loop Control of Discrete Deterministic Systems by Application of the Perch School Metaheuristic Optimization Algorithm." Algorithms 15, no. 5 (2022): 157. http://dx.doi.org/10.3390/a15050157.

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A new hybrid metaheuristic method for optimizing the objective function on a parallelepiped set of admissible solutions is proposed. It mimics the behavior of a school of river perch when looking for food. The algorithm uses the ideas of several methods: a frog-leaping method, migration algorithms, a cuckoo algorithm and a path-relinking procedure. As an application, a wide class of problems of finding the optimal control of deterministic discrete dynamical systems with a nonseparable performance criterion is chosen. For this class of optimization problems, it is difficult to apply the discrete maximum principle and its generalizations as a necessary optimality condition and the Bellman equation as a sufficient optimality condition. The desire to extend the class of problems to be solved to control problems of trajectory bundles and stochastic problems leads to the need to use not only classical adaptive random search procedures, but also new approaches combining the ideas of migration algorithms and swarm intelligence methods. The efficiency of this method is demonstrated and an analysis is performed by solving several optimal deterministic discrete control problems: two nonseparable problems (Luus–Tassone and LiHaimes) and five classic linear systems control problems with known exact solutions.

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22

Xu, Dengguo, Qinglin Wang, and Yuan Li. "Adaptive Optimal Robust Control for Uncertain Nonlinear Systems Using Neural Network Approximation in Policy Iteration." Applied Sciences 11, no. 5 (2021): 2312. http://dx.doi.org/10.3390/app11052312.

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In this study, based on the policy iteration (PI) in reinforcement learning (RL), an optimal adaptive control approach is established to solve robust control problems of nonlinear systems with internal and input uncertainties. First, the robust control is converted into solving an optimal control containing a nominal or auxiliary system with a predefined performance index. It is demonstrated that the optimal control law enables the considered system globally asymptotically stable for all admissible uncertainties. Second, based on the Bellman optimality principle, the online PI algorithms are proposed to calculate robust controllers for the matched and the mismatched uncertain systems. The approximate structure of the robust control law is obtained by approximating the optimal cost function with neural network in PI algorithms. Finally, in order to illustrate the availability of the proposed algorithm and theoretical results, some numerical examples are provided.

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23

Kryazhych, Olha, Victoria Itskovych, Kateryna Iushchenko, and Oleksii Kuprin. "Features in solving individual tasks to develop service-oriented networks using dynamic programming." Eastern-European Journal of Enterprise Technologies 1, no. 4 (121) (2023): 34–40. http://dx.doi.org/10.15587/1729-4061.2023.274144.

Full text

Abstract:

The object of this study is an approach to solving the problems of designing service-oriented networks that warn about emergencies using dynamic programming. The main issue is the complexity of algorithmization of processes that describe the achievement of an optimal solution in multi-stage nonlinear problems. The possibilities of applying the Bellman optimality principle for solving the set tasks for the purpose of their application in the field of engineering and technology are determined. Based on the Bellman functional equation, a model of the optimal number of sensors in the monitoring system for warning of emergencies was built. A feature of the design is that using the classical Bellman equation, it is proposed to solve problems of various technical directions, provided that the resource determines what exactly makes it possible to optimize work in any way. Important with this approach is the planning of the action as an element of some problem with the augmented state. After that, the proposed structure in formal form extends to other objects. A problem was proposed and considered, which confirmed the mathematical calculations, as a result of which an optimal plan for replacing the sensors of the system was obtained; and the possibilities of significant cost reduction were identified. In the considered example, an optimal plan for replacing the system sensors was compiled and the possibility of reducing costs by 31.9 % was proved. The proposed option was used in the development of information technology for modeling a service-oriented network based on energy-efficient long-range protocols; some of the identified features were further developed in the design of a recommendation system for issuing loans and developing an interactive personnel training system

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Xu, Dengguo, Qinglin Wang, and Yuan Li. "Adaptive optimal control approach to robust tracking of uncertain linear systems based on policy iteration." Measurement and Control 54, no. 5-6 (2021): 668–80. http://dx.doi.org/10.1177/00202940211007177.

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In this study, an optimal adaptive control approach is established to solve the robust output tracking problem of a class of continuous time uncertain linear systems based on the policy iteration (PI) in actor-critic algorithm. First, by augmenting the integral variables of the tracking error into state variables, the robust tracking problem is transformed into a robust control problem of an augmented uncertain linear system. It is proven that the robust control law of the augmented system enables the output of the considered system to track a polynomial time signal asymptotically. Second, an optimal control method in the corresponding auxiliary nominal system is established, and based on the Bellman optimality principle, PI algorithms are proposed to solve online tracking controllers for the matched and the mismatched uncertain systems. Finally, for testing the availability of the proposed approach and theoretical results, two numerical experiments are provided.

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Udrişte, Constantin, and Ionel Ţevy. "Minirobots Moving at Different Partial Speeds." Mathematics 8, no. 6 (2020): 1036. http://dx.doi.org/10.3390/math8061036.

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In this paper, we present the mathematical point of view of our research group regarding the multi-robot systems evolving in a multi-temporal way. We solve the minimum multi-time volume problem as optimal control problem for a group of planar micro-robots moving in the same direction at different partial speeds. We are motivated to solve this problem because a similar minimum-time optimal control problem is now in vogue for micro-scale and nano-scale robotic systems. Applying the (weak and strong) multi-time maximum principle, we obtain necessary conditions for optimality and that are used to guess a candidate control policy. The complexity of finding this policy for arbitrary initial conditions is dominated by the computation of a planar convex hull. We pointed this idea by applying the technique of multi-time Hamilton-Jacobi-Bellman PDE. Our results can be extended to consider obstacle avoidance by explicit parameterization of all possible optimal control policies.

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Mikhalov, Oleksandr Illich, Oleksandr Afrykanovych Stenin, Viktor Petrovych Pasko, Oleksandr Serhiiovych Stenin, and Yurii Opanasovych Tymoshyn. "Situational planning and operational adjustment of the route of the Autonomous robotic underwater vehicle." System technologies 3, no. 122 (2019): 3–11. http://dx.doi.org/10.34185/1562-9945-3-122-2019-01.

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Currently, missions (tasks) for the underwater robot formed using imperative programming methods (both text and graphic), describing in detail the sequence of robot actions that need performed to achieve the desired goal. At the same time, only the operator of the underwater robot, which makes up the mission, for example, the delivery of cargo to the target point, has an idea of the goal itself. Such technology is effective if the robot's mission carried out within a priori scenario. In other cases, it can either not be executed at all, or it can be executed with large violations and a threat to the safety of the device.When assessing the effectiveness of an underwater robot, the degree of its information autonomy, i.e. the ability to act independently in an unknown or insufficiently defined environment, is of fundamental importance. Therefore, the "intellectualization" of the Autonomous control system of the underwater robot is extremely important for the mission in unforeseen circumstances. For this propose to use intelligent decision support system. Two ways to implement optimal decision-making strategies based on the mathematical apparatus of the theory of Markov and semi-Markov processes using the Bellman optimality principle propose. The considered ways of implementation of optimal strategies of decision - making process relate to the strategy for a short finite time of cargo delivery, which is the most common in practice, and for a long interval of cargo delivery relative to the entire task. In addition, the article discusses ways to find optimal strategies when the time of making single decisions is fixed or when the time of translation is implement randomly.Hence, the situational approach to decision-making in the planning of the route ARPA is very relevant and allows not only to assess the possible situation on the route, but also to determine the control solutions for the operational adjustment of the route using the intelligent decision support system (ISPR). The development of models of the routing process based on the representation of the situational model in the form of nodes of the graph, the transitions of which correspond to the control solutions.The paper proposes two ways to implement optimal strategies of decision - making based on the mathematical apparatus of the theory of Markov and semi-Markov processes using the Bellman principle of optimality.

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Ma, Ruoxun, Lipo Mo, and Bokang Zhou. "Optimal Model-Free Mean-Square Consensus for Multi-Agents with Markov Switching Topology." Applied Sciences 14, no. 22 (2024): 10273. http://dx.doi.org/10.3390/app142210273.

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Due to the real applications, optimal consensus reinforcement learning with switching topology is still challenging due to the complexity of topological changes. This paper investigates the optimal consensus control problem for discrete multi-agent systems under Markov switching topologies. The goal is to design an appropriate algorithm to find the optimal control policies that minimize the performance index while achieving consensus among the agents. The concept of mean-square consensus is introduced, and the relationship between consensus error and tracking error to achieve mean-square consensus is studied. A performance function for each agent under switching topologies is established and a policy iteration algorithm using system data is proposed based on the Bellman optimality principle. The theoretical analysis shows that the consensus error realizes mean-square consensus and the performance function is optimized. The efficacy of the suggested approach is confirmed by numerical simulation using an actor–critic neural network. As a result, the value function is the optimum and the mean-square consensus can be reached using this technique.

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Koroleva, Valentina, Evgeny Filippov, Valeria Yachmeneva, and Bulat Ziganshin. "MATHEMATICAL MODEL OF THE PROBLEM OF EQUIPMENT REPLACEMENT." Vestnik of Kazan State Agrarian University 17, no. 3 (2022): 90–95. http://dx.doi.org/10.12737/2073-0462-2022-94-99.

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The study was carried out with the aim of solving the problem of replacing equipment and creating a mathematical model of the problem of controlled optimization, as the most complete and realistic description of the production process. The Bellman principle of optimality was applied in the study, since when solving dynamic programming problems, this principle is general. For the solution, the Bellman functional equation was chosen as the initial mathemat cal model:       The advantage of such a mathematical model is the possibility of modifying the problem. At the first step, it is proposed to sell the replaced equipment at the price (R(tk)-Z(tk))p, 0<p<1, p is the markdown coefficient for the sale. Further, the installed equipment may not be new and purchased at a price ((R(tk)-Z(tk))q, 0<q<1. Here q is the discount factor for purchase and is set at the input. Not new equipment is cheaper than new and sale of replaced equipment also contributes to profits New components must be added to the control vector - the age of the equipment being installed (not new) The control structure becomes more complex and realistic The effect of the "curse of dimensionality" does not appear during numerous tests of the program Three modifications of the problem have been developed on equipment replacement. Replaced equipment is thrown away. You can not throw it away, but sell it at a price (Rk(tk)-Zk(tk))p, where p is a markdown coefficient 0<p<1. You can increase profits if you buy not new equipment The paper considers the option of adding another component to the control vector - "rejuvenating" repair of equipment. After such repair, the age of the equipment becomes, for example, one year less. Repairs are taken from the graph of profit earned versus repair cost, which is obtained from the results of the program runs. Combining the above modifications into a single whole is a mathematical model of the problem of replacing equipment, which is the basis of the program.

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Chertov, V. A., and D. E. Orlova. "The search algorithm for the optimal control of multilevel hierarchical system based on the use of fuzzy logic and the principle of optimality of Bellman." Journal of Physics: Conference Series 1479 (March 2020): 012004. http://dx.doi.org/10.1088/1742-6596/1479/1/012004.

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Jere, Stanley, Elias R. Offen, and Othusitse Basmanebothe. "Optimal Investment, Consumption and Life Insurance Problem with Stochastic Environments." Journal of Mathematics Research 14, no. 4 (2022): 33. http://dx.doi.org/10.5539/jmr.v14n4p33.

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Optimal investment, consumption and life insurance problem with stochastic environments for a CRRA wage-earner is solved in this study.&nbsp; The wage-earner invests in the financial market with one risk-free security, one risky security, receives labor income and has a life insurance policy in the insurance market. A life insurance policy is purchased to hedge the financial wealth for the beneficiary in case of wage earner premature death. The interest rate and the volatility are stochastic. The stochastic interest rate dynamics of risk-free security follow a Ho-Lee model and the risky security follow Heston&rsquo;s model with stochastic volatility parameter dynamics following a Cox-Ingersoll-Ross (CIR) model. The objective of the wage-earner is to allocate wealth between risky security and risk-free security but also buy a life insurance policy during the investment period to maximize the expected discounted utilities derived from consumption, legacy and terminal wealth over an uncertain lifetime. By applying Bellman&#39;s optimality principle, the associated HJB PDE for the value function is established. The power utility function is employed for our analysis to obtain the value function and optimal policies. Finally, numerical examples and simulations are provided.

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Gmytrasiewicz, Piotr. "How to Do Things with Words: A Bayesian Approach." Journal of Artificial Intelligence Research 68 (August 17, 2020): 753–76. http://dx.doi.org/10.1613/jair.1.11951.

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   Communication changes the beliefs of the listener and of the speaker. The value of a communicative act stems from the valuable belief states which result from this act. To model this we build on the Interactive POMDP (IPOMDP) framework, which extends POMDPs to allow agents to model others in multi-agent settings, and we include communication that can take place between the agents to formulate Communicative IPOMDPs (CIPOMDPs). We treat communication as a type of action and therefore, decisions regarding communicative acts are based on decision-theoretic planning using the Bellman optimality principle and value iteration, just as they are for all other rational actions. As in any form of planning, the results of actions need to be precisely specified. We use the Bayes’ theorem to derive how agents update their beliefs in CIPOMDPs; updates are due to agents’ actions, observations, messages they send to other agents, and messages they receive from others. The Bayesian decision-theoretic approach frees us from the commonly made assumption of cooperative discourse – we consider agents which are free to be dishonest while communicating and are guided only by their selfish rationality. We use a simple Tiger game to illustrate the belief update, and to show that the ability to rationally communicate allows agents to improve efficiency of their interactions.  

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Zheglova, Yu G., and B. P. Titarenko. "METHOD FOR EVALUATING EXCAVATION SHORING DESIGN CONCEPTS." Herald of Dagestan State Technical University. Technical Sciences 47, no. 1 (2020): 86–92. http://dx.doi.org/10.21822/2073-6185-2020-47-1-86-92.

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Abstract. Aim.When conducting engineering and geological surveys at the conceptual stage of the feasibility study, it is necessary to create an automated system that allows the designer to take all possible factors into account and choose the most optimal design solution for the shoring of excavations.Method. The study is based on dynamic programming methods based on the principle of Bellman optimality. Logical convolution matrices are applied according to the dichotomy method.Results. As a result of the logical convolution of the aggregated criteria of the environmental parameters K12 along with the parameters of the surrounding buildings K34, a comprehensive assessment of the design solution for a foundation shoring fence is obtained such as to ensure the technical characteristics and safety of the development zone. However, a considerable influence on the choice of the design solution for the building envelope is exerted by the cost of its construction. Therefore, the indicator of a comprehensive assessment of the shoring fence considering the economic efficiency of the KOE is obtained by combining the indicator of a comprehensive assessment of the foundation shoring fence KO with the criterion of economic efficiency of the design decision K5.Conclusion. The analysis of methods for solving optimisation problems demonstrates that the method of complex estimates based on the theory of active systems is optimal for the task of evaluating design solutions for foundation shoring fences.

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Wu, Xiaoqing, and Lei Du. "Optimal Control of the Logistics Automation Transmission System Based on Partial Differential Equation." Mathematical Problems in Engineering 2022 (September 13, 2022): 1–12. http://dx.doi.org/10.1155/2022/1198954.

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In this paper, the finite difference scheme of the spatiotemporal fractional convection-diffusion equation is established, and its stability and convergence are proved. Furthermore, this discrete technique is extended to solve nonlinear spatiotemporal fractional convection-diffusion equations. By using the Krylov subspace method to solve the discrete system, the numerical solution of the spatiotemporal fractional convection-diffusion equation can be obtained quickly. In this paper, an efficient optimal control algorithm is proposed to solve the free control problem of a class of nonlinear time-delay systems. We obtained the optimal control law of the system through the Bellman optimality principle, obtained the asymptotic stability criterion of the system in the form of LMI under the optimal control input by using the Lyapunov stability theory, and discussed the effect of the delay parameter on the system stability. Using the principle of intelligent neural network approximation function, the evaluation neural network and the execution neural network are used to approximate the optimal performance index function and optimal control input, respectively, the optimal control strategy of the system is obtained, and the convergence of the weight estimation error is proved to be optimal. On the basis of optimal state adjustment, the optimal tracking control problem is further solved. Numerical example results verify the effectiveness of the proposed method in terms of stability analysis, optimal state control, and optimal tracking control for the nonlinear time-delay system proposed in this paper. We calculate the parameters of the conveyor and select a reasonable transmission and sorting mechanism to realize the speed regulation of the driving motor of each mechanism. Through the work of each part, the design scheme of the automatic transmission system is formed, and the reliability, practicability, and economy of the system are guaranteed.

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Panteleev, Andrei, and Elizaveta Khvoshnyanskaya. "Robust estimation of state vector coordinates in the controlled helicopter motion problem." E3S Web of Conferences 402 (2023): 02003. http://dx.doi.org/10.1051/e3sconf/202340202003.

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The problem of finding H∞ – a observer of the state vector of a linear continuous non-stationary dynamical system with a semi-infinite time of functioning is considered. It is assumed that a mathematical model of a closed-loop linear continuous deterministic dynamical system with an optimal linear regulator, found as a result of minimization of the quadratic quality criterion, is known. For solving the state observer synthesis problem the reduction of the problem to a min-max optimal control problem is used. In this problem, the minimum of the quality criterion is sought by the observer’s gain matrix, and the maximum – by the external influence, measurement noise, and initial conditions. To solve this problem, the extension principle is applied and sufficient optimality conditions are obtained that requires the choice of auxiliary functions of the Krotov–Bellman type. As a result of the implementation of the procedure for choosing an auxiliary function and using the rules of matrix differentiation, relations for the synthesis of the observer and formulas for finding the best matrix of observer gains, as well as the laws for choosing the worst external influences and noise, were obtained. We find a solution to the problem of state vector coordinates estimation in the presence of limited external influences and disturbances in a linear model of the measuring system. As an example, the equations of motion of the Raptor-type helicopter are used.

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Adamenko, Maryna, Ihor Afanasiev, Serhii Kapitula, and Alona Shahno. "INVESTMENT IN INNOVATIVE DEVELOPMENT OF COMPETITIVENESS OF RESOURCE AND PRODUCTION POTENTIAL OF MINING ENTERPRISES." Economic Analysis, no. 31(3) (2021): 105–14. http://dx.doi.org/10.35774/econa2021.03.105.

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The subject of the research is the processes of ensuring the competitiveness of the potential of mining enterprises based on certain key aspects of methodology and practice of rational management of investment distribution, which are inherent in the investment and innovation activities of subsidiaries within the parent company. The purpose and objectives of the study is to improve theoretical and methodological approaches to increase the competitiveness of the potential of mining enterprises on the basis of rational management of the allocation of investment resources in the process of their innovative development. Method (methodology). Solving issues of rational allocation of investment resources from the standpoint of proper competitiveness of mining enterprises is proposed to be carried out according to the method of optimizing multi-stage processes - the problem of dynamic programming of resource allocation between enterprises based on the principle of optimality R. Bellman. Results. As a result of research it is established that the most important criterion for optimizing the program of innovative development on the basis of rational distribution of investment resources is to choose the cost of sales, due to the dependence of financial performance of mining companies on total costs. The implementation of the proposed method of optimizing the allocation of investment resources is considered on the example of jointly operating in one company mining companies. Conclusions. Based on the generalization of research on theoretical and practical aspects related to improving the efficiency of managing the competitiveness of resource and production potential of mining enterprises, a methodological approach to prioritizing the allocation of investment resources in the context of innovation development working together in one company. The application of the developed methodological approach in practice allows the top management of large vertically integrated companies to rationally direct the allocation of investment resources to the stable innovative development of subsidiaries.

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Galperin, E. A. "Reflections on Bellman's optimality principle." Nonlinear Analysis: Theory, Methods & Applications 63, no. 5-7 (2005): e707-e713. http://dx.doi.org/10.1016/j.na.2005.01.069.

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37

Gross, Eitan. "On the Bellman’s principle of optimality." Physica A: Statistical Mechanics and its Applications 462 (November 2016): 217–21. http://dx.doi.org/10.1016/j.physa.2016.06.083.

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38

Sniedovich, M. "A new look at Bellman's principle of optimality." Journal of Optimization Theory and Applications 49, no. 1 (1986): 161–76. http://dx.doi.org/10.1007/bf00939252.

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Šimandl, Miroslav, and Marek Lešek. "Controlling of Pension Fund Investment by Using Bellman's Optimality Principle." IFAC Proceedings Volumes 36, no. 18 (2003): 419–24. http://dx.doi.org/10.1016/s1474-6670(17)34704-3.

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40

Wakuta, Kazuyoshi. "The Bellman's principle of optimality in the discounted dynamic programming." Journal of Mathematical Analysis and Applications 125, no. 1 (1987): 213–17. http://dx.doi.org/10.1016/0022-247x(87)90176-4.

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SHENG, LINXUE, and YUANGUO ZHU. "OPTIMISTIC VALUE MODEL OF UNCERTAIN OPTIMAL CONTROL." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21, supp01 (2013): 75–87. http://dx.doi.org/10.1142/s0218488513400060.

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Optimal control is an important field of study both in theory and in applications. Based on uncertainty theory, an expected value model of uncertain optimal control problem was studied by Zhu. In this paper, an optimistic value model for uncertain optimal control problem is investigated. Applying Bellman's principle of optimality, the principle of optimality for the model is presented. And then the equation of optimality is obtained for the optimistic value model of uncertain optimal control. Finally, a portfolio selection problem is solved by this equation of optimality.

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42

Garavello, Mauro, and Pierpaolo Soravia. "Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost." Nonlinear Differential Equations and Applications NoDEA 11, no. 3 (2004): 271–98. http://dx.doi.org/10.1007/s00030-004-1058-9.

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Luo, Qi, and Romesh Saigal. "Dynamic Multiagent Incentive Contracts: Existence, Uniqueness, and Implementation." Mathematics 9, no. 1 (2020): 19. http://dx.doi.org/10.3390/math9010019.

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Multiagent incentive contracts are advanced techniques for solving decentralized decision-making problems with asymmetric information. The principal designs contracts aiming to incentivize non-cooperating agents to act in his or her interest. Due to the asymmetric information, the principal must balance the efficiency loss and the security for keeping the agents. We prove both the existence conditions for optimality and the uniqueness conditions for computational tractability. The coupled principal-agent problems are converted to solving a Hamilton–Jacobi–Bellman equation with equilibrium constraints. Extending the incentive contract to a multiagent setting with history-dependent terminal conditions opens the door to new applications in corporate finance, institutional design, and operations research.

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44

Sanchez-Gomez, Luis Fernando, and Jose Antonio de la O Serna. "Dynamic Phasor Estimates Under the Bellman's Principle of Optimality: The Taylor-LQG-Fourier Filters." IEEE Transactions on Instrumentation and Measurement 62, no. 12 (2013): 3137–47. http://dx.doi.org/10.1109/tim.2013.2270316.

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Kwong, C. P. "Development of the Schrödinger equation and Klein-Gordon equation via Bellman's principle of optimality." Physics Letters A 124, no. 4-5 (1987): 220–22. http://dx.doi.org/10.1016/0375-9601(87)90624-4.

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SIRBILADZE, G. "ON FUZZY OPTIMAL CONTROLS IN THE WEAKLY STRUCTURABLE CONTINUOUS DYNAMIC SYSTEMS (WSCDS)." New Mathematics and Natural Computation 04, no. 01 (2008): 41–60. http://dx.doi.org/10.1142/s179300570800091x.

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We pose problems of the WSCDS optimal control and discuss the results developed by G. Sirbiladze in Ref. 7. Sufficient and necessary conditions are presented for the existence of an extremal fuzzy optimal control process, for which we use R. Bellman's optimality principle and take into consideration the gain-loss fuzzy process. An example of constructing the continuous extremal fuzzy dynamic systems optimal control is presented.

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Peralez, Johan, Aurélien Delage, Jacopo Castellini, Rafael F. Cunha, and Jilles S. Dibangoye. "Optimally Solving Simultaneous-Move Dec-POMDPs: The Sequential Central Planning Approach." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 22 (2025): 23276–85. https://doi.org/10.1609/aaai.v39i22.34494.

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The centralized training for decentralized execution paradigm emerged as the state-of-the-art approach to ϵ-optimally solving decentralized partially observable Markov decision processes. However, scalability remains a significant issue. This paper presents a novel and more scalable alternative, namely the sequential-move centralized training for decentralized execution. This paradigm further pushes the applicability of the Bellman’s principle of optimality, raising three new properties. First, it allows a central planner to reason upon sufficient sequential-move statistics instead of prior simultaneous-move ones. Next, it proves that ϵ-optimal value functions are piecewise linear and convex in such sufficient sequential-move statistics. Finally, it drops the complexity of the backup operators from double exponential to polynomial at the expense of longer planning horizons. Besides, it makes it easy to use single-agent methods, e.g., SARSA algorithm enhanced with these findings, while still preserving convergence guarantees. Experiments on two- as well as many-agent domains from the literature against ϵ-optimal simultaneous-move solvers confirm the superiority of our novel approach. This paradigm opens the door for efficient planning and reinforcement learning methods for multi-agent systems.

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Hendzel, Z., and P. Penar. "Application of Differential Games in Mechatronic Control System." International Journal of Applied Mechanics and Engineering 21, no. 4 (2016): 867–78. http://dx.doi.org/10.1515/ijame-2016-0051.

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Abstract Differential games are a combination of game theory and optimum control methods. Their solutions are based on Bellman's principle of optimality. In this paper, the zero-sum differential game theory has been used for the purposes of controlling a mechatronic object: a single-link manipulator. In this case, analytical solutions are unavailable, thus approximate solutions were used. Two approximation methods were compared with the use of numerical simulations and selected quality indicators. The results confirm previous assumptions and the connection between the differential game theory and H∞ control problems.

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Chen, Yuefen, and Liubao Deng. "Discrete-Time Uncertain LQ Optimal Control with Indefinite Control Weight Costs." Journal of Advanced Computational Intelligence and Intelligent Informatics 20, no. 4 (2016): 633–39. http://dx.doi.org/10.20965/jaciii.2016.p0633.

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This paper deals with a discrete-time uncertain linear quadratic (LQ) optimal control, where the control weight costs are indefinite . Based on Bellman’s principle of optimality, the recurrence equation of the uncertain LQ optimal control is proposed. Then, by using the recurrence equation, a necessary condition of the optimal state feedback control for the LQ problem is obtained. Moreover, a sufficient condition of well-posedness for the LQ problem is presented. Furthermore, an algorithm to compute the optimal control and optimal value is provided. Finally, a numerical example to illustrate that the LQ problem is still well-posedness with indefinite control weight costs.

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Popkov, Yuri S., Yuri A. Dubnov, and Alexey Yu Popkov. "Reinforcement Procedure for Randomized Machine Learning." Mathematics 11, no. 17 (2023): 3651. http://dx.doi.org/10.3390/math11173651.

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This paper is devoted to problem-oriented reinforcement methods for the numerical implementation of Randomized Machine Learning. We have developed a scheme of the reinforcement procedure based on the agent approach and Bellman’s optimality principle. This procedure ensures strictly monotonic properties of a sequence of local records in the iterative computational procedure of the learning process. The dependences of the dimensions of the neighborhood of the global minimum and the probability of its achievement on the parameters of the algorithm are determined. The convergence of the algorithm with the indicated probability to the neighborhood of the global minimum is proved.

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Journal articles on the topic 'Bellman optimality principle'

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