Optimal Control Theory: Applications to Management Science and Economics
Abstract
This fully revised 3rd edition offers an introduction to optimal control theory and its diverse applications in management and economics. It brings to students the concept of the maximum principle in continuous and discrete time by using dynamic programming and Kuhn-Tucker theory. While some mathematical background is needed, the emphasis of the book is not on mathematical rigor, but on modeling realistic situations faced in business and management. The book exploits optimal control theory to the functional areas of management science including finance, production and marketing and to economics of growth and of natural resources. In addition, this new edition features materials on stochastic Nash and Stackelberg differential games and an adverse selection model in the principal-agent framework. The book provides exercises for each chapter and answers to selected exercises to help deepen the understanding of the material presented. Also included are appendices comprised of supplementary material on the solution of differential equations, the calculus of variations and its relationships to the maximum principle, and special topics including the Kalman filter, certainty equivalence, singular control, a global saddle point theorem, Sethi-Skiba points, and distributed parameter systems.
Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as a foundation for the book, which the author has applied to business management problems developed from his research and classroom instruction. The new edition has been completely refined and brought up to date. Ultimately this should continue to be a valuable resource for graduate courses on applied optimal control theory, but also for financial and industrial engineers, economists, and operational researchers concerned with the application of dynamic optimization in their fields.
... Therefore, we deduce inequality (25). ...
... Inequality (25) means that the unemployment rate is a decreasing function of k. This means J o u r n a l P r e -p r o o f Journal Pre-proof that to increase the employment rate, the government can improve the success rate in recruitment process, for example by creating job placement agencies with the aim to identify job offers and, for a given offer, to direct the jobseekers whose profiles best correspond to the offer. ...
... We apply Pontryagin's Maximum Principle [25] and convert the optimization problem (31) to the problem of finding the point-wise minimum relative to u and v of the Hamiltonian. ...
... With the derivative of the equation ( ) 3.9 it is obtained: ...
... By substituting the equation ( ) 3.6 and ( ) 3.9 into equation ( ) 3.10 it is obtained: ...
... By substituting the equation ( ) 3.6 and ( ) 3.9 into equation ( ) 3.10 it is obtained: ...
This journal discusses optimal control of inventory problems that are increasing. The inventory in the company is needed, to meet every incoming demand. Inventory in minimum quantity can result in shortage of inventory. But inventory quantity maximum can result in losses, due to the minimum demand. The purpose of this research is to determine the level of optimal inventory in PT. Canang Indah. Using the optimal control theory model and analyzing the stability of the dynamic differential equation, to find the optimal inventory level. Obtained optimal inventory levels achieve stability at the time . For the planning length of 12 months includes: raw material inventory (logs sengon and rambung), production (finished materials in process) and finished particle board products that are in the warehouse. From this research optimal control theory can be applied in PT. Canang Indah to optimize inventory on the problem of increasing inventory.
... Proof. Once again, this proof can be seen as a generalized version of the one given in [15] for differentially driven mobile robots based on [20]. The idea is to show optimality of u (· | 0) ≡ 0 and uniqueness of the optimizer. ...
... degree −1 and is the same as for the kinematic car (20). Again, the boundedness of the residuum of the difference of this approximation and the original kinematics (22) can be shown as necessitated in [8]. ...
... This technical computation is skipped for the sake of brevity. Thus, the tailored stage cost for the unicycle with an attached trailer follows from the homogeneity of (20) and the transformation of the tandem into the privileged coordinates z, yielding (x, u) = q 1 x 12 1 + q 2 x 12 3 + q 3 (x 2 ) 6 + q 4 ( 1 (x 2 − 1 x 4 )) 4 + r 1 v 12 + r 2 ω 12 . ...
Non-holonomic vehicles are of immense practical value and increasingly subject to automation. However, controlling them accurately, e.g., when parking, is known to be challenging for automatic control methods, including model predictive control (MPC). Combining results from MPC theory and sub-Riemannian geometry in the form of homogeneous nilpotent system approximations, this paper proposes a comprehensive, ready-to-apply design procedure for MPC controllers to steer controllable, driftless non-holonomic vehicles into given setpoints. It can be ascertained that the resulting controllers nominally asymptotically stabilize the setpoint for a large-enough prediction horizon. The design procedure is exemplarily applied to four vehicles, including the kinematic car and a differentially driven mobile robot with up to two trailers. The controllers use a non-quadratic cost function tailored to the non-holonomic kinematics. Novelly, for the considered example vehicles, it is proven that a quadratic cost employed in an otherwise similar controller is insufficient to reliably asymptotically stabilize the closed loop. Since quadratic costs are the conventional choice in control, this highlights the relevance of the findings. To the knowledge of the authors, it is the first time that MPC controllers of the proposed structure are applied to non-holonomic vehicles beyond very simple ones, in particular (partly) on hardware.
... All control minimizers in problem (71) are in 0, 1 [ ] and therefore are essentially bounded. By the Pontryagin maximum principle (PMP) [29,30], we address the question of how to identify the solutions predicted by Theorem 3.1. Moreover, our optimal control problem (71) has only fixed initial conditions, with the state variables being free at the final time, that is, H t f ( ), D t f ( ), and C t f ( ) are free. ...
... Proof. Direct computations show that system (73) follows from the adjoint system of the PMP [29]. Similarly, the equalities in (74) are directly given by the transversality conditions of the PMP. ...
... Similarly, the equalities in (74) are directly given by the transversality conditions of the PMP. It remains to characterize the controls using the minimality condition of the PMP [29]. ...
The bank run phenomenon, mostly due to rumor spread about the financial health of given financial institutions, is prejudicious to the stability of financial systems. In this paper, by using the epidemiological approach, we propose a nonlinear model for describing the impact of rumor on the banking crisis spread. We establish conditions under which the crisis dies out or remains permanent. We also solve an optimal control problem focusing on the minimization, at the lowest cost, of the number of stressed banks, as well as the number of banks undergoing the restructuring process. Numerical simulations are performed to illustrate theoretical results obtained.
... This section uses optimal control theory to solve the optimal control strategies of the shipbuilder under the carbon quota policy; the trajectory evolution equation of the ship recovery rate is also solved. We refer the readers to Sethi [20] for the details of the method. Proposition 1 gives the optimal control strategy for the shipbuilder under the government carbon quota. ...
... Proof. We use the HJB (Hamilton-Jacobi-Bellman) control strategy (Sethi [20]) to solve the problem. Formulate the HJB function of the shipbuilder as: ...
Reducing carbon emissions is crucial for humanity. Recently, the International Maritime Organization has set carbon emission quotas to limit the extensive carbon emissions from the marine industry. This paper examines how shipbuilders can adopt the rebuilding of decommissioned ships to reduce carbon emissions and also make more profits. Incorporating the carbon emission quota policy, we formulate the dynamic rebuilding model of decommissioned ships and derive the optimal rebuilding and pricing control strategies for the shipbuilder. We investigate the evolutionary dynamics as well as the impact of carbon quotas and carbon emission savings on shipbuilder’s strategies. The study’s findings suggest that shipbuilders have the potential to improve their profitability while also contributing to energy conservation and emission reduction. This can be achieved through the implementation of technological innovations aimed at reducing carbon emissions from their production activities. The government has a crucial role to play in regulating and managing shipbuilders. In cases where the cost per unit of carbon quota is deemed excessively high, it may be necessary to establish appropriate regulations that prevent shipbuilders from directly benefiting from the trade of carbon quotas. This approach can also help ensure the positive development of the carbon trading market.
... where y ij are the components of y(2π) given in (39). ...
... yields, after integration, the required expression in (39). So in the same way as in the proofs of Lemmas 2-3 J ϕ (0) = y 1 (2π) y 2 (2π) , and once inverted the expression in (38) follows. ...
Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize four popular projection algorithms, namely the Method of Alternating Projections, Dykstra, Douglas--Rachford and Arag\'on Artacho--Campoy algorithms, to solve control-constrained linear-quadratic optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two and use projection methods to find a point in the intersection of the solution sets of these two subproblems hence giving the solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through case studies, we explore and propose desirable ranges of values of some algorithmic parameters which yield a smaller number of iterations.}
... Pramanik (2023) [10] assumes that the information regarding the transmission of COVID-19 is incomplete and imperfect, which can lead to multiple Skiba points or multiple solutions. Studies about Skiba points have been rigorously conducted in [4,11] and [12]. Although there is a growing literature on COVID-19 and its socioeconomic impacts related to extended lockdown time, the length of lockdown and the appropriate time to initiate lockdown have not been studied in depth [3]. ...
COVID-19, a pandemic that affected the whole world, claimed the lives of almost 1.1 million people in the United States and 7 million worldwide. Prior to the discovery of vaccines, many countries resorted to implementing lock downs to reduce the spread of the virus. Most economies have implemented this policy, except in essential sectors such as public health and safety. Different states in the US have imposed lock downs at different times, based on the severity of the outbreak in their respective regions. Lock downs involve reducing social interactions, leading to a decrease in the transmission of the virus. However, if lock downs remain in effect for too long, people may become hesitant to resume social activities for fear of contracting COVID-19 [3]. Hence, businesses are facing a reduction in the number of consumers and employees, resulting in a decrease in sustainable long-term protability. Furthermore, if a business fails to have enough inventory to with stand the crisis, it may eventually shut down. Since the government is not providing nancial support, it is easy to shut down a business, but dicult to restore it to its original employment levels [3]. It is recommended by the Centers for Disease Control and Prevention (CDC) that anyone infected with Omicron should isolate themselves for five days. This is because a person infected with the virus can spread it to others, so isolation helps reduce transmission. Similarly, if more people are vaccinated, the virus will spread less and fewer people will be affected, thus saving more lives. In their study, Pramanik (2023) [10] determined the best way to decide when to shut down an economy and what rate of vaccination is optimal. They used a healthcare cost function that was minimized while taking into account a stochastic susceptible infectious-recovered (SIR) dynamic, which was first introduced in Aron et al (1984) [1]. Most models of infectious disease transmission are based on the SIR model. Pramanik's construction can be extended to a generalized random surface to investigate unprecedented shocks, such as the emergence of a new COVID-19 variant, sudden infection due to random interactions caused by travel, and environmental calamities resulting in more exposure to the pandemic. The random surface replaces the jump diffusion of the stochastic differential equations.
... 6 of Sethi (2019) for some references. S Sethi and Thompson (1981) and Chapter 12 of Sethi (2019) consider the case when the demand is stochastic but the demand rate is constant. Fleming et al. (1987) allow the demand rate to be stochastic and model the demand as a continuous-time Markov chain with a finite state space. ...
We study a production problem in which the cumulative consumer demand for an item follows a Brownian motion with drift, with both the drift and the variance parameters modulated by a continuous-time Markov chain that represents the regime. The company wants to maintain the inventory level as close as possible to a target inventory level, but there is a linear cost of production. We assume that the production rate is nonnegative. The company is penalized for deviations from the inventory target level and the cost of production, and the objective is to minimize the total discounted penalty costs. We consider two models. In the first model, there is no upper bound for the production rate, and in the second model there is an upper bound for the production rate. We solve both problems analytically and obtain the optimal production policy and the minimal total expected discounted cost. Our solutions allow us to obtain interesting managerial insights.
... Note that we indicate timedependent scalar or vectorial functions with bold symbols to distinguish them from time-independent quantities. The OCP has an infinite time horizon, which is typical, for example, in mathematical economics when considering economic sustainability or economic growth [47]. Moreover, the dynamics of the trajectory x : [0, ∞) → R N is affine with respect to the control signal u. ...
Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context-knowledge by its structure. Especially, the value function surrogate is elegantly enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm is proposed in a matrix-free way, which circumvents the large-matrix-problem for multivariate Hermite interpolation. For finite time horizons, both convergence of the surrogate to the value function as well as for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model that has a scalable dimension and an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.
... As a result, one might have multiple non-unique Skiba points or multiple solutions. Rigorous studies about Skiba points have been done in Skiba (1978); Grass (2012) and Sethi (2019). Aspri et al. (2021) take into account a SEIRD model with population divided into susceptibles, exposed but asymptotic, infected, recovered and deceased, and they obtain multiple lock-downs as well as Skiba points. ...
In this paper a Feynman-type path integral control approach is used for a recursive formulation of a health objective function subject to a fatigue dynamics, a forward-looking stochastic multi-risk susceptible-infective-recovered (SIR) model with risk-group's Bayesian opinion dynamics toward vaccination against COVID-19. My main interest lies in solving a minimization of a policy-maker's social cost which depends on some deterministic weight. I obtain an optimal lock-down intensity from a Wick-rotated Schrödinger-type equation which is analogous to a Hamiltonian-Jacobi-Bellman (HJB) equation. My formulation is based on path integral control and dynamic programming tools facilitates the analysis and permits the application of algorithm to obtain numerical solution for pandemic control model.
... The evolution of this field is intertwined with the genesis of analytical mechanics in the 18th century, before the crowning contributions of L. Pontryagin [119] in the 1950s and later R. Bellman [120]. Applications of optimal control could hardly be more diverse, with examples in aerospace with satellite maneuvering [121,122], communication protocols in computer science [123], finance [124] and biology [125]. ...
The field of quantum simulation aims at emulating complex quantum systems on platforms that are easier to control and observe. In the last twenty years, ultracold atoms in optical lattices have established themselves as a controllable and versatile system for quantum simulation. The three experimental studies presented in this manuscript take place in the development of this field. They are performed using Bose-Einstein condensates (BECs) in a one-dimensional optical lattice that can be precisely controlled in amplitude and phase.
In the first study, we use the optimal control formalism to compute the way in which to continuously shift the lattice in order to arbitrarily shape the BEC distribution in the phase space of the system. We apply this method to different targets, among which squeezed Gaussian states more than four times narrower in position than the ground state of the system, as well as the ideal Floquet state superposition to perform quantum simulation of dynamical tunneling is the modulated lattice.
The second study concerns the realization of a non-diffusive Hamiltonian ratchet. The ratchet effect consists in the emergence of a directed current of particles in a system with no net force. In this second work, we correlate the amplitude and phase modulations of the lattice to produce, in the phase space of the system, a region of non-chaotic trajectories that travels between lattice sites, resting periodically in the center of each sites. We experimentally implement this system and observe non-diffusive ratchet transport of matter waves in the optical lattice.
Finally, we show how short-range interactions between atoms in the BECs lead to the emergence of a supercrystalline order in the modulated optical lattice for a modulation frequency coupling two energy levels. We develop a two-band tight-binding model which predicts that collisions occuring between the atoms of the BECs can lead to the growth of unstable Bogoliubov modes in the vicinity of avoided crossings in the quasi-energy spectrum of the modulated system. Interestingly, we experimentally demonstrate that the periodicity of the emergent order can be tuned through Floquet engineering of these crossings.
... At the same time, Optimal adopts Optimal Control Theory, the science of maximizing the results and minimizing the operating costs of physical, social, and economic processes. By developing prospect theory through Perception of Risk studies proposed by (Slovic, 1987), and Investment Risk (Olsen 1997), as well as Optimal Control Theory (Sethi, 2019) and Behavioral Portfolio Theory (Shefrin & Statman) illustrate that an investor has The main task are to provide investment returns that are by the expected objectives along with the risks and returns offered for each investment and portfolio. Therefore, how investment selection is realized becomes essential in optimizing risk-based portfolio investment on investor behavior, as in the study of Sachse, Jungermann, Belting (2012), that investment decisions and investment efficiency will depend on investment risk. ...
During the current pandemic, the number of investors in Indonesia is increasing rapidly. It is an exciting thing how novice investors make decisions and face risks. On investment to minimize risk can be done with an investment portfolio. This study tries to offer investment efficiency factors and risk-based investment portfolio optimization so that investors in making investment decisions will feel satisfied and can meet their investment goals. The method used is descriptive quantitative. The sampling technique used the purposive sampling technique.The study results indicate that risk-based investment optimization and investment efficiency have a positive influence on investment decisions. It can ensure that accuracy in optimizing investment portfolios and increasing investment efficiency by investors can increase the efficiency of Investment Experience in investing.
... From the discrete time maximum principle, we have that (see, e.g., [23] and [24,Chapter 8] for the continuous time version) that for every i ∈ N there exist co-state variables ...
In this paper, we study the time consistency of cooperative agreements in dynamic games with non-transferable utility. An agreement designed at the outset of a game is time-consistent (or sustainable) if it remains in place for the entire duration of the game, that is, if the players would not benefit from switching to their non-cooperative strategies. The literature has highlighted that, since side payments are not allowed, the design of such an agreement is very challenging. To address this issue, we introduce different notions for the temporal stability of an agreement and determine endogenously the duration of the agreement. We illustrate our general results with a linear-quadratic difference game and show that an agreement’s duration can be easily assessed using the problem data. We also study the effect of information structure on the endogenous duration of the agreement. We illustrate our results with a numerical example.
... The flushing operation involves controlling a dynamic system, i.e., the system that evolves with time. Optimal control theory is a branch of mathematics that finds optimal ways to control dynamic systems (Sethi 2019). Here, the system refers to the pipeline flushing operation. ...
Lube-oil industries use a complex network of pipelines for transporting thousands of high-value finished products successively in batches throughout the production plant. Each lube-oil is unique in regard to its properties, and its integrity is extremely crucial. Therefore, during a changeover operation, the lines are flushed using a high-value
finished product of the current batch that is desired to be processed. The existing flushing
operation typically rely on a trial-and-error procedure, resulting in the downgrading of
the finished product. Moreover, it leads to enormous economic losses to the industries. In
response to this problem, this work presents an approach for modeling and optimizing the
flushing operation by employing first-principles and optimal control strategies. We model
the flushing operation by integrating the Kendall and Monroe viscosity blending
equations with time-dependent component balance equations for lube-oil pipelines. The
models developed are validated against the data collected from well-designed flush-study
experiments, and a good agreement is observed. We generate theoretical optimal flowrate
profiles and provide insights for designing and controlling the flushing operation.
Product co-creation based on company-sponsored online community has come to be a paradigm of developing new products collaboratively with customers. In such a product co-creation campaign, the sponsoring company needs to interact intensively with active community members about the design scheme of the product. We call the collection of the rates of the company’s response to active community members at all time in the co-creation campaign as a company response policy (CRP). This article addresses the problem of finding a cost-effective CRP (the CRP problem). First, we introduce a novel community state evolutionary model and, thereby, establish an optimal control model for the CRP problem (the CRP model). Second, based on the optimality system for the CRP model, we present an iterative algorithm for solving the CRP model (the CRP algorithm). Third, through extensive numerical experiments, we conclude that the CRP algorithm converges and the resulting CRP exhibits excellent cost benefit. Consequently, we recommend the resulting CRP to companies that embrace product co-creation. Next, we discuss how to implement the resulting CRP. Finally, we investigate the effect of some factors on the cost benefit of the resulting CRP. To our knowledge, this work is the first attempt to study value co-creation through optimal control theoretic approach.
First, this paper defines a general nonlinear optimal control problem
with state/control constraints and its approximation problem as the Haar
wavelet Galerkin optimal control problem (HWGOCP). Then, a Haar wavelet-based Galerkin numerical method has been developed, which converts it
to a nonlinear optimization problem. We theoretically prove that a Haar
wavelet feasible solution of HWGOCP will exist. We also show that the
approximate solutions of HWGOCP are consistent and converge to the
optimal solution of the problem. A variety of application problems have
been considered, which include optimal control of tumour growth using
Chemotherapy drugs, optimal control of infection via the SIS model using
treatment, the Brachistochrone problem in mechanics, optimal control of
mold using a fungicide, optimal control of pH value of a chemical reaction
to determine the quality of a product, etc.
Nowadays, environmental issues have received increasing attention from experts. The main cause is the increase of carbon emissions in the atmosphere, so it is urgent to reduce carbon emissions. In order to establish the optimal pricing strategy as well as the emission reduction effort strategy for companies who produce and sell low carbon products, this paper proposes an optimal control model for low carbon products. The reduction of the carbon emission for the product is described dynamically by a differential equation, and the analytical expressions of the optimal pricing and the emission abatement strategies are derived using the Pontryagin’s maximum principle. Finally, the numerical experiments are used to explain the results obtained. The results show that companies producing and selling low-carbon products must pay more attention to the amount of carbon emission reduction in their products, and make more efforts to reduce emissions in order to make more profits. Additionally, the parametric analysis shows that expanding market size and reducing inventory depletion can be equally helpful in shortening the sales cycle and boosting profits.
The intertemporal consumption model is extended to the case of uncertainty in continuous time. Stochastic variables are described by Wiener processes and Brownian motions. We study the properties of these processes and show that although they are not differentiable in time, an approximation known as Ito’s Lemma can be used. This Lemma makes it possible to solve the stochastic Hamilton-Jacobi-Bellman equation in continuous time. We derive more general implications for stochastic optimal control of consumption and savings. Finally, we conclude the chapter by applying these methods to five examples of stochastic consumption.
We explore the relationship between the dual of a weighted minimum-energy control problem, a special case of linear-quadratic optimal control problems, and the Douglas--Rachford (DR) algorithm. We obtain an expression for the fixed point of the DR operator as applied to solving the optimal control problem, which in turn devises a certificate of optimality that can be employed for numerical verification. The fixed point and the optimality check are illustrated in two example optimal control problems.
Performance evaluation is an essential part of the management process. Performance appraisal provides the information needed for decision making as well as a competitive advantage for successive operations. Therefore, firms’ managers should look at the organization systematically to improve their performance. Data envelopment analysis is one of the most influential management techniques that provide managers a tool to test the performance of the organization’s subunits and to make decisions based on the results for a better future. The purpose of writing this article is to use the optimal control technique in setting controllable inputs of data envelopment analysis model. This setting is based on the subunits output of the previous period, so that their efficiencies of the current period are close to the desired levels. Optimal control considers the units outputs in each period and adjusts the inputs of the next period based on converging the efficiency of each to its desired level. This method’s cost function is modeled based on data envelopment analysis to evaluate the efficiency of decision-making units in a finite discrete time interval. In this research, a new method is used for two DMUs for three-time steps, which can undoubtedly be extended to a more significant number of units and time steps. This method implemented on a science and technology park to achieve its subunits’ performance to their desired level by facilities sharing as inputs. This method has been able to achieve well results with Monte Carlo approach.
Conventional harvesting problems for natural resources often assume physiological homogeneity of the body length/weight among individuals. However, such assumptions generally are not valid in real‐world problems, where heterogeneity plays an essential role in the planning of biological resource harvesting. Furthermore, it is difficult to observe heterogeneity directly from the available data. This paper presents a novel optimal control framework for the cost‐efficient harvesting of biological resources for application in fisheries management. The heterogeneity is incorporated into the resource dynamics, which is the population dynamics in this case, through a probability density that can be distorted from reality. Subsequently, the distortion, which is the model uncertainty, is penalized through a divergence, leading to a nonstandard dynamic differential game wherein the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation has a unique nonlinear partial differential term. Here, the existence and uniqueness results of the HJBI equation are presented along with an explicit monotone finite difference method. Finally, the proposed optimal control is applied to a harvesting problem with recreationally, economically, and ecologically important fish species using collected field data.
Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize the Douglas–Rachford (DR) algorithm to solve control-constrained minimum-energy optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the DR algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.
Commercial lubricant industries use a complex pipeline network for the sequential processing of thousands of unique products annually. Flushing is conducted between changeovers to ensure the integrity of each production batch. An upcoming product is used for cleaning the residues of the previous batch, resulting in the formation of a commingled/mixed oil that does not match the specifications of either of the two batches. The existing operations are based on the operator’s experience and trial and error. After a selected flush time, the samples are tested for their viscosity to determine the success of a flush. The approach results in long downtime, the generation of large commingled oil volumes, and huge economic losses. Hence, to overcome the drawback, our work introduces a solution strategy for systematically optimizing flushing operations and making more informed decisions to improve the resource-management footprint of these industries. We use the American Petroleum Institute-Technical Data Book (API-TDB) blending correlations for calculating the mixture viscosities in real-time. The blending correlations are combined with our first-principles models and validated against well-designed experimental data from the partnered lubricant facility. Next, we formulate an optimal control problem for predicting the optimum flushing times. We solve the problem using two solution techniques viz. Pontryagin’s maximum principle and discrete-time nonlinear programming. The results from both approaches are compared with well-designed experimental data, and the economic and environmental significance are discussed. The results illustrate that with the application of a discrete-time nonlinear programming solution approach, the flushing can be conducted at a customized flow rate, and the necessary flushing volume can be reduced to over 30% as compared to the trial-and-error mode of operation.
Ride-share platforms are contemporary businesses that match passengers with drivers, unlike taxis that can be hailed from the street. In the literature, the problem of optimizing the operations of such companies is mostly considered in static settings. We use in this paper a dynamic model and propose differential equations to model the evolution of the system. The objective is to maximize the profit during the planning horizon. Using optimal control theory, we determine the optimal rate of change in the ride price rate. An illustrative example along with sensitivity analyses shows the effect of the system parameters on the optimal solution obtained.
Existing methods for optimal control struggle to deal with the complexity commonly encountered in real-world systems, including dimensionality, process error, model bias and data heterogeneity. Instead of tackling these system complexities directly, researchers have typically sought to simplify models to fit optimal control methods. But when is the optimal solution to an approximate, stylized model better than an approximate solution to a more accurate model? While this question has largely gone unanswered owing to the difficulty of finding even approximate solutions for complex models, recent algorithmic and computational advances in deep reinforcement learning (DRL) might finally allow us to address these questions. DRL methods have to date been applied primarily in the context of games or robotic mechanics, which operate under precisely known rules. Here, we demonstrate the ability for DRL algorithms using deep neural networks to successfully approximate solutions (the "policy function" or control rule) in a non-linear three-variable model for a fishery without knowing or ever attempting to infer a model for the process itself. We find that the reinforcement learning agent discovers a policy that outperforms both constant escapement and constant mortality policies-the standard family of policies considered in fishery management. This DRL policy has the shape of a constant escapement policy whose escapement values depend on the stock sizes of other species in the model.
This work presents a comprehensive comparative analysis of four prominent swarm intelligence (SI) optimization algorithms: Ant Lion Optimizer (ALO), Bat Algorithm (BA), Grey Wolf Optimizer (GWO), and Moth Flame Optimization (MFO). When compared under the same conditions with other SI algorithms, the Particle Swarm Optimization (PSO) stands out. First, the Unscented Kalman Filter (UKF) parameters to be optimized are selected, and then each SI optimization algorithm is executed within an off-line simulation. Once the UKF initialization parameters P0, Q0, and R0 are obtained, they are applied in real-time in the decentralized neural block control (DNBC) scheme for the trajectory tracking task of a 2-DOF robot manipulator. Finally, the results are compared according to the criteria performance evaluation using each algorithm, along with CPU cost.
Motivated by the COVID‐19 pandemic, we study how a public health authority may allocate vaccines from a limited stockpile to different jurisdictions over time. We propose an epidemiological model with time‐varying contact rates determined by a stylized behavioral feedback mechanism to reflect multi‐wave transmission dynamics. We evaluate the performance of various information‐sensitive allocation policies (e.g., allocation proportional to local incidence) as alternatives to the widely used pro‐rata policy. We also obtain optimized allocation strategies under the proposed epidemiological model with fairness and implementable freeze‐period constraints. For the case of a multi‐wave epidemic as represented by our compartmental model with behavioral feedback, we find that none of the alternative policies offers consistently more efficient allocations than a simple pro‐rata policy across a broad range of behavioral parameter settings. In fact, in some cases the alternative policies may actually result in less efficient allocations than the pro‐rata policy. Thus our results support the conclusion that the widely used pro‐rata policy can be well justified because it is simple to explain/implement and does not cause unexpected adverse effects. However, if policy makers are willing to invest in more tailored strategies based on numerical optimization, then the identified optimized strategies are a more favorable option as they allow for a more efficient allocation of vaccines.
Many ordinary approaches in optimization are mathematical-based. Due to the limitations of such methods, optimal problems have been commonly defined in single-objective and quadratic forms, leading to non-optimal solutions based on the main design requirements. In order to address these issues, this paper presents a new type of genetic programming (GP) called “multi-objective archived-based genetic programming (MAGP)”. The absolute forms of cost functions, which are unsolvable using the conventional methods, could be assessed herein through the suggested algorithm. It is shown that in addition to the possibility of obtaining optimal single-objective solutions, It is shown that in addition to the possibility of obtaining optimal single-objective solutions, some very fruitful non-dominant solutions on the Pareto fronts would be acquired, providing the designer with a set of optimal analytical solutions which could be selected depending on the design requirements. For example, in the case studies examined in this paper, around 30,000 and 14,000 non-dominant solutions were obtained for linear and nonlinear problems, respectively, indicating the high performance of the novel proposed MAGP algorithm in acquiring non-dominant solutions in optimization problems. Generally, it is observed that obtaining and comparing optimal solutions in the quadratic form are basically unreliable and as a consequence, an optimal control problem must be analyzed in its absolute form of indices.
This paper is concerned with a discounted optimal control problem of partially observed forward-backward stochastic systems with jumps on infinite horizon. The control domain is convex and a kind of infinite horizon observation equation is introduced. The uniquely solvability of infinite horizon forward (backward) stochastic differential equation with jumps is obtained and more extended analyses, especially for the backward case, are made. Some new estimates are first given and proved for the critical variational inequality. Then a maximum principle is obtained by introducing some infinite horizon adjoint equations whose uniquely solvabilities are guaranteed necessarily. Finally, some comparisons are made with two kinds of representative infinite horizon stochastic systems and their related optimal controls.
The cybernetic viable system model (VSM) and viability theory (VT) are based on common assumptions. Both are tools for understanding and designing organizations. What, then, are the specific notions or concepts they share? What are their points of contact? To answer these questions, we identify the various mathematical concepts of control theory used in VSM and study the advances made in VT. We also summarize the various concepts in each approach. It is shown that VSM emphasizes general structures whereas VT puts the accent on the behaviour of particular dynamic systems. Our general conclusions are that: (1) VSM contributes important cybernetic concepts while VT provides precise notions of tychastic viability, viability and invariant sets as well as capture and absorption basins; and (2) control theory strongly enriched by system dynamics techniques is the most important theoretical and practical bridge between VSM and VT. These various characteristics of the two methods lend themselves to fruitful collaboration between them and form the basis of an attractive agenda for further research.
We explore the relationship between the dual of a weighted minimum-energy control problem, a special case of linear-quadratic optimal control problems, and the Douglas-Rachford (DR) algorithm. We obtain an expression for the fixed point of the DR operator as applied to solving the optimal control problem, which in turn devises a certificate of optimality that can be employed for numerical verification. The fixed point and the optimality check are illustrated in an example optimal control problem.
Thanks to the booming development of artificial intelligence, 5G technology, and intelligent manufacturing technology, numerous intelligent edge devices contained in the industrial Internet of Things (IIoT) are endowed with the ability to mine knowledge from perceived massive data. Knowledge-driven IIoT plays an unprecedented role in application fields such as cyber-physical systems and Industry 4.0. However, knowledge is generally scattered across the distributed edge devices of IIoT. Therefore, in order to further achieve the edge intelligence in IIoT, it is very important to explore an efficient edge knowledge sharing method. In this article, we establish a decentralized knowledge sharing platform in IIoT. First, for public knowledge, a dynamics model that can quantitatively describe its sharing process is established by using the system dynamics theory. Furthermore, a control method for maximizing public knowledge sharing under constraints based on the optimal control theory is presented. Second, for private knowledge, a trusted transaction control method based on blockchain technology is proposed. By developing both smart contract and lightweight consensus mechanism, the efficient peer-to-peer sharing of private knowledge is realized, and the integrity of knowledge and the privacy of participants are protected. The results of extensive experiments show that the proposed method can eliminate the obstacles of knowledge sharing among edge devices in IIoT, and further promote the development of edge intelligence empowered 5G-enabled IIoT applications.
For reliable epidemic monitoring and control, this paper proposes a feedback mechanism design to effectively cope with data and model uncertainties. Using past epidemiological data, we describe methods to estimate the parameters of general epidemic models. Because the data could be noisy, the estimated parameters may not be accurate. Therefore, under uncertain parameters and noisy measurements, we provide an observer design method for robust state estimation. Then, using the estimated model and state, we devise optimal control policies by minimizing a predicted cost functional. Finally, the effectiveness of the proposed method is demonstrated through its implementation on a modified SIR epidemic model.
The task of looking for the optimal allocation of resources in an economy is fraught with a number of severe restrictions. This is manifested in the complexity of the technical implementation of the solution even in the case of a low dimension of the problem. In this paper, we consider two approaches, analytical and numerical, for deriving the dynamical optimal allocation of resources in a three-sector economy and show that the use of modern artificial intelligence (AI) technologies such as genetic algorithms (GA), can be useful for expanding the range of effective tools and new contributions to this problem.
Blended learning networks (BLNs) based on the integration of online learning networks and offline learning environments provide new opportunities and platforms for people to acquire and update useful knowledge and carry out all kinds of learning activities anytime and anywhere. Effective modeling and regulation of the knowledge dissemination process can accurately grasp its dissemination process, promote knowledge innovation and collaborative sharing among learners, and accelerate the maximization of knowledge dissemination. However, it is a challenge to establish a comprehensive dynamics model and adopt the optimal regulation for the knowledge dissemination process under the constraints of a limited budget in large-scale BLNs with diverse learners. To this end, we first explore the evolution process of knowledge dissemination in BLNs and the blended learning interaction process of learners. Based on the system dynamics modeling theory, a dynamics model of knowledge dissemination is established. Second, two kinds of effective regulation strategies are proposed. We establish an optimal regulation system intending to maximize the dissemination of knowledge and use the optimal control theory to tackle the optimal solution distribution of regulation strategies. Then, we propose a knowledge dissemination regulation task allocation method based on the collaborative participation of users, and the reverse auction theory is used to quickly solve the task allocation scheme while ensuring performance. Finally, we demonstrate the effectiveness of proposed models and methods through extensive simulation experiments based on real datasets.
Operations Management (OM) has evolved to its current status due to the contributions of researchers in many fields including economics, finance, behavioral sciences, operations research/management science, mathematics, statistics, and computer science. Many of these contributors are Nobel laureates – winners of the highest academic award. The Nobel Memorial Prize in Economic Sciences is awarded to those with monumental works in economics. Yet some winners have also made contributions to operations management, the field of this journal. Others have developed important concepts and techniques that have made an enormous impact on OM research. Here we describe the OM contributions as well as the impactful concepts of these Laureates. This article is protected by copyright. All rights reserved
Metaverse, also known as the Internet of 3D worlds, has recently attracted much attention from both academia and industry. Each virtual sub-world, operated by a virtual service provider (VSP), provides a type of virtual service. Digital twins (DTs), namely digital replicas of physical objects, are key enablers. Generally, a DT belongs to the party that develops it and establishes the communication link between the two worlds. However, in an interoperable Metaverse, data-like DTs can be shared within the platform. Therefore, one set of DTs can be leveraged by multiple VSPs. As the quality of the shared DTs may not always be satisfying, in this paper we propose an agile solution, i.e., a dynamic hierarchical framework, in which a group of IoT devices in the lower-level are incentivized to collectively sense physical objects’ status information and VSPs in the upper-level determine synchronization intensities to maximize their payoffs. We adopt an evolutionary game approach to model the devices’ VSP selections and a simultaneous differential game to model the optimal synchronization intensity control problem. We further extend it as a Stackelberg differential game by considering some VSPs to be first movers. We provide open-loop solutions based on control theory for both formulations. We theoretically and experimentally show the existence, uniqueness, and stability of the equilibrium to the lower-level game and further provide a sensitivity analysis for various system parameters. Experiments show that the proposed dynamic hierarchical game outperforms the baseline.
We consider the optimal operating policies of a free-to-play multiplayer game with a premium subscription to maximize its lifetime operating profit. Accounting for social comparisons between free and premium players, we model the game attracting or losing players with a hybrid of the Bass diffusion model and the replicator equation in evolutionary game theory. Leveraging optimal control theory, we characterize optimal dynamic pricing and advertising policies and show that the developer should prioritize initial growth through aggressive advertising, while postponing the introduction of a premium subscription. Surprisingly, the optimal subscription price may start high and gradually decrease. We further show that the developer should strengthen social-comparison effects, that payment-based matchmaking can be an effective monetization driver, and that our main findings remain robust when allowing individual in-game item purchases/partial premium subscription. These findings are potentially instructive for game developers adopting the premium subscription model.
This paper was accepted by Víctor Martínez-de-Albéniz, operations management.
We consider a setting where two noncooperative players optimally influence the evolution of an initial spatial probability in a game-theoretic hierarchical fashion (Stackelberg differential game), so that at a specific final time the distribution of the state matches a given final target measure. We provide a sufficient condition for the existence and uniqueness of an optimal transport map and prove that it can be characterized as the gradient of some convex function. An important by-product of our formulation is that it provides a means to study a class of Stackelberg differential games where the initial and final states of the underlying system are uncertain, but drawn randomly from some probability measures.
Long‐term corporate social responsibility (CSR) projects are affected by the time span and the decay of CSR level. Different government tax policies and different status of supply chain members also have an impact on CSR performance. Considering these issues, multi‐period dynamic game processes with the supplier and the manufacturer as the Stackelberg leader, respectively, are established. By solving the Hamiltonian function and differential game model, it shows that supplier‐led supply chains have better CSR performance. Leaders and followers invest differently in CSR projects in different periods. Single tax return improves the CSR level more than system tax return.
In this paper a Feynman-type path integral control approach is used for a recursive formulation of a health objective function subject to a fatigue dynamics, a forward-looking stochastic multi-risk susceptible-infective-recovered (SIR) model with risk-group's Bayesian opinion dynamics towards vaccination against COVID-19. My main interest lies in solving a minimization of a policy-maker's social cost which depends on some deterministic weight. I obtain an optimal lock-down intensity from a Wick-rotated Schrodinger-type equation which is analogous to a Hamiltonian-Jacobi-Bellman (HJB) equation. My formulation is based on path integral control and dynamic programming tools facilitates the analysis and permits the application of algorithm to obtain numerical solution for pandemic control model. Feynman path integral is a quantization method which uses the quantum Lagrangian function, while Schrodinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent but, this equivalence has not fully proved mathematically. As the complexity and memory requirements of grid-based partial differential equation (PDE) solvers increase exponentially as the dimension of the system increases, this method becomes impractical in the case with high dimensions. As an alternative path integral control solves a class a stochastic control problems with a Monte Carlo method for a HJB equation and this approach avoids the need of a global grid of the domain of the HJB equation.
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