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The N–H photodissociation dynamics of the pyridinyl radical upon continuous excitation to the optically bright, first excited ππ* electronic state by an ultra-violet (UV) laser pulse has been investigated within the mathematical framework of optimal control theory. The genetic algorithm (GA) is employed as the optimization protocol. We considered a three-state and three-mode model Hamiltonian, which includes the reaction coordinate, R (a1 symmetry); the coupling coordinates (namely, out-of-plane bending coordinate of the hydrogen atom of azine group), Θ (b1 symmetry); and the wagging mode, Q9 (a2 symmetry). The three electronic states are the ground, ππ*, and πσ* states. The πσ* state crosses both the ground state and the ππ* state, and it is a repulsive state on which N–H dissociation occurs upon photoexcitation. Different vibrational wave functions along the coupling coordinates, Θ and Q9, of the ground electronic state are used as the initial condition for solving the time-dependent Schrödinger equation. The optimal UV laser pulse is designed by applying the GA, which maximizes the dissociation yield. We obtained over 95% dissociation yield through the πσ* asymptote using the optimal pulse of a time duration of ∼30 000 a.u. (∼725.66 fs). [ABSTRACT FROM AUTHOR]

Controlling molecular reactivity by shaped laser pulses is a long-standing goal in chemistry. Here, we suggest a direct optimal control approach that combines external pulse optimization with other control parameters arising in the upcoming field of vibro-polaritonic chemistry for enhanced controllability. The direct optimal control approach is characterized by a simultaneous simulation and optimization paradigm, meaning that the equations of motion are discretized and converted into a set of holonomic constraints for a nonlinear optimization problem given by the control functional. Compared with indirect optimal control, this procedure offers great flexibility, such as final time or Hamiltonian parameter optimization. A simultaneous direct optimal control theory will be applied to a model system describing H-atom transfer in a lossy Fabry–Pérot cavity under vibrational strong coupling conditions. Specifically, optimization of the cavity coupling strength and, thus, of the control landscape will be demonstrated. [ABSTRACT FROM AUTHOR]

Quantum walks are the quantum counterpart of classical random walks and have various applications in quantum information science. Polar molecules have rich internal energy structure and long coherence time and thus are considered as a promising candidate for quantum information processing. In this paper, we propose a theoretical scheme for implementing discrete-time quantum walks on a circle with dipole–dipole coupled SrO molecules. The states of the walker and the coin are encoded in the pendular states of polar molecules induced by an external electric field. We design the optimal microwave pulses for implementing quantum walks on a four-node circle and a three-node circle by multi-target optimal control theory. To reduce the accumulation of decoherence and improve the fidelity, we successfully realize a step of quantum walk with only one optimal pulse. Moreover, we also encode the walker into a three-level molecular qutrit and a four-level molecular ququart and design the corresponding optimal pulses for quantum walks, which can reduce the number of molecules used. It is found that all the quantum walks on a circle in our scheme can be achieved via optimal control fields with high fidelities. Our results could shed some light on the implementation of discrete-time quantum walks and high-dimensional quantum information processing with polar molecules. [ABSTRACT FROM AUTHOR]

We study the additive eigenvalues on changing domains, along with the associated vanishing discount problems. We consider the convergence of the vanishing discount problem on changing domains for a general scaling type Ω λ = (1 + r (λ)) Ω with a continuous function r and a positive constant λ. We characterize all solutions to the ergodic problem on Ω in terms of r. In addition, we demonstrate that the additive eigenvalue λ ↦ c Ω λ on a rescaled domain Ω λ = (1 + λ) Ω possesses one-sided derivatives everywhere. Additionally, the limiting solution can be parameterized by a real function, and we establish a connection between the regularity of this real function and the regularity of λ ↦ c Ω λ . We provide examples where higher regularity is achieved. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the optimal actuator location for achieving minimum norm controls in the context of approximate controllability for degenerate parabolic equations. We propose a formulation of the optimization problem that encompasses both the actuator location and its associated minimum norm control. Specifically, we transform the problem into a two-person zero-sum game problem, resulting in the development of four equivalent formulations. Finally, we establish the crucial result that the solution to the relaxed optimization problem serves as an optimal actuator location for the classical problem. [ABSTRACT FROM AUTHOR]

In this paper we advocate for Isaacs' method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin's canonical optimal control example, which entails a double integrator plant. However, rather than controlling the state to the origin, we require the end state to reach a terminal set that contains the origin in its interior. Indeed, in practice, it is required to control to a prescribed tolerance rather than reach a desired end state; constraining the end state to a terminal manifold of co-dimension n − 1 renders the optimal control problem easier to solve. The global solution of the optimal control problem is obtained and the synthesized optimal control law is in state feedback form. In this respect, two target sets are considered: a smooth circular target and a square target with corners. Closed-loop state-feedback control laws are synthesized that drive the double integrator plant from an arbitrary initial state to the target set in minimum time. This is accomplished using Isaacs' method for the solution of differential games, which entails dynamic programming (DP), working backward from the usable part of the target set, as opposed to obtaining the optimal trajectories using the necessary conditions for optimality provided by Pontryagin's Maximum Principle (PMP). In this paper, the case is made for Isaacs' method for the solution of differential games to be applied to the solution of optimal control problems by way of the juxtaposition of the PMP and DP methods. [ABSTRACT FROM AUTHOR]

The Lunar Gateway will represent a primary space system useful for the Artemis program, Earth–Moon transportation, and deep space exploration. It is expected to serve as a staging location and logistic outpost on the way to the lunar surface. This study focuses on low-thrust transfer dynamics, from the Near-Rectilinear Halo Orbit traveled by Gateway to a specified Low-altitude Lunar Orbit (LLO). More specifically, this research addresses two closely-related problems: (i) determination of the minimum-time low-thrust trajectory and (ii) design, implementation, and testing of a guidance and control architecture, for a space vehicle that travels from Gateway to LLO. Orbit dynamics is described in terms of modified equinoctial elements, with the inclusion of all the relevant perturbations, in the context of a high-fidelity multibody ephemeris model. The minimum-time trajectory from Gateway to a specified lunar orbit is detected through an indirect heuristic approach, which uses the analytical conditions arising in optimal control theory in conjunction with a heuristic technique. However, future missions will pursue a growing level of autonomy, and this circumstance implies the mandatory design and implementation of an efficient feedback guidance scheme, capable of compensating for nonnominal flight conditions. This research proposes nonlinear orbit control as a viable and effective option for autonomous explicit guidance of low-thrust transfers from Gateway to LLO. This approach allows defining a feedback law that enjoys quasi-global stability properties without requiring any offline reference trajectory. The overall spacecraft dynamics is modeled and simulated, including attitude control and actuation. The latter is demanded to an array of reaction wheels, arranged in a pyramidal configuration. Guidance, attitude control, and actuation are implemented in an iterative scheme. Monte Carlo simulations demonstrate that the guidance and control architecture at hand is effective in nonnominal flight conditions, i.e. with random starting point from Gateway as well as in case of temporary unavailability of the propulsion system. The numerical results also point out that only a modest propellant penalty is associated with the use of feedback guidance and control in comparison to the minimum-time optimal trajectory. • Minimum-time low-thrust orbit transfer from the Gateway to a low lunar orbit. • The optimal transfer is obtained using the indirect heuristic method. • Feedback guidance is investigated, leveraging nonlinear orbit control. • Nonlinear attitude control and actuation through reaction wheels is designed. • Numerical simulations prove effectiveness of guidance and control architecture. [ABSTRACT FROM AUTHOR]

This paper takes a reaction-diffusion predator-prey system with ratio-dependent Holling III functional response function and Leslie-Gower term into consideration. First of all, the system model is proposed on the basis of basic biological assumptions and previous work, and the existence conditions of the equilibrium point of the system are discussed. Secondly, under the assumption of the existence of equilibrium point, the Turing instability necessary conditions induced by diffusion are investigated. Thirdly, optimal control theory is derived, and the adjoint system and the first-order optimization condition are established. Fourthly, parameter identification based on optimal control is studied, and the technique is extended to network structure. Finally, extensive numerical simulations, including Turing pattern, Normalized Population High Distribution Area (NPHDA) diagram and parameter identification, are carried out to illustrate and validate the analytical results. For the system dynamics phenomena, the results from different perspectives effectively demonstrate that the theoretical findings, numerical simulations and natural reality are identical. In terms of the parameter identification of continuous model and network model, the efficiency and accuracy of various algorithms are fully tested. • This paper proposes and investigates index normalized population height distribution area. • The optimal control problem of reaction-diffusion predator-prey system is formally derived. • The parameter identification research is extended to population network structure space. [ABSTRACT FROM AUTHOR]

The track tracking effect of intelligent vehicle directly affects the safety of vehicle and passengers. In the process of intelligent vehicle track tracking, the track tracking accuracy is related to many factors such as track curvature, road friction coefficient, and longitudinal speed change. In this paper, a new lateral and longitudinal coupling control algorithm is proposed. Based on the vehicle dynamics model and the optimal control theory, combined with the fuzzy control theory, the Fuzzy Linear Quadratic Regulator (FLQR) lateral optimal controller is designed, and feed-forward control and predictive controller are added. According to the real-time tracking lateral error and fuzzy control algorithm fed back by the system, the weight coefficient of the lateral displacement deviation in the cost function is dynamically adjusted; considering the coupling effect of lateral and longitudinal controllers of vehicle trajectory tracking control, a model predictive control (MPC) longitudinal speed controller is designed based on MPC theory, considering acceleration constraint and acceleration variation constraint, and taking lateral stability as evaluation index. A joint simulation platform is built based on CarSim and Simulink. The simulation results show that the designed lateral and longitudinal coupling controller of FLQR + MPC has better track tracking accuracy and can improve the driving stability of the vehicle; finally, the tracking effect of the designed algorithm is verified by real vehicle experiments. The maximum error of the designed controller algorithm in real vehicle tracking is 0.56 m, and the tracking effect is good. [ABSTRACT FROM AUTHOR]

• A data-driven optimal control framework is proposed for TSR deployment. • A linear representation of TSR's dynamics is derived with Koopman operator. • An enhanced deep learning method is proposed for finding embedding functions. To avoid complex constraints of the traditional nonlinear method for tethered space robot (TSR) deployment, a data-driven optimal control framework with an improve deep learning based Koopman operator is proposed in this work. In consideration of nonlinearity of tethered space robot dynamics, its finite dimensional global linear representation called lifted linear system is derived with the Koopman operator theory. A deep learning scheme is adopted to find the embedding functions associate with Koopman operator. And an auxiliary neural network is developed to encode the nonlinear control term of finite dimensional lifted system. Then a controllability constraint is considered for learning a controllable lifted linear system. Besides two loss functions that relate to reconstruction and prediction ability of lifted linear system are designed for training the deep neural network. With the learned lifted linear dynamics, Linear Quadratic Regulator (LQR) is applied to derive the optimal control policy for the tethered space robot deployment. Finally, simulation results verify the effectiveness of proposed framework and show that it could deploy tethered space robot more quickly with less swing of in-plane angle. [ABSTRACT FROM AUTHOR]

Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to locations on the boundary of the support of the distribution or spatial infinity for distributions with unbounded support. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an approximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed a nonlinear mathematical model to study the dynamics of atmospheric carbon dioxide $({\rm{CO}}{{\kern 1pt} _{\rm{2}}})$(CO2) and atmospheric temperature due to energy sectors. Energy sector is a major contributor of atmospheric carbon dioxide. We have studied the effects of energy consumption and human population on ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2 level and temperature. Burning of fossil fuels results in an increase of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2 level and temperature. To curb the ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2 level and temperature, efficient mitigation options are required. Mitigation options, which reduce emission rate of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2, energy consumption rate and rate of increase in temperature, are considered in this model. By taking the efficiencies of mitigation options as control variables, the optimality system is derived. Numerical simulations are carried out to validate our analytical findings. The optimal profiles of control variables are plotted for different values of emission rate of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2, energy consumption rate and rate of increase in temperature. Maximum efficiencies of mitigation options are also plotted against to reduce emission rate of ${\rm{CO}}{{\kern 1pt} _{\rm{2}}}$CO2, energy consumption rate and rate of increase in temperature. It is found that mitigation cost of implementation of mitigation options can be minimized by implementing more efficient mitigation options. [ABSTRACT FROM AUTHOR]

To ensure the stable operation of a wind turbine generator system when the wind speed exceeds the rated value and address the issue of excessive rotor speed during high wind speeds, this paper proposes a novel variable pitch controller strategy based on a back-propagation neural network and optimal control theory to solve this problem. Firstly, a mathematical model for the wind turbine is established and linearized. Then, each optimal sub-controller is designed for different wind speed conditions by optimal theory. Subsequently, a back-propagation neural network is utilized to learn the variation pattern of controller parameters with respect to wind speed. Finally, real-time changes in wind speed are applied to evaluate and adjust controller parameters using the trained back-propagation neural network. The model is simulated in MATLAB 2019b, real-time data are observed, and the control effect is compared with that of a Takagi–Sugeno optimal controller, firefly algorithm optimal controller and fuzzy controller. The simulation results show that the rotor speed overshoot of the optimal controller under the step wind speed is the smallest, only 0.05 rad/s. Under other wind speed conditions, the rotor speed range fluctuates around 4.35 rad/s, and the fluctuation size is less than 0.2 rad/s, which is much smaller than the fluctuation range of other controllers. It can be seen that the back-propagation optimal controller can ensure the stability of the rotor speed above the rated wind speed. At the same time, it has better control accuracy compared to other controllers. [ABSTRACT FROM AUTHOR]

Innovative products entering the market will cause dynamic changes in market demand, and consumers' purchase regret and their return behavior make the market environment more and more complex, which in turn affects the dynamic decision-making in the supply chain. In this paper, under the situation of discrete decision time, combining with the objective reality, we make discrete modification to the classical Bass diffusion model (Bass model), construct a manufacturer-led, retailer-followed supply chain differential game model, analyze the optimal decision-making of the manufacturer and the retailer by combining with the theory of discrete optimal control, and then verify the conclusions by numerical simulation. The results show that: when retailers purchase directly from the manufacturer and sell in the market, the optimal pricing of the innovative product can make the supply chain as a whole, realizing Pareto optimality; consumer's purchase regret will increase the amount of returns, which will lead to the decrease of product sales and the profits of the manufacturer and the retailer; when the innovative product accounts for a different share of the market, the impact of purchase regret on the wholesale price and the retail price are also different. Therefore, manufacturers need to have an extensive comprehension of the market to minimize the negative effects of consumer regret and returns, and to formulate a reasonable pricing strategy for their products to gain as much profit as possible. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds for exact asymptotic error behaviour. For that reason, we analyse separately two classes of admissible algorithms: based on equidistant, and possibly not equidistant meshes. Our results indicate that in both cases, decrease of any method error requires significant increase of the cost term, which is illustrated by the product of cost and error diverging to infinity. This is, however, not visible in the finite-dimensional case. In addition, we propose an implementable, path-independent Euler algorithm with adaptive step-size control, which is asymptotically optimal among algorithms using specified truncation levels of the underlying Wiener process. Our theoretical findings are supported by numerical simulation in Python language. [ABSTRACT FROM AUTHOR]

Malaria remains a global threat and the conventional methods used for combating the disease leave out mosquitoes that feed outdoors. This study addresses the challenge posed by such mosquitoes based on a tool called ivermectin drug which is lethal to mosquitoes that ingest bloodmeal containing a concentration of it. We formulated a mathematical model with three control tools (insecticide treated nets, treatment of infective individuals and ivermectin drug on livestock and humans) for the transmission and control of malaria under optimal condition. The model’s basic reproduction number, R0 was estimated and the local and global stability analyses of the disease-free and endemic equilibrium points of the model were carried out. Sensitivity analysis carried out showed that R0 is most sensitive to the mosquito biting rate and to the proportion of blood meal on human with cattle availability in such a way that any percent increase in the value of any of these parameters will lead to an equal percent increase in the value of R0. The result of an optimal control analysis based on three time dependent controls suggests that the combination of all three controls gives the best result followed by the strategy that combines the use of ivermectin drug and the treatment of infective human. Depending on available resources, any of these is recommended to be adopted in malaria intervention programmes because of their effectiveness on both the infective human and mosquito populations with the potential of contributing significantly to the disease elimination within a minimal time frame. [ABSTRACT FROM AUTHOR]

This paper presents a novel optimization model that explores the optimal investment strategies for DC pension plans with return of premium clauses. We have assumed that the financial market consists of a risk-free asset and a risky asset, where the price of the risky asset follows the CEV model. Under the expected utility criterion, the optimal investment strategies were derived by employing stochastic optimal control theory and the Legendre transformation method. Explicit expressions of the optimal investment strategy were provided when the utility function was specified as exponential, power, or logarithmic. Finally, numerical analysis was conducted to examine the impact of factors such as interest rate, return rate, and volatility of the risky asset on the optimal strategies. [ABSTRACT FROM AUTHOR]

The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first-order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second-order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented. [ABSTRACT FROM AUTHOR]

Quantum optimal control (QOC) is a set of methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. This tutorial paper introduces the basic elements of this theory based on the Pontryagin maximum principle, in a physicist-friendly way. An analogy with classical Lagrangian and Hamiltonian mechanics is proposed to present the main results used in this field. Emphasis is placed on the different numerical algorithms to solve a QOC problem. Several examples ranging from the control of two-level quantum systems to that of Bose–Einstein condensates (BECs) in a one-dimensional optical lattice are studied in detail, using both analytical and numerical methods. Codes based on shooting method and gradient-based algorithms are provided. The connection between optimal processes and the quantum speed limit is also discussed in two-level quantum systems. In the case of BEC, the experimental implementation of optimal control protocols is described, both for two-level and many-level cases, with the current constraints and limitations of such platforms. This presentation is illustrated by the corresponding experimental results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of optimal pairs for optimal control problems with their states constrained pointwise to Riemannian manifolds. For this purpose, by means of the Riemannian geometric tool, we introduce a crucial Cesari-type property, which is an extension of the classical Cesari property (see Definition 3.3, p. 51 in [L. D. Berkovitz, Optimal Control Theory, Appl. Math. Sci. 12, Springer-Verlag, New York, Heidelberg, 1974]) from the setting of Euclidean spaces to that of Riemannian manifolds. Moreover, we show the efficiency of our result by a concrete example. [ABSTRACT FROM AUTHOR]

The low-thrust fuel-optimal transfer problem between periodic orbits of two circular restricted three-body systems has been considered. The focus is on utilizing invariant manifolds of the respective systems. A power-limited, variable-specific impulse engine model has been employed. The system-to-system transfer problem is modeled within the framework of optimal control theory through the application of the Calculus of Variations and Pontryagin's maximum principle (PMP). Along with unknown co-states, the departure and arrival points on respective periodic orbits and manifold coast durations are chosen as design variables in the problem formulation. The solution to the associated two-point boundary value problem (TPBVP) is obtained using the shooting method. Low-thrust fuel-optimal Sun-Earth-L2 halo to Sun-Mars-L1 halo transfers are obtained. The outlined result shows a significant amount of fuel savings. Such transfer trajectories may be helpful for transferring a spacecraft or cargo mission from a near-Earth station on a periodic orbit to a similar station near Mars. • Employing a low-thrust engine model results in propellant mass saving due to their high specific impulse. • A single powered arc is required for transfers with a variable specific impulse (VSI) engine. • The system-to-system transfer problem is modeled within the framework of optimal control theory. • The goal of optimization problem is to minimize fuel consumption. • Shooting method is used to converge to the desired initial co-states of the problem. [ABSTRACT FROM AUTHOR]

This paper explores stochastic differential equation models to characterize the growth dynamics of a stock modeled by generalized logistic growth under a general harvesting function. The latter function adds diminishing marginal productivity to effort increases, while the former incorporates several well‐known growth functions as particular instances. In order to obtain optimal policies, a Crank–Nicolson discretization scheme is derived for the general models. We employ the optimal control theory to determine the optimal harvesting policy for a specific harvested stock and compare the expected net present value of the harvester's total profit earned across harvesting policies. The main conclusion is that models with a higher growth rate result in greater profit values. [ABSTRACT FROM AUTHOR]

Deep operator networks is a popular machine learning approach. Some problems require multiple inputs and outputs. In this work, a multi-input and multi-output operator neural network (MIMOONet) for solving optimal control problems was proposed. To improve the accuracy of the numerical solution, a physics-informed MIMOONet was also proposed. To test the performance of the MIMOONet and the physics-informed MIMOONet, three examples, including elliptic (linear and semi-linear) and parabolic problems, were presented. The numerical results show that both methods are effective in solving these types of problems, and the physics-informed MIMOONet achieves higher accuracy due to its incorporation of physical laws. [ABSTRACT FROM AUTHOR]

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods. [ABSTRACT FROM AUTHOR]

The purpose of this study is to introduce a computational method for solving the two-dimensional optimal control problem of the large membrane vibration equation of fractional order. Our proposed approach utilizes Chebyshev cardinal functions in two dimensions to obtain a finite dimensional optimal control problem. This problem is then solved using the Lagrange multipliers method by substituting the expansions of the state and control variables in terms of two-dimensional Chebyshev cardinal functions into the objective function, dynamical system, and initial conditions. The utilization of Chebyshev nodes offers a distinct advantage by effectively mitigating the Runge phenomenon, characterized by the undesirable oscillation and divergence of high-degree polynomial interpolants in close proximity to the interval's endpoints. We extensively discuss the convergence of our method and evaluate its accuracy using three test problems. Our results demonstrate that this technique provides highly accurate numerical solutions for these types of problems. [ABSTRACT FROM AUTHOR]

Forest resources are renewable, and the rational exploitation and utilization of forest resources are not only conducive to sustainable development on a population scale, they can also lead to higher economic benefits. Based on the actual timber harvest problem, this paper establishes the joint harvest model of timber and non-timber with nonlinear harvest items. In the numerical simulation, by comparing the existing proportional harvest model, it is concluded that the optimal harvest strategy of nonlinear harvest items in this paper can obtain larger ecological benefits and be more conducive to the sustainable development of a population. Firstly, using the qualitative theory of ordinary differential equations, the dynamic behavior of the model is studied, and the existence and stability of the equilibrium point of the model are proven. Secondly, the optimal control solution is obtained by using the optimal control theory. Finally, the optimal harvesting strategy of timber and non-timber products is given based on the numerical simulation results, and a comparison of the effects of different parameters on the optimal harvest strategy, which provides a certain theoretical basis for the sustainable development of the ecological economy of forestry, is carried out. [ABSTRACT FROM AUTHOR]

This paper presents an analytical method to solve the optimal control problem for affine nonlinear systems with unknown drift dynamics. A new non‐quadratic cost function over an infinite horizon is presented that considers input constraints and includes the cost of the feed‐forward component of the control law. The mean value theorem for vector‐valued functions has been used to derive an integral form of this theorem. Based on this theorem, a rigorous proof is provided demonstrating that the cost function can be converted into another form. In the presence of input constraints, this converted form enables extracting the optimal control solution without solving the HJB equation. Additionally, unknown nonlinearity effects in drift dynamics are compensated in the control input. This is accomplished by estimating the unknown drift dynamics via an adaptive neural network (NN) approach. It is proven that the states and weights of NN are uniformly ultimately bounded based on a Lyapunov technique. The necessary and sufficient conditions are provided that ensure the optimality of the infinite horizon optimal control problem with a discount factor. As a result, it is demonstrated that the proposed approach satisfies the optimality criteria. To evaluate the effectiveness of the proposed approach, simulation examples are provided. [ABSTRACT FROM AUTHOR]

This study presents a novel approach for solving the economic dispatch (ED) problem in groups of generating units communicating through a communication network. The suggested strategy is a consensus-based dynamic event-triggered (ET) distributed optimization method. Our methodology considers the sharing of the local information between generators and their convex cost functions to address the total cost function and offers a decentralized optimization solution over a network. The proposed distributed method addresses the ED problem by considering the criterion of optimal cost and by offering efficient communication. Generating units are grouped according to their generation operational limits, that is, total capacity and dynamic ET distributed protocols are developed to ensure the consensus of cost variables among generating units, operating under normal capacity conditions. The remaining generating agents work on their operating limits, which are segregated through the sharing of flag information through a switching mechanism. Consequently, in contrast to the existing methods, the recommended protocol allows nodes to function in groups, based on the power supply, for ED with geographical clustering and capacity restrictions, in addition to handling the system constraints. Furthermore, the proposed technique employs a dynamic triggering method to manage bandwidth and guarantee the elimination of Zeno behavior. The simulation results validate the efficacy of the proposed approach. [ABSTRACT FROM AUTHOR]

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler-Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical Euler-Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimum power for an RLC circuit. [ABSTRACT FROM AUTHOR]

This work presents an innovative application of optimal control theory to the strategic scheduling of battery storage in the day-ahead electricity market, focusing on enhancing profitability while factoring in battery degradation. This study incorporates the effects of battery degradation on the dynamics in the optimisation framework. Considering this cost in economic analysis and operational strategies is essential to optimise long-term performance and economic viability. Neglecting degradation costs can lead to suboptimal operation and dispatch strategies. We employ a continuous-time representation of the dynamics, in contrast with many other studies that use a discrete-time approximation with rather coarse intervals. We adopt an equivalent circuit model coupled with empirical degradation parameters to simulate a battery cell's behaviour and degradation mechanisms with good support from experimental data. Utilising direct collocation methods with mesh refinement allows for precise numerical solutions to the complex, nonlinear dynamics involved. Through a detailed case study of Belgium's day-ahead electricity market, we determine the optimal charging and discharging schedules under varying objectives: maximising net revenues, maximising profits considering capacity degradation, and maximising profits considering both capacity degradation and internal resistance increase due to degradation. The results demonstrate the viability of our approach and underscore the significance of integrating degradation costs into the market strategy for battery operators, alongside its effects on the battery's dynamic behaviour. Our methodology extends previous work by offering a more comprehensive model that empirically captures the intricacies of battery degradation, including a fine and adaptive time domain representation, focusing on the day-ahead market, and utilising accurate direct methods for optimal control. This paper concludes with insights into the potential of optimal control applications in energy markets and suggestions for future research avenues. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a prey–predator model with the functional response of Beddington–DeAngelis and the fear effect that have infection only in prey populations. We determine existence criteria of several equilibria, and the stability at different equilibria are presented. We exert pesticide control over prey and additional food control over predators, the optimal control is obtained by the Pontryagin maximum principle. We confirm that adding controls to the predator and prey yields better results. Further we enrich our analysis with the inclusion of the existence and uniqueness of the optimal control. Finally, some numerical results to illustrate our analysis are presented. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an epidemic switched system. A susceptible–infected–treated model is considered. The course of an epidemic is profoundly influenced by the allocation of resources. If these resources are limited, then we need to devise an optimal distribution strategy. One significant case to study is when the drug supply is insufficient. We study a control problem that minimizes the total outbreak size of the epidemic and optimizes the rate of vaccination/isolation control by minimizing the suitable functional subject to resource constraints. In the end, simulations are performed for illustrations. [ABSTRACT FROM AUTHOR]

The article is devoted to the outstanding mechanical scientist and pedagogue Yakov Grigoryevich Sapunkov (04.11.1938–23.10.2020), whose important scientific results range from hypersonic aerodynamics, fluid mechanics and computational methods to the theory of optimal control, mechanics of space flight and the theory of strapdown inertial navigation systems (SINS). An outline of his scientific activity and list of his most significant scientific works are given.

This article presents a non-linear mathematical model that captures the dynamics of drinking prevalence within a population. The model is analyzed under an optimal control framework, dividing the total population into four compartments: susceptible, heavy drinker, drinker in treatment, and recovered classes. The model’s validity is affirmed through considerations of positivity, boundedness, reproduction number, stability, and sensitivity analysis. Stability theory is employed to explore both local and global stabilities. Sensitivity analysis identifies parameters with a significant impact on the reproduction number (R0{R}_{0}), with maximum sensitivity observed in parameters related to drinking transmission and transitions from heavy drinking to treatment stages. These parameters exhibit sensitivity indices of (0.538,1)\left(0.538,1), indicating that a 10% increase in these parameters would result in a (5.38,1)\left(5.38,1) increase in the threshold quantity. The study introduces an optimal control strategy that involves awareness campaigns and treatment as control variables. These controls aim to minimize the number of heavy drinkers while maximizing the number of recovered individuals. Pontryagin’s maximum principle is used to solve optimal control problems. Additionally, the research explores various parametric settings for each compartment, enriching the study environment. The effectiveness of the proposed control scheme is evaluated through rigorous numerical simulations, highlighting its competitive edge. The results, validated using MATLAB simulations, are detailed throughout the article.

Historically, advertising allocation decisions have operated mostly at a macro level, comprising determination of the total budget followed by apportionment among media channels. Owing to the rapid and sustained rise of digital media, an additional decision now operates at a micro level, namely, which specific customers to target with advertising. In the macro case, optimal control theory provides a powerful framework for firm profit maximization, specifically allowing for ad response, cost per medium, and discount rate, all in the presence of multiple competing brands. However, optimal control theory has never been applied to the situation of microtargeting individual customers. Consequently, in this study the author shows how optimal control theory can be adapted for application to individual customers. The author estimates a multinomial logit model with individual-specific advertising response parameters. In turn, these parameters are used to determine the optimal number of exposures each customer should receive for each advertising medium. An empirical example demonstrates that using the optimal microtargeting method improves firm profits over existing ad scheduling methods by between 150% and 183%. [ABSTRACT FROM AUTHOR]

In this study, we apply optimal control theory to an immunoepidemiological model of HIV and opioid epidemics. For the multiscale model, we used four controls: treating the opioid use, reducing HIV risk behaviour among opioid users, entry inhibiting antiviral therapy, and antiviral therapy which blocks the viral production. Two population-level controls are combined with two within-host-level controls. We prove the existence and uniqueness of an optimal control quadruple. Comparing the two population-level controls,wefind that reducing the HIV risk of opioid users has a stronger impact on the population who is both HIV-infected and opioid-dependent than treating the opioid disorder. The within-host-level antiviral treatment has an effect not only on the co-affected population but also on the HIV-only infected population. Our findings suggest that the most effective strategy for managing the HIV and opioid epidemics is combining all controls at both within-host and between-host scales. [ABSTRACT FROM AUTHOR]

The Sun–Earth Lagrange points, L1 and L2, are excellent locations for space based observatories, due to their favorable dynamical and environmental properties. Space telescopes, however, come with an extremely high price tag, and without refueling and servicing the typical lifetime is only about ten to fifteen years. The return on investment of a space telescope could be significantly increased by developing on-orbit servicing and refueling capabilities, assuming that the servicing vehicle can be launched at a fraction of the cost of the original mission and that the lifetime of the space telescope can be significantly extended to allow for continued scientific activity. In this paper we demonstrate the feasibility of this class of servicing mission from a trajectory design perspective, using novel techniques that we have developed for quick sampling of the solution space, as well as an efficient and reliable means to obtained high-fidelity, time-optimal, end-to-end transfer trajectories. As candidates for this investigation we consider Gaia and the James Webb Space Telescope. Our trajectory design methodology is, however, general enough to be applicable to future space observatories that are designed to be serviceable in space. • The lifetime of space assets at SEL2 can be greatly extended via servicing missions. • Poincare maps are leveraged to explore the design space and obtain initial guesses. • Indirect methods are used to refine the initial guesses and design optimal transfers. [ABSTRACT FROM AUTHOR]

Optimal control theory is a branch of mathematics. It is developed to find optimal ways to control a dynamic system. In 1957, R.Bellman applied dynamic programming to solve optimal control of discrete-time systems. His procedure resulted in closed-loop, generally nonlinear, and feedback schemes. Optimal control problems which will be tackled involve the minimization of a cost function subject to constraints on the state vector and the control. Lagrange multipliers provide a method of converting a constrained minimization problem into an unconstrained minimization problem of higher order. The necessary condition for optimality can be obtained as the solution of the unconstrained optimization problem of the Lagrange function and the bordered Hessian matrix is used for the second-derivative test. A spectral method for solving optimal control problems is presented. The method is based on Bernoulli polynomials approximation. By using the Bernoulli operational matrix of integration and the Lagrangian function, stochastic optimal control is transformed into an optimisation problem, where the unknown Bernoulli coefficients are determined by using Newton’s iterative method. The convergence analysis of the proposed method is given. The simulation results based on the Monte-Carlo technique prove the performance of the proposed method. Some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

A pandemic can wreak havoc in supply chains, as witnessed in the COVID-19 context. As workers get infected, production level drops and demand from customers goes unfulfilled. Combining in a novel way an epidemic model with optimal control theory, our model provides a plant manager with the optimal level of prophylactic effort she needs to deploy over a planning horizon to protect the workforce from a pandemic in its early stage and so maintain production levels. Given the production planning problem, the effort in terms of prophylactic measures can be optimally determined in closed form, balancing worker protection against production requirements in a single step. The manager must initially implement the strictest measures before relaxing them in time. Three extensions are presented:, (1) determine the optimal period of time over which the prophylactic measures should be maintained; (2) determine the optimal effort in terms of prophylactic measures in the case of an endemic disease; and (3) assess the effect of a stochastic exogenous shock on the total number of infected. This research provides a production planning model that allows a decision-maker to mitigate the impact of worker absenteeism at the onset of a pandemic, thus improving supply chain resilience. [ABSTRACT FROM AUTHOR]

This Review discusses the main results obtained in training end-to-end neural architectures for guidance and control of interplanetary transfers, planetary landings, and close-proximity operations, highlighting the successful learning of optimality principles by the underlying neural models. Spacecraft and drones aimed at exploring our solar system are designed to operate in conditions where the smart use of onboard resources is vital to the success or failure of the mission. Sensorimotor actions are thus often derived from high-level, quantifiable, optimality principles assigned to each task, using consolidated tools in optimal control theory. The planned actions are derived on the ground and transferred on board, where controllers have the task of tracking the uploaded guidance profile. Here, we review recent trends based on the use of end-to-end networks, called guidance and control networks (G&CNets), which allow spacecraft to depart from such an architecture and to embrace the onboard computation of optimal actions. In this way, the sensor information is transformed in real time into optimal plans, thus increasing mission autonomy and robustness. We then analyze drone racing as an ideal gym environment to test these architectures on real robotic platforms and thus increase confidence in their use in future space exploration missions. Drone racing not only shares with spacecraft missions both limited onboard computational capabilities and similar control structures induced from the optimality principle sought but also entails different levels of uncertainties and unmodeled effects and a very different dynamical timescale. [ABSTRACT FROM AUTHOR]

In this work, we present a method for the Global Asymptotic Stabilization (GAS) of an affine control chaotic Lorenz system, via admissible (bounded and regular) feedback controls, where the control bounds are given by a class of (convex) polytopes. The proposed control design method is based on the control Lyapunov function (CLF) theory introduced in [Artstein, 1983; Sontag, 1998]. Hence, we first recall, with parameters including those in [Lorenz, 1963], that these equations are point-dissipative, i.e. there is an explicit absorbing ball ℬ given by the level set of a certain Lyapunov function, V ∞ (x). However, since the minimum point of V ∞ (x) does not coincide with any rest point of Lorenz system, we apply a modified solution to the "uniting CLF problem" (to unify local (possibly optimal) controls with global ones, proposed in [Andrieu & Prieur, 2010]) in order to obtain a CLF V (x) for the affine system with minimum at a desired equilibrium point. Finally, we achieve the GAS of "any" rest point of this system via bounded and regular feedback controls by using the proposed CLF method, which also contains the following controllers: (i) damping controls outside ℬ , so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) feedback controls inside ℬ that stabilize the control system at "any" desired rest point of the (unforced) Lorenz system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite element approximation of an optimal control problem governed by a time fractional wave equation. We first prove the well-posedness and regularity of the optimal control problem. Furthermore, we discuss the fully discrete scheme and analyze its stability and a priori error estimate. Numerical experiments are given to illustrate the theoretical findings. • The weak solution to the time fractional wave equation is studied under low regular data. • The well-posedness and regularity of the optimal control problem are proven. • Numerical schemes exist the first-order and second-order accuracy in time and space respectively. [ABSTRACT FROM AUTHOR]

Tumour growth is a complex process influenced by various factors, including cell proliferation, migration, and chemotaxis. In this study, a biphasic chemotaxis model for tumour growth is considered, and the effect of chemotherapy on the growth process is investigated. We use optimal control theory to derive the optimized treatment strategy that minimises the tumour size while minimising the toxicity associated with chemotherapy. Moreover, the existence, uniqueness, and strong solution estimates for the biphasic chemotaxis model subsystem in one dimension are derived. These results are achieved through semigroup theory and the truncation method. In addition, the research provides evidence of the existence of an optimal pair through the utilization of the minimising sequence technique. It also demonstrates the differentiability of the mapping from control variable to state variable and establishes the first-order necessary optimality condition. Lastly, a sequence of numerical simulations are presented to showcase the impact of chemotherapy and the influence of parameters in restraining tumour growth when applied in an optimized manner. Our results show that optimal control can provide a more effective and personalised treatment for cancer patients, and the approach can be extended to other tumour growth models. [ABSTRACT FROM AUTHOR]

In this paper, an improved retractable body model (IRBM) is established, which has an advantage in simulating the flexion-and-extension motion of skier's legs during carved turning and straight gliding. The trajectory optimization problem for the nonlinear alpine skiing system is transformed into a multi-phase optimal control (MPOC) problem. Subsequently, a constrained multi-phase trajectory optimization model is developed based on the optimal control theory, where the optimization target is to minimize the total skiing time. The optimization model is discretized by using the Radau pseudospectral method (RPM), which transcribes the MPOC problem into a nonlinear programming (NLP) problem that is then solved by SNOPT solver. Through numerical simulations, the optimization results under different constraints are obtained using MATLAB. The variation characteristics of the variables and trajectories are analyzed, and four influencing factors related to the skiing time are investigated by comparative experiments. It turns out that the small turning radius can reduce the total skiing time, the flexion-and-extension motion of legs is beneficial to skier's performance, and the large inclination angle can shorten skier's turning time, while the control force has a slight effect on the skiing time. The effectiveness and feasibility of the proposed models and trajectory optimization strategies are validated by simulation and experiment results. [ABSTRACT FROM AUTHOR]

This study aims to investigate a novel nonlinear optimization problem that incorporate a partial differential equation (PDE) constraint for image denoising in the context of mixed noise removal. Based on H−s$$ {H}^{-s} $$‐norm and decomposition approach, we develop a nonlinear system that involves the fractional Laplacian operator. Based on Schauder's fixed point theorem, we establish the existence and uniqueness of weak solution for the direct problem. Furthermore, we study the well‐posedness of the optimal control, and we also prove the existence of the weak solutions for the adjoint problem by using Galerkin's method. In order to numerically compute the solution of the proposed model, we introduce the numerical discretization scheme and the primal–dual algorithm used to solve our problem. Finally, we provide comparative numerical experiments to evaluate the efficiency and effectiveness of our proposed model. [ABSTRACT FROM AUTHOR]

Topology optimization is an appealing technique for optimal layout of active controllers of structures. However, the existing methods are mainly restricted to deterministic excitations, and the optimal control law involved is determined by use of the classical optimal control (COC) method, in which the sensitivities of the gain matrix with respect to the design variables need to be determined by solving the Riccati sensitivity equation numerically. In this study, a reliability-based topology optimization framework is proposed for optimal layout of active controllers of structures under nonstationary random excitations. The optimization problem is formulated as the minimization of the failure probability of the structure subjected to a specified maximum number of controllers. To avoid solving the Riccati equation, an explicit optimal control (EOC) method is first employed to derive the closed-form optimal control law in terms of the position parameters of controllers. The statistical moments of the optimal control forces and structural responses under random excitations are then obtained explicitly by the operation rules of moments, and the first-passage dynamic reliability of the structure can be formulated using the level-crossing theory. On this basis, the sensitivities of the structural failure probability with respect to the position parameters of controllers can be derived analytically by the direct differential method. Finally, the explicit formulations of the response statistics and the relevant sensitivities are incorporated into a gradient-based method of moving asymptotes (MMA) for topology optimization of the layout of controllers in conjunction with the solid isotropic material with penalization (SIMP) technique. Two numerical examples are presented to demonstrate the feasibility of the proposed topology optimization framework. [ABSTRACT FROM AUTHOR]

The problem is considered to find the time-optimal control that transfers a fourth-order system that consists of a double integrator and a harmonic oscillator, coupled by one control channel, between two arbitrary states. That fourth-order system is equivalent to the Hill-Clohessy-Wiltshire model of the relative dynamics of an orbiting spacecraft in proximity of a second spacecraft on a circular reference orbit, subjected to a thrust parallel to the orbital velocity vector and having time-continuous amplitude. A new method is here introduced to determine the time-optimal control problem stated above. This method combines two previously discovered optimal control synthesis methods: the method by Romano and Curti (2020), that enables to find (analytically, in some case) the optimal control transferring a general Linear Time Invariant Normal system between two arbitrary states; and, the method by Belousova and Zarkh (1996), that enables to find the optimal control transferring a fourth-order system consisting of a double integrator and a harmonic oscillator, coupled by one control channel, from an arbitrary initial state to the origin of the state space, if the optimal control is a priori known, that transfers the same system from a reference state to the origin. The here proposed combined method utilizes two phases. During the first phase, a number of reference minimum-time controls are obtained that transfer the system from specific reference states to the state space origin; this is achieved by exploiting Pontryagin's principle together with back-propagation from the origin of the state space. During the second phase, a search is run along a particular curve in the state space (named extremal search path) that depends on the boundary states of the problem at hand. In particular, a minimum-time control problem is iteratively solved to find the optimal control history that steers the system from a state on that curve to the origin, by exploiting Belousova and Zarkh method, until a particular state is found which satisfies an equivalency condition that, as demonstrated by Romano and Curti, guarantees that the optimal control history pertaining to the problem of transferring the system from that state to the origin is the same optimal control history that transfers the system between the arbitrarily set initial and final states. The new results, substantiated by numerical experiments, have both a theoretical and a practical value, as they could be applied for the optimal guidance of spacecraft performing autonomous proximity maneuvers. [ABSTRACT FROM AUTHOR]

This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart (Numer Math 58(1):299–328, 1990). FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge–Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time. [ABSTRACT FROM AUTHOR]