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In this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of differentiability results for probability functions in infinite dimensions usable in this context. These results are subsequently combined with the optimal control setting to derive a novel Pontryagin’s optimality principle. [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]

Corruption is a global problem that affects many countries by destroying economic, social, and political development. Therefore, we have formulated and analyzed a mathematical model to understand better control measures that reduce corruption transmission with optimal control strategies. To verify the validity of this model, we examined a model analysis showing that the solution of the model is positive and bounded. The basic reproduction number Rq was computed by using the next-generation matrix. The formulated model was studied analytically and numerically in the context of corruption dynamics. The stability analysis of the formulated model showed that the corruption-free equilibrium is locally and globally asymptotically stable for Rq < 1, but the corruption-endemic equilibrium is globally asymptotically stable for Rq > 1. Furthermore, the optimal control strategy was expressed through the Pontryagin Maximum Principle by incorporating two control variables. Finally, numerical simulations for the optimal control model were performed using the Runge- Kutta fourth order forward and backward methods. This study showed that applying both mass education and law enforcement is the most efficient strategy to reduce the spread of corruption. [ABSTRACT FROM AUTHOR]

We investigate optimal control problems with L 0 constraints, which restrict the measure of the support of the controls. We prove necessary optimality conditions of Pontryagin maximum principle type. Here, a special control perturbation is used that respects the L 0 constraint. First, the maximum principle is obtained in integral form, which is then turned into a pointwise form. In addition, an optimization algorithm of proximal gradient type is analyzed. Under some assumptions, the sequence of iterates contains strongly converging subsequences, whose limits are feasible and satisfy a subset of the necessary optimality conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $ \mathsf{X} $-valued left Caputo fractional evolution equation with the analytic semigroup, where $ \mathsf{X} $ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $ \mathsf{X} $. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $ \mathsf{X}^* $, the dual space of $ \mathsf{X} $. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.

The hybrid minimum principle (HMP) is established for the optimal control of deterministic hybrid systems with both autonomous and controlled switchings and jumps where state jumps at the switching instants are permitted to be accompanied by changes in the dimension of the state space and where the dynamics, the running and switching costs as well as the switching manifolds and the jump maps are permitted to be time varying. First-order variational analysis is performed via the needle variation methodology and the necessary optimality conditions are established in the form of the HMP. A feature of special interest in this work is the explicit presentations of boundary conditions on the Hamiltonians and the adjoint processes before and after switchings and jumps. Analytic and numerical examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In multiplying systems, such as nuclear reactors and criticality experiments, it is desirable to place the fissile material in the optimal or 'best' way to reduce the critical mass of the system as well as to achieve uniform fuel burnup. This paper considers two methods, namely Pontryagin's maximum principle (PMP) and Monte Carlo (MC) perturbation for estimating a minimum critical mass configuration. These methods are applied to an elementary multizone model of a pressurized water reactor (PWR) and a criticality experiment to estimate the minimum critical mass. It is found that while twogroup diffusion theory with PMP predicts a minimum critical mass, more detailed MC simulations with MCNP5 show a consistent reduction in critical mass when fissile fuel is placed in inner zones. Such a distribution reduces the fissile material requirement but is undesirable due to the higher power peaking. MC simulations show that for a three-zone model of the KORI 1 PWR, a uniform fissile distribution gives criticality for 1.09 atomic percent (at.%) enrichment, whereas non-uniform fissile distribution (0.6, 1.6, 0.6 at.%) reduces the critical mass by 14%. The changes found from MC simulations were subsequently predicted from first-and second-order derivative sampling. It was found that substantial computational savings can be achieved for large-scale optimization problems. In the case of a criticality experiment, MC derivative sampling was also used to estimate optimal fissile distribution for minimizing the critical mass. [ABSTRACT FROM AUTHOR]

In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the -valued left Caputo fractional evolution equation with the analytic semigroup, where is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of , the dual space of. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper. [ABSTRACT FROM AUTHOR]

Establishing a scientific ecological compensation mechanism for air pollution is crucial for air protection. This study models the ecological compensation mechanism of the Stackelberg differential game between the local regulator and an enterprise with a competitor by introducing the air quality index and the social welfare benefits of the local regulator. Using the Pontryagin maximum principle, this study obtains dynamic strategies for the local regulator and the enterprise while maximizing the benefits. The evolution of the shadow price is analyzed with the inverse differential equation method. Then, the effects of the shadow price on the optimal dynamic strategies are analyzed using numerical simulation, together with the effects of the introduction of social welfare benefits on the efforts of the local regulator to protect the air environment. The conclusions show that introducing social welfare benefits as an ecological compensation criterion for air pollution promotes air protection by the local regulator. [ABSTRACT FROM AUTHOR]

Corruption is a global problem that affects many countries by destroying economic, social, and political development. Therefore, we have formulated and analyzed a mathematical model to understand better control measures that reduce corruption transmission with optimal control strategies. To verify the validity of this model, we examined a model analysis showing that the solution of the model is positive and bounded. The basic reproduction number R0 was computed by using the next-generation matrix. The formulated model was studied analytically and numerically in the context of corruption dynamics. The stability analysis of the formulated model showed that the corruption-free equilibrium is locally and globally asymptotically stable for R0 < 1, but the corruption-endemic equilibrium is globally asymptotically stable for R0 > 1. Furthermore, the optimal control strategy was expressed through the Pontryagin Maximum Principle by incorporating two control variables. Finally, numerical simulations for the optimal control model were performed using the Runge-Kutta fourth order forward and backward methods. This study showed that applying both mass education and law enforcement is the most efficient strategy to reduce the spread of corruption.

In this article, we are concerned with minimal-time optimal problems for the class of controllable linear control system on low-dimensional nonnilpotent solvable Lie groups and their homogeneous spaces.

A three-dimensional reachable set for a nonlinear controlled object "Dubins car" is investigated. The control is the angular velocity of rotation of the linear velocity vector. An integral quadratic constraint is imposed on the control. Based on the Pontryagin maximum principle, a description of the motions generating the boundary of the reachable set is given. The motions leading to the boundary are optimal Euler elasticae. Simulation results are presented. [ABSTRACT FROM AUTHOR]

The task of managing unstable systems is a critically important management problem, as an unstable object can pose significant danger to humans and the environment when it fails. In this paper, a neural network was trained to determine the optimal control for an unstable system, based on a comparative analysis of two control methods: the implicit Euler method and the linearization method. This neural network identifies the optimal control based on the position of a point on the phase plane. [ABSTRACT FROM AUTHOR]

In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method. [ABSTRACT FROM AUTHOR]

The problem of maximizing the horizontal coordinate of a mass point moving in a vertical plane driven by gravity, viscous drag, curve reaction force, and thrust is considered. It is assumed that inequality-type constraints are imposed on the angle of inclination of the trajectory. The system of equations belongs to a certain type that allows us to reduce the optimal control problem with constraints on the state variable to the optimal control problem with control constraints. The sequence and the number of switchings of the state constraints along the optimal trajectory are determined, and a scheme for optimal control is designed. [ABSTRACT FROM AUTHOR]

In this article, we consider the chemostat system with n≥1$$ n\ge 1 $$ species, one limiting substrate, and mutations between species. Our objective is to globally stabilize the corresponding dynamical system around a desired equilibrium point. Doing so, we introduce auxostat feedback controls which are controllers allowing the regulation of the substrate concentration. We prove that such feedback controls globally stabilize the resulting closed‐loop system near the desired equilibrium point. This result is obtained by combining the theory of asymptotically autonomous systems and an explicit computation of solutions to the limit system. The performance of such controllers is illustrated on an optimal control problem of Lagrange type which consists in maximizing the production of species over a given time period w.r.t. the dilution rate chosen as control variable. [ABSTRACT FROM AUTHOR]

There are no common methods of the search for optimal solutions to problems of economic dynamics. Application of variational methods is possible for a quite narrow class of problems. In this paper, we discuss application of the Pontryagin maximum principle, which significantly expands the class of problems considered and allows one to obtain numerical solutions. [ABSTRACT FROM AUTHOR]

Vaccination strategies remain one of the most effective and feasible preventive measures in combating infectious diseases, particularly during the COVID-19 pandemic. With the passage of time, continuous long-term lockdowns became impractical, and the effectiveness of contact-tracing procedures significantly declined as the number of cases increased. This paper presents a mathematical assessment of the dynamics and prevention of COVID-19, taking into account the constant and time-varying optimal COVID-19 vaccine with multiple doses. We attempt to develop a mathematical model by incorporating compartments with individuals receiving primary, secondary, and booster shots of the COVID-19 vaccine in a basic epidemic model. Initially, the model is rigorously studied in terms of qualitative analysis. The stability analysis and mathematical results are presented to demonstrate that the model is asymptotically stable both locally and globally at the COVID-19-free equilibrium state. We also investigate the impact of multiple vaccinations on the COVID-19 model's results, revealing that the infection risk can be reduced by administrating the booster vaccine dose to those individuals who already received their first vaccine doses. The existence of backward bifurcation phenomena is studied. A sensitivity analysis is carried out to determine the most sensitive parameter on the disease incidence. Furthermore, we developed a control model by introducing time-varying controls to suggest the optimal strategy for disease minimization. These controls are isolation, multiple vaccine efficacy, and reduction in the probability that different vaccine doses do not develop antibodies against the original virus. The existence and numerical solution to the COVID-19 control problem are presented. A detailed simulation is illustrated demonstrating the population-level impact of the constant and time-varying optimal controls on disease eradication. Using the novel concept of human awareness and several vaccination doses, the elimination of COVID-19 infections could be significantly enhanced. [ABSTRACT FROM AUTHOR]

In the paper, we study an optimal control problem connected with a fractional Fornasini-Marchesini model described by partial Caputo derivatives. Using a new version of the increment method, the necessary and sufficient optimality condition is obtained in the form of the Pontryagin maximum principle for the posed fractional optimal control problem. The paper is concluded with illustrating example. [ABSTRACT FROM AUTHOR]

We consider the terminal state-constrained optimal control problem for Volterra integral equations with singular kernels. A singular kernel introduces abnormal behavior of the state trajectory with respect to the parameter of α ∈ (0, 1). Our state equation covers various state dynamics such as any types of classical Volterra integral equations with nonsingular kernels, (Caputo) fractional differential equations, and ordinary differential state equations. We prove the maximum principle for the corresponding state-constrained optimal control problem. In the proof of the maximum principle, due to the presence of the (terminal) state constraint and the control space being only a separable metric space, we have to employ the Ekeland variational principle and the spike variation technique, together with the intrinsic properties of distance function and the generalized Gronwall's inequality, to obtain the desired necessary conditions for optimality. The maximum principle of this paper is new in the optimal control problem context and its proof requires a different technique, compared with that for classical Volterra integral equations studied in the existing literature. [ABSTRACT FROM AUTHOR]

This paper studies a single degree of freedom system under free vibration and controlled by a general semi-active damping. A general integral of squared error is considered as the performance index. A one-time switching damping controller is proposed and optimized. The pontryagin maximum principle is used to prove that no other form of semi-active damping can provide the better performance than the proposed one-time switching damping.

We consider the terminal state-constrained optimal control problem for Volterra integral equations with singular kernels. A singular kernel introduces abnormal behavior of the state trajectory with respect to the parameter of $ \alpha \in (0, 1) $. Our state equation covers various state dynamics such as any types of classical Volterra integral equations with nonsingular kernels, (Caputo) fractional differential equations, and ordinary differential state equations. We prove the maximum principle for the corresponding state-constrained optimal control problem. In the proof of the maximum principle, due to the presence of the (terminal) state constraint and the control space being only a separable metric space, we have to employ the Ekeland variational principle and the spike variation technique, together with the intrinsic properties of distance function and the generalized Gronwall's inequality, to obtain the desired necessary conditions for optimality. The maximum principle of this paper is new in the optimal control problem context and its proof requires a different technique, compared with that for classical Volterra integral equations studied in the existing literature.

Efficient train driving plays a vital role in reducing the overall energy consumption in the railway sector. An energy minimising control strategy can be computed using the framework given by optimal control theory; in particular, the Pontryagin maximum principle can be used. Our optimisation approach is based on an algorithm presented by Khmelnitsky that considers electric trains equipped with regenerative braking. A derivation of Khmelnitsky's theory from a more general formulation of the maximum principle is given in this article, and a complete list of switching cases between different driving regimes is included that is essential for practical application. A number of numerical examples are added to visualise the various switching cases. Energy consumption data from real-life operation of passenger trains are compared to the calculated energy minimum. In the presented study, the optimised strategy was able to save 37 percent of the average energy demand of the train in operation. The sensitivity of the energy consumption to deviations of the train speed from the optimum speed profile is studied in an example. Another example illustrates that the efficiency of regenerative braking has an effect on the optimum speed profile. [ABSTRACT FROM AUTHOR]

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE 2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part. [ABSTRACT FROM AUTHOR]

The paper deals with an infinite-horizon optimal control problem with general asymptotic endpoint constraints. The fulfillment of constraints of this type can be viewed as the minimal necessary condition for the sustainability of solutions. A new version of the Pontryagin maximum principle with an explicitly specified adjoint variable is developed. The proof of the main results is based on the fact that the restriction of the optimal process to any finite time interval is a solution to the corresponding finite-horizon problem containing the conditional cost of the phase vector as a terminal term. [ABSTRACT FROM AUTHOR]

The turnpike phenomenon stipulates that the solution of an optimal control problem in large time remains essentially close to a steady-state of the dynamics, itself being the optimal solution of an associated static optimal control problem. Under general assumptions, it is known that not only the optimal state and the optimal control, but also the adjoint state coming from the application of the Pontryagin maximum principle, are exponentially close to that optimal steady-state, except at the beginning and at the end of the time frame. In such a result, the turnpike set is a singleton, which is a steady-state. In this paper, we establish a turnpike result for finite-dimensional optimal control problems in which some of the coordinates evolve in a monotone way, and some others are partial steady-states of the dynamics. We prove that the discrepancy between the optimal trajectory and the turnpike set is linear, but not exponential: we thus speak of a linear turnpike theorem. [ABSTRACT FROM AUTHOR]

In this article we establish first and second order sufficient optimality conditions for a class of single-input control-affine problems, in presence of a scalar state constraint. We consider strong-local optimality (that is, the C 0 topology in the state space). The minimum-time and the Mayer problem are addressed. We restrict our analysis to extremals containing a bang arc, a single boundary arc, followed by a finite number of bang arcs. The sufficient conditions are expressed as a strengthened version of the necessary ones, plus the coerciveness of a suitable finite-dimensional quadratic form. The sufficiency of the given conditions is proven via Hamiltonian methods. [ABSTRACT FROM AUTHOR]

In this research, for a given time interval, which is the general period of melanoma treatment, a bilinear control model is considered, given by a system of differential equations, which describes the interaction between drug-sensitive and drug-resistant cancer cells both during drug therapy and in the absence of it. This model also contains a control function responsible for the transition from the stage of such therapy to the stage of its absence and vice versa. To find the optimal moments of switching between these stages, the problem of minimizing the cancer cells load both during the entire period of melanoma treatment and at its final moment is stated. Such a minimization problem has a nonconvex control set, which can lead to the absence of an optimal solution to the stated minimization problem in the classes of admissible modes traditional for applications. To avoid this problem, the control set is imposed to be convex. As a result, a relaxed minimization problem arises, in which the optimal solution exists. An analytical study of this minimization problem is carried out using the Pontryagin maximum principle. The corresponding optimal solution is found in the form of synthesis and may contain a singular arc. It shows that there are values of the parameters of the bilinear control model, its initial conditions, and the time interval for which the original minimization problem does not have an optimal solution, because it has a sliding mode. Then for such values it is possible to find an approximate optimal solution to the original minimization problem in the class of piecewise constant controls with a predetermined number of switchings. This research presents the results of the analysis of the connection between such an approximate solution of the original minimization problem and the optimal solution of the relaxed minimization problem based on numerical calculations performed in the Maple environment for the specific values of the parameters of the bilinear control model, its initial conditions, and the time interval. [ABSTRACT FROM AUTHOR]

We consider a harvesting problem in which the dynamics of the renewable resource is described by a generalized population model, which requires only a very basic knowledge about the evolution of the resource and the governing equations. We assume that the catch is bilinear in both the control and the stock of the resource, and solve an undiscounted infinite horizon optimal control problem, where the profit functional is assumed to be linear in the catch. We further suggest a discrete scheme that—in addition to being easily implementable—closely approximates the computed optimal control, and illustrate the effect of the proposed discrete scheme with a numerical example. [ABSTRACT FROM AUTHOR]

A combined flight profile is being considered, in which the Falcon 9 launch vehicle launches the spacecraft into elliptical intermediate orbit from the Cape Canaveral launch site. In the process of analyzing the problem, the value of the height of the perigee and the inclination of the intermediate orbit are fixed, and the height of the apogee of the intermediate orbit varies. After separation from the last stage of the launch vehicle, the spacecraft carries out transfer to geostationary orbit using electric propulsion system. At the stage of insertion spacecraft from intermediate orbit to geostationary orbit using electric propulsion system, the problem of minimizing the mass of the propellant, multi-revolutionary transfer is considered. The number of revolution and the height of the apogee of the intermediate orbit vary in order to analyze the effect of these parameters on the duration of the transfer and the delivered mass of the spacecraft into geostationary orbit. The main purpose of this paper is to calculate the optimal values of the apogee height of the intermediate orbit and the optimal number of revolution that ensure the delivery of the maximum mass of the spacecraft to the geostationary orbit in a given time delta(t)*. To solve the optimization problem, the Pontryagin maximum principle is applied. After apply ing the maximum principle, the optimization problem is reduced to solving the boundary value problem, which is solved by the continuation method by parameter. The paper presents the results of the optimization problem of multi-revolutionary transfer and analysis of the energy characteristics of combined flight profile for insertion of spacecraft into geostationary orbit.

This study aimed to minimize the tumor cell population using minimal medicine for chemotherapy treatment, while maintaining the effector-immune cell population at a healthy threshold. Therefore, a mathematical model was developed in the form of ordinary differential equations (ODE), and the solution to the Multi-Objective Optimal Control Problem (MOOCP) was obtained using Multi-Objective Optimization algorithms. In this study, the interaction of the tumor cell and effector cell populations with chemotherapy was investigated using Pure MOOCP and Hybrid MOOCP methods. The handling of constraints and the Pontryagin Maximum Principle (PMP) differ among these methods. Swarm Intelligence (SI) and Evolutionary Algorithms (EA) were used to process the results of these methods. The numerical outcomes of SI and EA are displayed via the Pareto Optimal Front. In addition, the solutions from these algorithms were further analyzed using the Hypervolume Indicator. The findings of this study demonstrate that the Hybrid Method outperforms Pure MOOCP via Multi-Objective Differential Evolution (MODE). MODE produces a point on the Pareto Optimal Front with a minimal distance to the origin, where the distance represents the best solution.

Optimal control of two-qubit quantum systems attracts high interest due to applications ranging from two-qubit gate generation to optimization of receiver for transferring coherence matrices along spin chains. State preparation and manipulation are among important tasks to study for such systems. Typically coherent control, e.g., a shaped laser pulse, is used to manipulate two-qubit systems. However, the environment can also be used—as an incoherent control resource. In this article, we consider optimal state manipulation for a two-qubit system whose dynamics is governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation, where coherent control enters into the Hamiltonian and incoherent control into both the Hamiltonian (via Lamb shift) and the superoperator of dissipation. We exploit two physically different classes of interaction with coherent control and optimize the Hilbert–Schmidt overlap between final and target density matrices, including optimization of its steering to a given value. We find the conditions when zero coherent and incoherent controls satisfy the Pontryagin maximum principle and, in addition, when they form a stationary point of the objective functional. Moreover, we find a case when this stationary point provides the globally minimal value of the overlap. Using upper and lower bounds for the overlap, we develop one- and two-step gradient projection methods operating with functional controls. [ABSTRACT FROM AUTHOR]

The paper analyses dynamical growth models predicting the cyclic development of investigated economic factors. The provided research deals with an optimal control problem based on the economic growth model with the production function of Cobb–Douglas type. Following the Pontryagin maximum principle, we derived the Hamiltonian system and conducted its qualitative analysis, which reveals conditions for the cyclic behaviour of the optimal solutions around the isolate steady state. Numerical experiments visually illustrated the obtained results by demonstrating a phase portrait corresponding to a steady state of the focal type. [ABSTRACT FROM AUTHOR]

In this study, we develop a new mathematical model with vaccination to properly comprehend dynamics of the Lumpy Skin Disease (LSD) ailment. We analyze the model for the existence of a unique positive and bounded solution. To assess the contagiousness of the disease and to test the proposed model for local and global stability at the disease-free and endemic equilibrium points, we determine the reproduction number R 0 . We also investigate the influence of model parameters on reproduction number R 0 by performing sensitivity analysis. The main objective of this study is to carry out different disease control techniques to determine the optimal one. As a first strategy, we analyze the effect of different constant vaccination rates and constant exposure rates on disease control. Secondly, we construct an optimal control problem to investigate the influence of vaccination on disease control with possible elimination from society. The numerical findings reveal that the proposed optimal control strategy for control of LSD is more effective in lowering the number of infected animals. [ABSTRACT FROM AUTHOR]

We consider a time-optimal problem for a car model that can move forward on a plane and turn with a given minimum turning radius. Trajectories of this system are applicable in image processing for the detection of salient lines. We prove the controllability and existence of optimal trajectories. Applying the necessary optimality condition given by the Pontryagin maximum principle, we derive a Hamiltonian system for the extremals. We provide qualitative analysis of the Hamiltonian system and obtain explicit expressions for the extremal controls and trajectories. [ABSTRACT FROM AUTHOR]

In this paper, we derive first-order Pontryagin optimality conditions for risk-averse stochastic optimal control problems subject to final time inequality constraints whose costs are general, possibly nonsmooth finite coherent risk measures. Unlike preexisting contributions covering this situation, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. In addition, it presents the advantages of neither involving second-order adjoint equations nor leading to the so-called weak version of the Pontryagin maximum principle, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the sufficient optimality conditions for the optimal controls under some convexity assumptions. For the Bolza problem, under concavity of the Hamiltonian and convexity of the cost functional, the extreme control must be optimal control. However, for the Mayer problem, the cost functional can be relaxed to quasi‐convex and pseudo‐convex. Finally, some examples illustrate these theoretical results and some counter examples show that the convexity assumptions of these results cannot be further weakened in some sense. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient method to solve linear time-delay optimal control problems with a quadratic cost function. In this regard, first, by employing the Pontryagin maximum principle to time-delay systems, the original problem is converted into a sequence of two-point boundary value problems (TPBVPs) that have both advance and delay terms. Then, using the continuous Runge–Kutta (CRK) method, the resulting sequences are recursively solved by the shooting method to obtain an optimal control law. This obtained optimal control consists of a linear feedback term, which is obtained by solving a Riccati matrix differential equation, and a forward term, which is an infinite sum of adjoint vectors, that can be obtained by solving sequences of delay TPBVPs by the shooting CRK method. Finally, numerical results and their comparison with other available results illustrate the high accuracy and efficiency of our proposed method.

The focus of this article is on the study of discrete optimal control problems (DOCPs) governed by time-varying systems, including time-varying delays in control and state variables. DOCPs arise naturally in many multi-stage control and inventory problems where time enters discretely in a natural fashion. Here, the Euler--Lagrange formulation (which are two-point boundary values with time-varying multi-delays) is employed as an efficient technique to solve DOCPs with time-varying multi-delays. The main feature of the procedure is converting the complex version of the discrete-time optimal control problem into a simple form of differential equations. Since the main problem is in discrete form, then the Euler--Lagrange equation changes to an algebraic system with initial and final conditions. The graphic representation of numerical simulation results shows that the proposed method can effectively and reliably solve DOCPs with time-varying multi-delays.

The object of research is the temperature regime of melting in a cupola. The synthesis of optimal control of such an object is associated with the presence of a problem consisting in the complexity of its mathematical description and the absence of procedures that allow one to obtain optimal control laws. These problems are due to the presence of links with a pure delay, non-additive random drift, and difficulties in controlling the process parameters, in particular, accurately determining the temperature profile along the horizons and the periphery of the working space of the cupola. The proposed conceptual solution for the synthesis of optimal temperature control allows the use of two levels of control: the level controller solves the problem of maintaining the constant height of the idle charge, and the problem of increasing the temperature of cast iron is solved by controlling the air supply to the tuyere box. It is shown that the problem of regulating the upper level of an idle charge can be solved by reducing the model of the regulation process to a typical form, followed by the use of the Pontryagin maximum principle. A procedure for the synthesis of optimal air flow control is proposed, which makes it possible to obtain the temperature regime control law on the basis of experimental industrial studies preceding the synthesis process. This takes into account the time delay between the impact on the object and its reaction, which makes it possible to predict the temperature value one step acharge, equal to the time interval during which the lower surface of the fuel charge reaches the upper surface of the level of the idle charge. A procedure for temperature profile control based on the use of D-optimal plans for selecting sensor installation points is proposed. Due to this, it becomes possible to determine the temperature profile of the cupola according to its horizons and the periphery of the working space of the cupola with maximum accuracy. The proposed synthesis method can be used in iron foundries equipped with cupolas, as it is a tool for studying a real production process, taking into account its specific conditions. This will allow developing or improving control systems for cupola melting, implementing different control modes: manual, automated or automatic. [ABSTRACT FROM AUTHOR]

This paper is devoted to sufficient condition for Strong Metric sub‐Regularity (SMsR for short) of the set‐valued mapping corresponding to the local description of Pontryagin maximum principle for the Mayer‐type optimal control problems with convexity condition of the Hamiltonian and functional. In particular, stability property of optimal control for the Mayer‐type problem has been established for the occasion of a polyhedral control set and entirely bang‐bang solution structure. Moreover, based on the sufficiency of SMsR and stability property of optimal control, we give the approximate errors of Euler discretization methods utilized to such problems. [ABSTRACT FROM AUTHOR]

We consider an infinite-horizon optimal control problem with an asymptotic terminal constraint. For the weakly overtaking criterion and the overtaking criterion, necessary boundary conditions on co-state arcs are deduced, these conditions need no assumptions about the asymptotic behavior of the motion, co-state arc, cost functional, and its derivatives. In the absence of an asymptotic terminal constraint, these boundary conditions with the Pontryagin Maximum Principle allow raising the co-state arcs, corresponding to some asymptotic subdifferentials of the cost functional (fixing the optimal control) at infinity. If this set is a singleton, these conditions coincide with the co-state arc representation proposed by Aseev and Kryazhimskii. These results are illustrated by several examples. [ABSTRACT FROM AUTHOR]