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We present a gradient-based calibration algorithm to identify the system matrices of a linear port-Hamiltonian system from given input–output time data. Aiming for a direct structure-preserving approach, we employ techniques from optimal control with ordinary differential equations and define a constrained optimization problem. The input-to-state stability is discussed which is the key step towards the existence of optimal controls. Further, we derive the first-order optimality system taking into account the port-Hamiltonian structure. Indeed, the proposed method preserves the skew symmetry and positive (semi)-definiteness of the system matrices throughout the optimization iterations. Numerical results with perturbed and unperturbed synthetic data, as well as an example from the PHS benchmark collection [17] demonstrate the feasibility of the approach. [ABSTRACT FROM AUTHOR]

Crimes are social ills that make a society unsafe to live in. Since a child is born into a family and a society is made up of different families, we develop a model to study the influence of family background on the spread of crime in the society. Five compartments are considered including: those who can commit crime (S), those who are resistant to committing crime (R), those who are committing crime alone (I), those who are committing crime in group (G) and the criminals who are under rehabilitation (J). The validity of the model was established. Crime reproductive ratio C R was computed and used to derive the necessary and sufficient conditions for crime eradication and persistence. Simulation was conducted using the computer-in-built Runge–Kutta scheme implemented in software maple and the results showed that the effectiveness of crime education together with strict security and legal system up to 70% were sufficient to eradicate criminal activities in a community. [ABSTRACT FROM AUTHOR]

In the realm of sub-Riemannian manifolds, a relevant question is: what are the metric lines (isometric embedding of the real line)? The space of kk-jets of a real function of one real variable xx, denoted by Jk(R,R){J}^{k}\left({\mathbb{R}},{\mathbb{R}}), admits the structure of a Carnot group. Every Carnot group is sub-Riemannian manifold, so is Jk(R,R){J}^{k}\left({\mathbb{R}},{\mathbb{R}}). This study aims to present a partial result about the classification of the metric lines within Jk(R,R){J}^{k}\left({\mathbb{R}},{\mathbb{R}}). The method is to use an intermediate three-dimensional sub-Riemannian space RF3{{\mathbb{R}}}_{F}^{3} lying between the group Jk(R,R){J}^{k}\left({\mathbb{R}},{\mathbb{R}}) and the Euclidean space R2{{\mathbb{R}}}^{2}.

In this article, we study the role of awareness and its impact on the control of infectious diseases. We analyze a susceptible-infected-recovered model with a media awareness compartment. We find the effective reproduction number R0{R}_{0}. We observe that our model exhibits transcritical forward bifurcation at R0=1{R}_{0}=1. We also performed the sensitivity analysis to determine the sensitivity of parameters of the effective reproduction number R0{R}_{0}. In addition, we study the corresponding optimal control problem by considering control in media awareness and treatment. Our studies conclude that we can reduce the rate of spread of infection in the population by increasing the treatment rate along with media awareness.

A class of parametric optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is investigated. The perturbations appear in the objective functional, the state equation and in mixed pointwise constraints. By analyzing regularity and establishing stability condition of Lagrange multipliers we prove that, if the unperturbed problem satisfies the strong second-order sufficient condition, then the solution map and the associated Lagrange multipliers are locally Lipschitz continuous functions of parameters. [ABSTRACT FROM AUTHOR]

We study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a minimal time control problem for a class of homogeneous evolution equations with impulse controls. We provide a systematical method to study the existence, uniqueness and bang-bang property of optimal controls for the problem. Moreover, we also establish the continuous dependence of the optimal control by the continuity of the minimal time function of this problem with respect to the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general Mayer optimal control problem governed by a mono-input affine control system whose optimal solution involves a second-order singular arc (leading to chattering). The objective of the paper is to present a numerical scheme to approach the chattering control by controls with a simpler structure (concatenation of bang-bang controls with a finite number of switching times and first-order singular arcs). Doing so, we consider a sequence of vector fields converging to the drift such that the associated optimal control problems involve only first-order singular arcs (and thus, optimal controls necessarily have a finite number of bang arcs). Up to a subsequence, we prove convergence of the sequence of extremals to an extremal of the original optimal control problem as well as convergence of the value functions. Next, we consider several examples of problems involving chattering. For each of them, we give an explicit family of approximated optimal control problems whose solutions involve bang arcs and first-order singular arcs. This allows us to approximate numerically solutions (with chattering) to these original optimal control problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the Hamilton–Jacobi equation with phase constraints arising in molecular biology. The problem has no classical solutions, and the Hamiltonian does not satisfy the conditions guaranteeing the existence of minimax or viscous generalized solutions. A new continuous generalized solution is obtained. We present sufficient conditions for the existence of a global solution preserving the structure given by the initial manifold. The behavior of such solutions is examined for large values of time. [ABSTRACT FROM AUTHOR]

We consider a method for solving the problem of optimal control of a composite dynamic system with stochastic resource constraints. The method is based on the method of hierarchical optimization, which includes iterative solution of local problems and the coordination problem. To solve local problems, we apply the model predictive control. Taking into account stochastic resource constraints leads to the solution of one-stage problems of stochastic programming. [ABSTRACT FROM AUTHOR]

How to free a road from vehicle traffic as efficiently as possible and in a given time, in order to allow for example the passage of emergency vehicles? We are interested in this question which we reformulate as an optimal control problem. We consider a macroscopic road traffic model on networks, semi-discretized in space and decide to give ourselves the possibility to control the flow at junctions. Our target is to smooth the traffic along a given path within a fixed time. A parsimony constraint is imposed on the controls, in order to ensure that the optimal strategies are feasible in practice. We perform an analysis of the resulting optimal control problem, proving the existence of an optimal control and deriving optimality conditions, which we rewrite as a single functional equation. We then use this formulation to derive a new mixed algorithm interpreting it as a mix between two methods: a descent method combined with a fixed point method allowing global perturbations. We verify with numerical experiments the efficiency of this method on examples of graphs, first simple, then more complex. We highlight the efficiency of our approach by comparing it to standard methods. We propose an open source code implementing this approach in the Julia language. [ABSTRACT FROM AUTHOR]

Sustainable forest management is one of the warming issues in the present century. In this manuscript, we have employed the model of control theory to control the consequence of toxicity and illegal logging of mature trees in the ecosystem of Sundarbans, the largest mangrove forest in the world. In this investigation, we have momentarily mentioned some of the fields in which these challenges are present. These fields especially consist of sustainable forestry management of ecosystem. We have reflected on the modified Leslie-Gower response function to set up as the alternative resource for industries when forestry resources are divested. The boundedness, persistence, equilibria and stability are examined.Our main aim is to investigate the spans and applications of control theory to control the effect of toxicity and illegal logging. [ABSTRACT FROM AUTHOR]

We explicitly characterize boundary conditions that are compatible with low order variational principles. The freedom afforded by adding in a null Lagrangian without altering the Euler–Lagrange equation significantly expands the range of variationally admissible boundary conditions, although not all possibilities are permitted. Applications to several fundamental problems arising in elastostatics, including bars, beams, and plates, are presented. [ABSTRACT FROM AUTHOR]

Consider a Conservation Law and a Hamilton-Jacobi equation with a flux/Hamiltonian depending also on the space variable. We characterize first the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed profile. An explicit example then shows the deep differences between the cases of x-independent and x-dependent fluxes/Hamiltonians. [ABSTRACT FROM AUTHOR]

We prove that if the total factor productivity A of an aggregative economy is right at the barrier $ \sigma +\lambda $ σ + λ , with σ being the growth rate of labor force and λ the real interest rate, then the unique policy to optimally control the economy is the same as the one for optimally controlling weak economies, where $ A \lt \sigma +\lambda $ A < σ + λ. This result gives a complete answer for the interesting open question raised by Vu Thi Huong in her recent paper [Optimal economic growth problems with high values of total factor productivity. Appl Anal. 2022;101:1315–1329]. [ABSTRACT FROM AUTHOR]

We propose an extension of the anisotropic interaction model which allows for collision avoidance in pairwise interactions by a rotation of forces (Totzeck (2020) Kinet. Relat. Models 13 (6), 1219–1242.) by including the agents' body size. The influence of the body size on the self-organisation of the agents in channel and crossing scenarios as well as the fundamental diagram is studied. Since the model is stated as a coupled system of ordinary differential equations, we are able to give a rigorous well-posedness analysis. Then we state a parameter calibration problem that involves data from real experiments. We prove the existence of a minimiser and derive the corresponding first-order optimality conditions. With the help of these conditions, we propose a gradient descent algorithm based on mini-batches of the data set. We employ the proposed algorithm to fit the parameter of the collision avoidance and the strength parameters of the interaction forces to given real data from experiments. The results underpin the feasibility of the method. [ABSTRACT FROM AUTHOR]

We consider three optimal control problems (for linear, bilinear, and quadratic functionals) for a special bilinear system with a matrix of rank 1. For the first problem, we obtain two versions of conditions for initial data of the system and functional such that the maximum principle becomes a sufficient optimality condition. In this case, the problem becomes very simple: the optimal control is determined in the integration process of the phase or adjoint system (one Cauchy problem). Next, the optimization problem for a bilinear functional is considered. Sufficient optimality conditions for the boundary controls without switching points are obtained. These conditions are represented as inequalities for functions of one variable (time). The optimal control problem with the quadratic functional is reduced to the bilinear case on the basis of special increment formula. [ABSTRACT FROM AUTHOR]

The vector or pest control is essential to reduce the risk of vector-borne diseases or crop losses. Among the available biological control tools, the sterile insect technique (SIT) is one of the most promising. However, SIT-control campaigns must be carefully planned in advance in order to render desirable outcomes. In this paper, we propose SIT-control intervention programs that can avoid the real-time monitoring of the wild population and require to mass-rear a minimal overall number of sterile insects, in order to induce a local elimination of the wild population in the shortest time. Continuous-time release programs are obtained by applying an optimal control approach, and then laying the groundwork of more practical SIT-control programs consisting of periodic impulsive releases. [ABSTRACT FROM AUTHOR]

We consider control strategies for large-scale interacting agent systems under uncertainty. The particular focus is on the design of robust controls that allow to bound the variance of the controlled system over time. To this end we consider H ∞ control strategies on the agent and mean field description of the system. We show a bound on the H ∞ norm for a stabilizing controller independent on the number of agents. Furthermore, we compare the new control with existing approaches to treat uncertainty by generalized polynomial chaos expansion. Numerical results are presented for one-dimensional and two-dimensional agent systems. [ABSTRACT FROM AUTHOR]

A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton–Jacobi–Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach. [ABSTRACT FROM AUTHOR]

We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g. because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton–Jacobi–Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of Regular Lagrangian Flows. From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small. [ABSTRACT FROM AUTHOR]

In this paper, we examine the classical (so-called unrealistic) Ramsey–Cass–Koopmans model. For the corresponding differential equation, we apply the method of analytical approximation of its solutions. In subsequent works, we develop this method and apply it to problems in a more general statement. [ABSTRACT FROM AUTHOR]

This paper investigates a size-structured vermin contraception control model with nonlinear fertility and mortality, in which the control variable appears in both the state equation and the boundary condition. First, we show that the model has a unique non-negative solution, which has a separable form. Then, some continuity results are established, which are important for the purpose of optimal control. In order to make the final size of vermin as small as possible or as close as possible to the ideal distribution under the condition of the lowest control cost, we consider the least cost-size problem and the least cost-deviation problem. For each of the control problems, the optimality conditions are obtained via the adjoint system and tangent-normal cones. For the first problem, we show the existence of the optimal strategy by compactness and extremal sequence. However, for the second problem, we derive the existence of an optimal control via Ekeland's variational principle. Some numerical simulations have been performed to demonstrate the feasibility of the obtained theoretical results. Numerical results also suggest that decreasing the reproductive rate instead of increasing the mortality is an effective way of managing the impact of vermin. [ABSTRACT FROM AUTHOR]

Worldwide, the recent SARS-CoV-2 virus disease outbreak has infected more than 691,000,000 people and killed more than 6,900,000. Surprisingly, Sub-Saharan Africa has suffered the least from the SARS-CoV-2 pandemic. Factors that are inherent to developing countries and that contrast with their counterparts in developed countries have been associated with these disease burden differences. In this paper, we developed data-driven COVID-19 mathematical models of two 'extreme': Cameroon, a developing country, and New York State (NYS) located in a developed country. We then identified critical parameters that could be used to explain the lower-than-expected COVID-19 disease burden in Cameroon versus NYS and to help mitigate future major disease outbreaks. Through the introduction of a 'disease burden' function, we found that COVID-19 could have been much more severe in Cameroon than in NYS if the vaccination rate had remained very low in Cameroon and the pandemic had not ended. [ABSTRACT FROM AUTHOR]

In this paper, we ameliorate the model proposed in [13], by incorporating the influence of hepatitis B e antigen (HBeAg) status of mothers on vertical transmission. We use the improved model to fit reported HBV new infections in the Zhejiang Province of China. Also to predict the course of the Hepatitis B (HBV) infection in this Chinese area, and in Tokombere, located in sub-Saharan Africa(SSA). Furthermore, we apply optimal control techniques in view to re-examine the effects of the newborn vaccination, the universal vaccination and the treatment of chronic carriers in preventing the HBV infection. Simulation results show that treatment slightly steps in the optimal strategy, while immunisation is an effective measure. On the other hand, they indicate that the control measures and immunization programs implemented in Zhejiang Province are effective. Besides, they suggest that in SSA, a package of several policies centred on birth dose vaccination should be implemented. [ABSTRACT FROM AUTHOR]

In this paper we derive error estimates for Runge–Kutta schemes of optimal control problems subject to index one differential–algebraic equations (DAEs). Usually, Runge–Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge–Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers. [ABSTRACT FROM AUTHOR]

The article considers a high-order optimal control problem and its dual problems described by high-order differential inclusions. In this regard, the established Euler–Lagrange type inclusion, containing the Euler–Poisson equation of the calculus of variations, is a sufficient optimality condition for a differential inclusion of a higher order. It is shown that the adjoint inclusion for the first-order differential inclusions, defined in terms of a locally adjoint mapping, coincides with the classical Euler–Lagrange inclusion. Then the duality theorems are proved. [ABSTRACT FROM AUTHOR]

In this present paper, we concern a non-smooth higher-order extension of Noether's symmetry theorem for variational isoperimetric problems with delayed arguments. The result is proven to be valid in the class of Lipschitz functions, as long as the delayed higher-order Euler–Lagrange extremals are restricted to those that satisfy the delayed higher-order DuBois-Reymond necessary optimality condition. The important case of delayed isoperimetric optimal control problems is considered as well. [ABSTRACT FROM AUTHOR]

In this paper, the effect of inhibition of monocarboxylate transporters on intracellular and capillary lactate concentrations is investigated using an optimal control problem. A control term representing the concentration of the inhibitor is used in an ODE model that models lactate kinetics between the cell and the capillary. Finally, some numerical simulations were performed to confirm the efficiency of the control term for the problem. [ABSTRACT FROM AUTHOR]

Tokamaks are the most promising devices to prove the feasibility of energy production using nuclear fusion on Earth which is foreseen as a possible source of energy for the next centuries. In large tokamaks with superconducting poloidal field (PF) coils, the problem of avoiding saturation of the currents is of paramount importance, especially for a reactor such as the European demonstration fusion power plant DEMO. Indeed, reaching the current limits during plasma operation may cause a loss of control of the plasma shape and/or current, leading to a major disruption. Therefore, a current limit avoidance (CLA) system is essential to assure safe operation. Three different algorithms to be implemented within a CLA system are proposed in this paper: two are based on online solutions of constrained optimization problems, while the third one relies on dynamic allocation. The performance assessment for all the proposed solutions is carried out by considering challenging operation scenarios for the DEMO reactor, such as the case where more than one PF current simultaneously saturates during the discharge. An evaluation of the computational burden needed to solve the allocation problem for the various proposed alternatives is also presented, which shows the compliance of the optimization-based approaches with the envisaged deadlines for real-time implementation of the DEMO plasma magnetic control system. [ABSTRACT FROM AUTHOR]

The left-invariant sub-Lorentzian problem on the Heisenberg group is considered. An optimal synthesis is constructed, the sub-Lorentzian distance and spheres are described. [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]

A controlled sweeping process with prox-regular set, W 1 , 2 -controls, and separable endpoints constraints is considered in this paper. Existence of optimal solutions is established and local optimality conditions are derived via strong converging continuous approximations, whose state entirely resides in the interior of the prox-regular set. Consequently, subdifferentials smaller than the standard ones are now employed in the optimality results. [ABSTRACT FROM AUTHOR]

The local stability of the solution map to a parametric boundary control problem governed by semilinear elliptic equations with finite mixed pointwise constraints is considered in this paper. We prove that the solution map is locally Hölder continuous in L ∞ -norm of control variable when the strictly nonnegative second-order optimality conditions are satisfied for the unperturbed problem. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem with a finite number of mixed constraints G i (x , u) ≤ 0 that are nonregular, i.e. when the gradients G iu ′ (x , u) of active constraints can be positively dependent at some points of the reference process. All data of the problem are assumed to be smooth. Using the Dubovitskii–Milyutin theorem on the approximative separation of convex cones, we prove first order necessary condition for a weak minimum in the form of so-called "local minimum principle", which is formulated in terms of integrable functions and Lebesgue–Stieltjes measures, without using functionals from (L ∞) ∗. Some illustrative examples are given. The work is based on Milyutin's book (Maximum Principle in the General Problem of Optimal Control. Fizmatlit, Moscow, 2001) and is an extension of our paper Dmitruk and Osmolovskii (SIAM J. Control Optim. 60(4):1919-1941, 2022) that admits only one mixed inequality constraint. [ABSTRACT FROM AUTHOR]

The transmission dynamics of COVID-19 is investigated through the prism of the Atangana-Baleanu fractional model with acquired immunity. Harmonic incidence mean-type aims to drive exposed and infected populations towards extinction in a finite time frame. The reproduction number is calculated based on the next-generation matrix. A disease-free equilibrium point can be achieved globally using the Castillo-Chavez approach. Using the additive compound matrix approach, the global stability of endemic equilibrium can be demonstrated. Utilizing Pontryagin's maximum principle, we introduce three control variables to obtain the optimal control strategies. Laplace transform allows simulating the fractional-order derivatives analytically. Analysis of the graphical results led to a better understanding of the transmission dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we study an isospectral problem of a weighted Sturm–Liouville equation with the Dirichlet boundary condition, which lies at the basis of the integrability of the Camassa–Holm equation. We will choose the general setting of the so-called measure differential equations to solve the optimization problem on eigenvalues. It should be noticed that our technique in this paper can be used to deal with other self-adjoint boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of minimizers to a parametric optimal control problem governed by semilinear elliptic equations with mixed pointwise control-state constraints. Under the strictly nonnegative second-order optimality condition assumption, we show that the solution map is locally Lipschitz continuous in L 2 - norm as well as in L ∞ - norm of the control variable. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlinear Lagrange dynamics associated with two classes of constrained controlled optimization problems involving second- order derivatives. More precisely, we formulate and prove necessary conditions of optimality for the considered variational control problems governed by simple integral functionals and second-order ordinary differential equation (ODE)/ isoperimetric constraints. Moreover, we propose an algorithm to synthesize the concrete steps to be followed for solving constrained controlled optimization problems. [ABSTRACT FROM AUTHOR]