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Forecasts of demands or prices become increasingly unreliable as the future becomes more distant. It is, therefore, beneficial to show that optimal decisions during an initial time interval are either partially or wholly independent of the forecasted data from some future time onwards. Using a commodity trading model as an example, we obtain conditions that allow us to make optimal buying and selling decisions for a commodity in some initial time interval without knowing its price forecast beyond some future time. Such an initial time interval is called a decision horizon and the time up to which the forecasted data is required to make the optimal decisions during the decision horizon is called a forecast horizon. We use the maximum principle to solve the example and show that the decision and forecast horizons in the problem arise from lower and upper bounds imposed on the on-hand inventory of the commodity. [ABSTRACT FROM AUTHOR]

Motivated by a problem of optimal harvesting of natural resources, we study a control problem for Volterra type dynamics driven by time-changed Lévy noises, which are in general not Markovian. To exploit the nature of the noise, we make use of different kind of information flows within a maximum principle approach. For this we work with backward stochastic differential equations (BSDE) with time-change and exploit the non-anticipating stochastic derivative introduced in Di Nunno and Eide (Stoch Anal Appl 28:54-85, 2009). We prove both a sufficient and necessary stochastic maximum principle. [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 consider a constrained optimal control problem and an extension of it, in which the set of strict-sense trajectories is enlarged. Extension is a common procedure in optimal control used to derive necessary and sufficient optimality conditions for the original problem from the extended one, which usually admits a minimizer and has a more regular structure. However, this procedure fails if the two problems have different infima. Therefore, it is relevant to identify such situations. Following on from earlier work by Warga but adopting perturbation techniques developed in nonsmooth analysis, we investigate the relation between the occurrence of an infimum gap and the abnormality of necessary conditions. For the notion of a local minimizer based on control distance and an extension, including the impulsive one, we prove that (i) a local extended minimizer that is not a local minimizer of the original problem, and (ii) a local strict-sense minimizer that is not a local minimizer of the extended problem both satisfy the extended maximum principle in abnormal form. The main novelty is result (ii), as until now, it has only been shown that a strict-sense minimizer that is not an extended minimizer is abnormal for an 'averaged version' of the maximum principle. [ABSTRACT FROM AUTHOR]

This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, D a + α D b - β , 1 < α + β < 2 . Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of D a + α D b - β u (x) at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf's Lemma and maximum principles. As α → 1 - or α , β → 1 - , those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as "elliptic diffusion operators", however, without any physical interpretation. [ABSTRACT FROM AUTHOR]

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [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]

We consider generalized solutions of the Perona--Malik equation in dimension one, defined as all possible limits of solutions to the semidiscrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times and, in particular, many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely, the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub-/super-solutions. The monotonicity results are proved for a more general class of evolution curves, that we call uv-evolutions. [ABSTRACT FROM AUTHOR]

Analyzing the sustainable decision-making mechanism between household consumption and education investment can theoretically develop education. This study uses the continuous-time utility model to demonstrate the independent characteristics of consumption and education investment, as well as the principle of decision incompatibility in the decision-making process of the utility maximization problem. Then, we establish a three-phase logarithmic utility model to obtain the intertemporal decision-making path of a family. The analysis shows that the investment allocation ratio between the two phases depends on the expected and discounted level of the offsprings' abilities, while the total investment level is related to parental altruism. When parents, with foresight, factor in prospective transfer payments from progeny, the optimal decision is to maximize their children's ultimate human capital within a given total investment. Education investment not only squeezes out consumption but also promotes consumption in various periods due to future transfer payments. The decision-making process of three typical growth stages indicates that as offspring mature and their human capital increases, parents' willingness to invest in education decreases while self-consumption escalates. This study provides a new perspective and theoretical basis for studying household education expenditure, motivation, and related policy formulation. [ABSTRACT FROM AUTHOR]

This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux (F*) and a non-oscillatory flux (Fs). This novel approach paves a way to construct non-oscillatory entropy stable flux (F ˆ) as a simple combination of (F* and Fs) which inherently optimize the numerical diffusion in the entropy stable flux (F ˆ) such that it reduces to the underlying non-oscillatory flux (Fs) in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair (F*, Fs), (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that fully discrete schemes using these entropy stable fluxes do not exhibit nonphysical spurious oscillations in approximating the discontinuities and its non-oscillatory nature is comparable to the non-oscillatory schemes using underlying fluxes (Fs) only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair. [ABSTRACT FROM AUTHOR]

This paper investigates the discrete‐time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward‐backward stochastic differential equations (SG‐FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O (Δ t + h ¯ 4) , where Δ t is the time step and h ¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method. [ABSTRACT FROM AUTHOR]

We study a repulsion-diffusion equation with immigration and linear diffusion, whose asymptotic behaviour is related to stability of long-term dynamics in spatial population models and other branching particle systems. We prove well-posedness and find sharp conditions on the repulsion under which a form of the maximum principle and a strong notion of global boundedness of solutions hold. The critical asymptotic strength of the repulsion is $ |x|^{1-d} $, that of the Newtonian potential. [ABSTRACT FROM AUTHOR]

The following article presents the task of optimizing the control of an autonomous object within a group of other passing objects using Pontryagin's bounded maximum principle. The basis of this principle is a multidimensional nonlinear model of the control process, with state constraints reflecting the motion of passing objects. The analytical synthesis of optimal multi-object control became the basis for the algorithm for determining the optimal and safe object trajectory. Simulation tests of the algorithm on the example of real navigation situations with various numbers of objects illustrate their safe trajectories in changing environmental conditions. The optimal object trajectory obtained using Pontryagin's maximum principle was compared with the trajectory calculated using the Bellman dynamic programming method. The analysis of the research allowed for the formulation of valuable conclusions and a plan for further research in the field of autonomous vehicle control optimization. The maximum principle algorithm allows one to take into account a larger number of objects whose data are derived from ARPA anti-collision radar systems. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal control problem of a mean-field forward--backward stochastic system with random jumps in progressive structure, where both regular and singular controls are considered in our formula. In virtue of the variational technology, the related stochastic maximum principle (SMP) has been obtained, and it is essentially different from that in the classical predictable structure. Specifically, there are three parts in our SMP, that is, continuous part, jump part, and impulse part, and they are, respectively, used to characterize the characteristics of the optimal controls at continuous time, jump time, and impulse time. This shows that the progressive structure can more accurately describe the characteristics of the optimal control at the jump time.We also give two linear--quadratic examples to show the significance of our results. [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]

We consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian (double phase problem) and with a reaction term which exhibits asymmetric behavior as x → ± ∞ . Using variational tools, truncation, and comparison techniques and critical groups, we prove a multiplicity theorem producing four nontrivial solutions all with sign information and ordered. [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.

In this paper, based on the Lagrange multiplier method, we construct a maximum principle preserving scheme for the time-fractional Allen-Cahn equation of 2- α (0 < α < 1) order. The correction energy of this scheme is increased by a term compared to the original energy, which is O (τ α). We prove that our scheme is unconditionally stable related to the corrected energy and verify the convergence, maximum principle, and energy conservation properties of the algorithm through numerical examples. We also find that the larger the α , the faster the evolution of the time-fractional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field E and the a-priori L ∞ estimate, 0 < 2 δ ≤ ρ ε , δ ≤ M (t) , | u ε , δ | ≤ M (t) , where M (t) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions (ρ ε , δ , u ε , δ) ; Second, the compensated compactness theory is applied to prove the pointwise convergence of (ρ ε , δ , u ε , δ) as ε , δ go to zero, and that the limit (ρ (x , t) , u (x , t)) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫ − ∞ x ρ ε , δ (x , t) − 2 δ d x , deduces the uniform L ∞ estimate, 0 < 2 δ ≤ ρ ε , δ ≤ M , | u ε , δ | ≤ M , independent of the time t and ε , δ ; Finally, as a by-product, the known compactness framework [14,29] is applied to show the relaxation limit, as the relation time τ and ε , δ go to zero, for general pressure P (ρ). [ABSTRACT FROM AUTHOR]

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example. [ABSTRACT FROM AUTHOR]

The stabilization approach has been known to permit large time-step sizes while maintaining stability. However, it may “slow down the convergence rate” or cause “delayed convergence” if the time-step rescaling is not well resolved. By considering a fourth-order-in-space viscous Cahn–Hilliard (VCH) equation, we propose a class of up to the fourth-order single-step methods that are able to capture the correct physical behaviors with high-order accuracy and without time delay. By reformulating the VCH as a system consisting of a second-order diffusion term and a nonlinear term involving the operator (I - ν Δ) - 1 , we first develop a general approach to estimate the maximum bound for the VCH equation equipped with either the Ginzburg–Landau or Flory–Huggins potential. Then, by taking advantage of new recursive approximations and adopting a time-step-dependent stabilization, we propose a class of stabilization Runge–Kutta methods that preserve the maximum principle for any time-step size without harming the convergence. Finally, we transform the stabilization method into a parametric Runge–Kutta formulation, estimate the rescaled time-step, and remove the time delay by means of a relaxation technique. When the stabilization parameter is chosen suitably, the proposed parametric relaxation integrators are rigorously proven to be mass-conserving, maximum-principle-preserving, and the convergence in the l ∞ -norm is estimated with pth-order accuracy under mild regularity assumption. Numerical experiments on multi-dimensional benchmark problems are carried out to demonstrate the stability, accuracy, and structure-preserving properties of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we describe a new nonlinear finite-volume scheme that preserves the discrete maximum principle (DMP) for the two-dimensional sub-diffusion equation on distorted meshes. One distinguishing feature of our method is its ability to uphold the DMP for the anisotropic sub-diffusion problems, thereby ensuring the absence of spurious oscillations in numerical solutions and maintaining the physical bounds of various quantities, such as concentration, temperature, and density. Notably, our scheme offers the advantage of being applicable to distorted meshes without stringent constraints. Numerical results demonstrate that our scheme successfully preserves maximum principle on various randomly distorted meshes. [ABSTRACT FROM AUTHOR]

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance's concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance's concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions. [ABSTRACT FROM AUTHOR]

3D Burgers equation is an important model for turbulence. It is natural to expect the long-time behaviour for this hydrodynamics equation. However, there is no result about the long-time behaviour for this deterministic model. Surprisingly, if the system is perturbed by stochastic noise, we establish the existence and uniqueness of invariant measure for 3D stochastic Burgers equation. [ABSTRACT FROM AUTHOR]

Let (M,g) be a compact Riemann surface, h be a positive smooth function on M. It is well known that the functional \begin{equation*} J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^{u}dv_g \end{equation*} achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative h by Yang and the author. Later, Sun and Zhu [ Existence of Kazdan-Warner equation with sign-changing prescribed function , arXiv: 2012.12840 , 2020] showed the Ding-Jost-Li-Wang condition is also sufficient when h changes sign, which was reproved later by Wang and Yang [J. Funct. Anal. 282 (2022), Paper No. 109449] and Li and Xu [Calc. Var. Partial Differential Equations 61 (2022), Paper No. 143] respectively using a flow approach. The aim of this note is to give a new proof of Sun and Zhu's result. Our proof is based on the variational method and the maximum principle. [ABSTRACT FROM AUTHOR]

Two new numerical schemes to approximate the Cahn-Hilliard equation with degenerate mobility (between stable values 0 and 1) are presented, by using two different non-centered approximation of the mobility. We prove that both schemes are energy stable and preserve the maximum principle approximately, i.e. the amount of the solution being outside of the interval [ 0 , 1 ] goes to zero in terms of a truncation parameter. Additionally, we present several numerical results in order to show the accuracy and the well behavior of the proposed schemes, comparing both schemes and the corresponding centered scheme. [ABSTRACT FROM AUTHOR]

The paper studies a multiplicative control problem for the reaction–diffusion equation in which the reaction coefficient nonlinearly depends on the substance concentration, as well as on spatial variables. The role of multiplicative controls is played by the coefficients of diffusion and mass transfer. The solvability of the extremum problem is proved, and optimality systems are derived for a specific reaction coefficient. Based on the analysis of these systems, the relay property of multiplicative and distributed controls is established, and estimates of the local stability of optimal solutions to small perturbations of both the quality functionals and one of the given functions of the boundary value problem are derived. [ABSTRACT FROM AUTHOR]

In this paper we study the stochastic control problem of a partially observed (multidimensional) stochastic system driven by both Brownian motions and fractional Brownian motions. In the absence of the powerful tool of Girsanov transformation, we introduce and study new stochastic processes which are used to transform the original problem to a "classical one"". The adjoint backward stochastic differential equations and the necessary condition satisfied by the optimal control (maximum principle) are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study elliptic equations in which the reaction (right hand side) exhibits an asymmetric behavior as x → ± ∞ . More precisely, we assume that we have resonance as x → - ∞ , while as x → + ∞ the equation is superlinear. Using variational tools combined with the theory of critical groups, we prove several multiplicity theorems for nonlinear, nonhomogeneous equations and for semilinear equations (driven by the Laplacian). [ABSTRACT FROM AUTHOR]

In this article, we are concerned with optimal control of a frictionless contact problem with multiple unilateral constraints for a two-dimensional bar. The existence of an optimal trajectory-control pair is firstly proven under the framework of general cost functional. The Pontryagin maximum principle is then established for the investigational system equipped with many equality and inequality constraints in fixed final horizon case, owing to the Dubovitskii and Milyutin functional analytical approach. A remark concludes the article with the discussion, which address the utilization of obtained necessary optimality condition. [ABSTRACT FROM AUTHOR]

The purpose of the paper is devoted to proving the solvability of the fourth order boundary value problem. Firstly, we build a maximum principle for the corresponding linear equation, by the use of this maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to solve the nonlinear equation, secondly, the existence and uniqueness results for the problem is obtained. In addition, an example is presented to show the application of our main results. [ABSTRACT FROM AUTHOR]

We prove a weak maximum principle for nonlocal symmetric stable operators including the fractional Laplacian. The main focus of this work is on minimal regularity assumptions of the functions under consideration. [ABSTRACT FROM AUTHOR]

In this article, we study a risk‐sensitive stochastic optimal control problem driven by forward‐backward systems in which the control variable consists of two components: the continuous control and the impulse control. The control domain is assumed to be convex. We establish the maximum principle (i.e., necessary condition) for this kind of control problem. Under some additional assumptions, the necessary optimality conditions turn out to be sufficient. To explain the theoretical results, a linear‐quadratic risk‐sensitive control problem is solved by using the maximum principle derived and the optimal control is obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to the real analysis methods for the problem of construction of Markov processes with boundary conditions in probability. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called Ventcel' (Wentzell) boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by continuous paths and by jumps in the state space and it obeys the Ventcel' boundary condition that consists of six terms corresponding to a diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first order Ventcel' boundary value problems for second order elliptic Waldenfels integro-differential operators. By using the Calderón–Zygmund theory of singular integrals in real analysis, we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces, which extend earlier theorems due to Bony–Courrège–Priouret to the VMO (vanishing mean oscillation) case. Our proof is based on various maximum principles for second order elliptic Waldenfels operators with discontinuous coefficients in the framework of Sobolev spaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with an oblique derivative boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking (or viscosity) phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderón–Zygmund theory of singular integral operators with non-smooth kernels. [ABSTRACT FROM AUTHOR]

A reaction–diffusion model, known as the Sel'kov–Schnakenberg model, is considered. The nonlinear stability of the constant steady state is studied by using a special Liapunov functional and a maximum principle for regular solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic optimal control problem for moving average control system. The corresponding moving average stochastic differential equation is a kind of integral differential equations. We prove the existence and uniqueness of the solution of the moving average stochastic differential equations. We obtain the stochastic maximum principle of the moving average optimal control system by introducing a kind of generalized anticipated backward stochastic differential equations. We prove the existence and uniqueness of the solution of this adjoint equation, which is singular at 0. As an application, the linear quadratic moving average control problem is investigated to illustrate the main results. [ABSTRACT FROM AUTHOR]

We consider a Lagrange optimal control problem for a Volterra integral equation of fractional potential type. We prove a theorem on the existence of an optimal solution and derive a maximum principle. The proof of the existence theorem is based on the lower closure theorem for orientor fields due to Cesari and Filippov‐type selection theorem due to Rockafellar. The proof of the maximum principle is based on an extremum principle for smooth problems proved in Idczak and Walczak (Games. 2020;11:56). [ABSTRACT FROM AUTHOR]

In this article, the local and remote stochastic nonzero‐sum game for a multiplicative noise system with inconsistent information is investigated, in which the multiplicative noise can cause nonlinear characteristics of linear systems, making it difficult to solve the optimal linear feedback Nash equilibrium. For the considered local and remote stochastic nonzero‐sum game, the local player and the remote player obtain different information sets, which leads to inconsistent information between the two players. The goal is that each player is desired to minimize their own cost function. Our approach is based on a combination of orthogonal decomposition and completing square techniques, which allow us to derive a set of coupled Riccati equations that characterize the optimal feedback explicit (closed‐loop) Nash equilibrium. The contributions of this article are summarized as follows. First, the optimal open‐loop Nash equilibrium is obtained in terms of the forward and backward stochastic difference equations (FBSDEs) by adopting the Pontryagin maximum principle. Second, the closed‐loop Nash equilibrium of this local and remote stochastic nonzero‐sum game for a multiplicative noise system with inconsistent information is obtained by using the orthogonal decomposition methods. Finally, a simulation example is given to illustrate the validity of theoretical results and discuss potential extensions to more complex systems. [ABSTRACT FROM AUTHOR]

In this article, we mainly study the oblique value problem for the generalized parabolic mean curvature type equations. Using the maximum principle, we prove the uniform gradient estimate of the solution. As applications, the long time existence and asymptic behavior are both obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-mass controlled mechanical system consisting of a supporting disk rotating around its axis fixed in space and a carried ring attached to the disk using weightless elastic elements. There are no dampers in the system. Suppression of radial vibrations is considered from the perspective of optimal control theory. On sufficiently large time intervals, Newton's numerical method is used to solve the boundary value problem of Pontryagin's maximum principle. The properties of the phase trajectories of the system are studied depending on the initial states of the disk and ring and the number of springs in a complex model of elastic interaction. Under certain initial conditions and parameters of the system, the trajectory of the center of mass of the ring is shown to tend to a circle due to the radiality of the elastic force and the conservation law of kinetic momentum. The indicated tendency to reach the circular motion mode is not uniform and depends on the number of springs. It is demonstrated that, at a small number of elastic elements, the trajectory of the ring does not take the form of a circle, while almost complete damping of radial vibrations occurs. For the system parameters considered during the numerical experiment, the control is established to be relay with a sufficiently large number of switchings. In this case, the entire system is simultaneously spinning-up. [ABSTRACT FROM AUTHOR]

The influence of phase and other constraints on the method of searching for the trajectories of a civil supersonic aircraft that are optimal in terms of fuel consumption is considered. Based on the solutions found by the dynamic programming method, which take into account the numerous conditions that must be satisfied by the flight altitude, pitch angle, normal overload, aircraft speed, engine thrust, etc., it is shown that almost all of these conditions can be ignored during the initial stage of calculations, since these constraints do not affect the optimal solutions. Therefore, one can first apply the maximum principle and use the dynamic programming method only in those cases where a sizeable part of the constraints turns out to be significant. [ABSTRACT FROM AUTHOR]

A new product's introduction to the market is greatly influenced by effective and enough advertising. In this paper, we have considered a problem in which the firm partitions the market into various segments to reduce costs associated with advertising and targets to maximize the total profit. By integrating single channel and differentiated advertising to segmented market, we extend Nerlove-goodwill Arrow's dynamic model in which advertising variables are control variables. It is assumed that the whole available budget is imprecise and fuzzy in nature in order to create a realistic model. Using necessity and possibility constraints, the optimal control model with fuzzy parameters is transformed into crisp form, and the Pontryagin Maximum principle is then used to solve the problem. Numerical examples are provided to support the theoretical analysis.

It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge 0$ for $\lambda\in [-\Lambda ,\Lambda ]\setminus\{0\},$ where $y$ is the unique solution of $Ly+\lambda y=f,$ $BC y =0$. So, the boundary value problem satisfies a maximum principle for $\lambda\in [-\Lambda ,0)$ and the boundary value problem satisfies an anti-maximum principle for $\lambda\in (0, \Lambda ]$. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y=f,$ $BC y =0$ where $D_{0}^{\alpha}$ is a Riemann-Liouville fractional differentiable operator of order $\alpha$, $10$ such that $f\ge 0$ implies $\beta D_{0}^{\alpha -1}y \ge 0$ for $\beta \in [-\mathcal{B} ,\mathcal{B} ]\setminus\{0\},$ where $y$ is the unique solution of $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y =f,$ $BC y =0$. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of $\beta D_{0}^{\alpha -1}y.$ The boundary conditions are chosen so that with further analysis a sign property of $\beta y$ is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.

In this study, we investigate a maximum principle of the Fourier spectral method (FSM) for diffusion equations. It is well known that the FSM is fast, efficient and accurate. The maximum principle holds for diffusion equations: A solution satisfying the diffusion equation has the maximum value under the initial condition or on the boundary points. The same result can hold for the discrete numerical solution by using the FSM when the initial condition is smooth. However, if the initial condition is not smooth, then we may have an oscillatory profile of a continuous representation of the initial condition in the FSM, which can cause a violation of the discrete maximum principle. We demonstrate counterexamples where the numerical solution of the diffusion equation does not satisfy the discrete maximum principle, by presenting computational experiments. Through numerical experiments, we propose the maximum principle for the solution of the diffusion equation by using the FSM.

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O(Δt+h¯4), where Δt is the time step and h¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method.