Search Results (54)

Periodic solutions of high-order nonlinear differential equations are fundamental in dynamical systems, yet they remain challenging to establish with traditional methods. This paper addresses the existence of periodic solutions in general $2n$-order autonomous and nonautonomous ordinary differential equations. By extending Carathéodory’s variational technique from the calculus of variations, we reformulate the original periodic solution problem as an equivalent higher-order variational problem. The approach constructs a convex function and introduces an auxiliary transformation to enforce convexity in the highest-order term, enabling a tractable operator-theoretic analysis. Within this framework, we prove two main theorems that provide sufficient conditions for periodic solutions in both autonomous and nonautonomous cases. These results generalize the known theory for second-order equations to arbitrary higher-order systems and highlight a connection to the Hamilton–Jacobi theory, offering new insights into the underlying variational structure. Finally, numerical examples validate our theoretical results by confirming the periodic solutions predicted by the theory and demonstrating the approach’s practical applicability. Full article

In this paper, the problem of pitching attitude finite-horizon optimization for aircraft is posed with system uncertainties, external disturbances, and input constraints. First, a neural network (NN) and a nonlinear disturbance observer (NDO) are employed to estimate the value of system uncertainties and external disturbances. Taking input constraints into account, an auxiliary system is designed to compensate for the constrained input. Subsequently, the backstepping control containing NN and NDO is used to ensure the stability of systems and suppress the adverse effects caused by the system uncertainties and external disturbances. In order to avoid the derivation operation in the process of backstepping, a dynamic surface control (DSC) technique is utilized. Simultaneously, the estimations of the NN and NDO are applied to derive the backstepping control law. For the purpose of achieving finite-horizon optimization for pitching attitude control, an adaptive method termed adaptive dynamic programming (ADP) with a single NN-termed critic is applied to obtain the optimal control. Time-varying feature functions are applied to construct the critic NN in order to approximate the value function in the Hamilton–Jacobi–Bellman (HJB) equation. Furthermore, a supplementary term is added to the weight update law to minimize the terminal constraint. Lyapunov stability theory is used to prove that the signals in the control system are uniformly ultimately bounded (UUB). Finally, simulation results illustrate the effectiveness of the proposed finite-horizon optimal attitude control method. Full article

The shipping industry, a cornerstone of global trade, faces emissions reduction challenges amid tightening environmental policies. Blockchain technology, leveraging distributed symmetric architectures, enhances supply chain transparency by reducing information asymmetry, yet its dynamic interplay with emissions strategies remains underexplored. This study employs symmetry-driven differential game theory to model four blockchain scenarios in port-shipping networks: no blockchain (N), port-led (PB), shipping company-led (CB), and a joint platform (FB). By solving Hamilton–Jacobi–Bellman equations, we derive optimal emissions reduction efforts, green investments, and blockchain strategies under symmetric and asymmetric decision-making frameworks. Results show blockchain adoption improves emissions reduction and service quality under cost thresholds, with port-led systems maximizing low-cost profits and shipping firms gaining asymmetrically in high-freight contexts. Joint platforms achieve symmetry in profit distribution through fee-trust synergy, enabling win–win outcomes. Integrating graph-theoretic principles, we have designed dynamic state equations for emissions and service levels, segmenting shippers by low-carbon preferences. This work bridges dynamic emissions strategies with blockchain’s network symmetry, fostering economic–environmental synergies to advance sustainable maritime supply chains. Full article

In recent years, high-frequency trading has become increasingly popular in financial markets, making the dynamic modeling of the limit book and the optimization of market maker strategies become key topics. However, existing studies often lacked detailed descriptions of order books and failed to fully characterize the optimal decisions of market makers in complex market environments, especially in China’s A-share market. Based on Markov queue theory, this paper proposes the dynamic model of the limit order and the optimal strategy of the market maker. The model uses a state transition probability matrix to refine the market diffusion state, order generation, and trading process and incorporates indicators such as optimal quote deviation and restricted order trading probability. Then, the optimal control model is constructed and the reference strategy is derived using the Hamilton–Jacobi–Bellman (HJB) equation. Then, the key parameters are estimated using the high-frequency data of Ping An Bank for a single trading day. In the empirical aspect, the six-month high-frequency trading data of 114 representative stocks in different market states such as the bull market and bear market in China’s A-share market were selected for strategy verification. The results showed that the proposed strategy had robust returns and stable profits in the bull market and that frequent capture of market fluctuations in the bear market can earn relatively high returns while maintaining 50% of the order coverage rate and 66% of the stable order winning rate. Our study used Markov queuing theory to describe the state and price dynamics of the limit order book in detail and used optimization methods to construct and solve the optimal market maker strategy. The empirical aspect broadens the empirical scope of market maker strategies in the Chinese market and studies the stability and effectiveness of market makers in different market states. Full article

Optimal control theory aims to find an optimal protocol to steer a system between assigned boundary conditions while minimizing a given cost functional in finite time. Equations arising from these types of problems are often non-linear and difficult to solve numerically. In this article, we describe numerical methods of integration for two partial differential equations that commonly arise in optimal control theory: the Fokker–Planck equation driven by a mechanical potential for which we use the Girsanov theorem; and the Hamilton–Jacobi–Bellman, or dynamic programming, equation for which we find the gradient of its solution using the Bismut–Elworthy–Li formula. The computation of the gradient is necessary to specify the optimal protocol. Finally, we give an example application of the numerical techniques to solving an optimal control problem without spacial discretization using machine learning. Full article

This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox–Ingersoll–Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton–Jacobi–Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures. Full article

The covariant Hamilton–Jacobi formulation of electrodynamics is systematically derived from the first-order (Palatini-like) Lagrangian. This derivation utilizes the De Donder–Weyl covariant Hamiltonian formalism with constraints incroporating generalized Dirac brackets of forms and the associated polysymplectic reduction, which ensure manifest covariance and consistency with the field dynamics. It is also demonstrated that the canonical Hamilton–Jacobi equation in variational derivatives and the Gauss law constraint are derived from the covariant De Donder–Weyl Hamilton–Jacobi formulation after space + time decomposition. Full article

We present a finite difference method working on sparse grids to solve higher dimensional heterogeneous agent models. If one wants to solve the arising Hamilton–Jacobi–Bellman equation on a standard full grid, one faces the problem that the number of grid points grows exponentially with the number of dimensions. Discretizations on sparse grids only involve $\mathcal{O}\left(N{(logN)}^{d-1}\right)$ degrees of freedom in comparison to the $\mathcal{O}\left(N^d\right)$ degrees of freedom of conventional methods, where N denotes the number of grid points in one coordinate direction and d is the dimension of the problem. While one can show convergence for the used finite difference method on full grids by using the theory introduced by Barles and Souganidis, we explain why one cannot simply use their results for sparse grids. Our numerical studies show that our method converges to the full grid solution for a two-dimensional model. We analyze the convergence behavior for higher dimensional models and experiment with different sparse grid adaptivity types. Full article

In this study, we use the dynamic programming method introduced by Mirică (2004) to solve the well-known war game of attrition and attack as formulated by Isaacs (1965). By using this modern approach, we extend the classical framework to explore optimal strategies within the differential game setting, offering a complete, comprehensive and theoretically robust solution. Additionally, the study identifies and analyzes feedback strategies, which represent a significant advancement over other strategy types in game theory. These strategies dynamically adapt to the evolving state of the system, providing more robust solutions for real-time decision-making in conflict scenarios. This novel contribution enhances the application of game theory, particularly in the context of warfare models, and illustrates the practical advantages of incorporating feedback mechanisms into strategic decision-making. The admissible feedback strategies and the corresponding value function are constructed through a refined application of Cauchy’s Method of characteristics for stratified Hamilton–Jacobi equations. Their optimality is proved using a suitable Elementary Verification Theorem for the associated value function as an argument for sufficient optimality conditions. Full article

Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs’ solutions, where we denoise the loss-by-moment calculations of the solutions. The integral pattern of the models transforms the source of the essential learning error from the discrepancy between the modified Hamiltonian and the true Hamiltonian in the classical HNN models into that between the integrals and their quadrature approximations. This transforms the challenging task of deriving the relation between the modified and the true Hamiltonians from the (stochastic) Hamilton–Jacobi PDEs, into the one that only requires invoking results from the numerical quadrature theory. Meanwhile, denoising via moments calculations gives a simpler data fitting method than, e.g., via probability density fitting, which may imply better generalization ability in certain circumstances. Numerical experiments validate the proposed learning strategy on several concrete Hamiltonian systems. The experimental results show that both the learned Hamiltonian function and the predicted solution of our quadrature-based model are more accurate than that of the corrected symplectic HNN method on a harmonic oscillator, and the three-point Gaussian quadrature-based model produces higher accuracy in long-time prediction than the Kramers–Moyal method and the numerics-informed likelihood method on the stochastic Kubo oscillator as well as other two stochastic systems with non-polynomial Hamiltonian functions. Moreover, the Hamiltonian learning error ${\epsilon }_{\mathcal{H}}$ arising from the Gaussian quadrature-based model is lower than that from Simpson’s quadrature-based model. These demonstrate the superiority of our approach in learning accuracy and long-time prediction ability compared to certain existing methods and exhibit its potential to improve learning accuracy via applying precise quadrature formulae. Full article

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory. Full article

In our study, we investigate reinsurance issues and optimal investment related to derivatives trading for a mean-variance insurer, employing game theory. Our primary objective is to identify strategies that are time-consistent. In particular, the insurer has the flexibility to purchase insurance in proportion to its needs, explore new business, and engage in capital market investments. This is under the assumption that insurance companies’surplus capital adheres to the classical Cramér-Lundberg model. The capital market is made up of risk-free bonds, equities, and derivatives, with pricing dependent on the underlying stock’s basic price and volatility. To obtain the most profitable expressions and functions for the associated investment strategies and time guarantees, we solve a system of expanded Hamilton–Jacobi–Bellman equations. In addition, we delve into scenarios involving optimal investment and reinsurance issues with no derivatives trading. In the end, we present a few numerical instances to display our findings, demonstrating that the efficient frontier in the case of derivative trading surpasses that in scenarios where derivative trading is absent. Full article

The reachable set of controlled dynamical systems is the set of all reachable states from an initial condition over a certain time horizon, subject to operational constraints and exogenous disturbances. In astrodynamics, rapid approximation of reachable sets is invaluable for trajectory planning, collision avoidance, and ensuring safe and optimal performance in complex dynamics. Leveraging the connection between minimum-time trajectories and the boundary of reachable sets, we propose a sampling-based method for rapid and efficient approximation of reachable sets for finite- and low-thrust spacecraft. The proposed method combines a minimum-time multi-stage indirect formulation with the celebrated primer vector theory. Reachable sets are generated under two-body and circular restricted three-body (CR3B) dynamics. For the two-body dynamics, reachable sets are generated for (1) the heliocentric phase of a benchmark Earth-to-Mars problem, (2) two scenarios with uncertainties in the initial position and velocity of the spacecraft at the time of departure from Earth, and (3) a scenario with a bounded single impulse at the time of departure from Earth. For the CR3B dynamics, several cislunar applications are considered, including L1 Halo orbit, L2 Halo orbit, and Lunar Gateway 9:2 NRHO. The results indicate that low-thrust spacecraft reachable sets coincide with invariant manifolds existing in multi-body dynamical environments. The proposed method serves as a valuable tool for qualitatively analyzing the evolution of reachable sets under complex dynamics, which would otherwise be either incoherent with existing grid-based reachability approaches or computationally intractable with a complete Hamilton–Jacobi–Bellman method. Full article

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton’s equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton’s equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton’s and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton’s, variational, and Schrödinger equations are described. Employing the Hénon–Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton’s equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes. Full article

In this paper, we will review two analytical approaches to the construction of non-homogeneous Baryonic condensates in the low-energy limit of QCD in (3+1) dimensions. In both cases, the minimal coupling with the Maxwell $U\left(1\right)$ gauge field can be taken explicitly into account. The first approach (which is related to the generalization of the usual spherical hedgehog ansatz to situations without spherical symmetry at a finite Baryon density) allows for the construction of ordered arrays of Baryonic tubes and layers. When the minimal coupling of the Pions to the $U\left(1\right)$ Maxwell gauge field is taken into account, one can show that the electromagnetic field generated by these inhomogeneous Baryonic condensates is of a force-free type (in which the electric and magnetic components have the same size). Thus, it is natural to wonder whether it is also possible to analytically describe magnetized hadronic condensates (namely, Hadronic distributions generating only a magnetic field). The idea of the second approach is to construct a novel BPS bound in the low-energy limit of QCD using the theory of the Hamilton–Jacobi equation. Such an approach allows us to derive a new topological bound which (unlike the usual one in the Skyrme model in terms of the Baryonic charge) can actually be saturated. The nicest example of this phenomenon is a BPS magnetized Baryonic layer. However, the topological charge appearing naturally in the BPS bound is a non-linear function of the Baryonic charge. Such an approach allows us to derive important physical quantities (which would be very difficult to compute with other methods), such as how much one should increase the magnetic flux in order to increase the Baryonic charge by one unit. The novel results of this work include an analysis of the extension of the Hamilton–Jacobi approach to the case in which Skyrme coupling is not negligible. We also discuss some relevant properties of the Dirac operator for quarks coupled to magnetized BPS layers. Full article