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Using phase-field simulations, we investigate the bulk coarsening dynamics of ternary polymer solutions undergoing a glass transition for two models of phase separation: diffusion only and with hydrodynamics. The glass transition is incorporated in both models by imposing mobility and viscosity contrasts between the polymer-rich and polymer-poor phases of the evolving microstructure. For microstructures composed of polymer-poor clusters in a polymer-rich matrix, the mobility and viscosity contrasts significantly hinder coarsening, effectively leading to structural arrest. For microstructures composed of polymer-rich clusters in a polymer-poor matrix, the mobility and viscosity contrasts do not impede domain growth; rather, they change the transient concentration of the polymer-rich phase, altering the shape of the discrete domains. This effect introduces several complexities to the coarsening process, including percolation inversion of the polymer-rich and polymer-poor phases—a phenomenon normally attributed to viscoelastic phase separation. [ABSTRACT FROM AUTHOR]

In this paper, we study a kind of optimal control problem for forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type via the dynamic programming principle (DPP for short) motivated by studying the infinite dimensional Hamilton–Jacobi–Bellman (HJB for short) equation derived from the decoupling field of the FBSDEs posed by Carmona and Delarue (2015) [28]. At the beginning, the value function is defined by the solution to the controlled BSDE alluded to earlier. On one hand, we can prove the value function is deterministic function with respect to the initial random variable; On the other hand, we can show that the value function is law-invariant , i.e., depending on only via its distribution by virtue of BSDE property. Afterward, utilizing the notion of differentiability with respect to probability measures introduced by P.L. Lions [50] , we are able to establish a DPP for the value function in the Wasserstein space of probability measures based on the application of BSDE approach, particularly, employing the notion of stochastic backward semigroups associated with stochastic optimal control problems and Itô formula along a flow property of the conditional law of the controlled forward state process. We prove that the value function is the unique viscosity solutions of the associated generalized HJB equations in some separable Hilbert space. Finally, as an application, we formulate an optimal control problem for linear stochastic differential equations with quadratic cost functionals of McKean-Vlasov type under nonlinear expectation, g -expectation introduced by Peng [43] and derive the optimal feedback control explicitly by means of several groups of Riccati equations. [ABSTRACT FROM AUTHOR]

This paper studies the optimal investment and benefit payment strategies for target benefit (TB) pension plans. The pension fund receives contributions from active members and pays benefits to retirees. Meanwhile, the accumulated wealth is invested in financial market consisting of one risk-free asset and one risky asset, in which the risk-free interest rate is described by the Vasicek model. The general hyperbolic absolute risk aversion (HARA) utility is adopted to describe pension fund managers' risk preferences. Using the dynamic programming approach, we construct the Hamilton–Jacobi–Bellman (HJB) equation and obtain explicit expressions for optimal investment and benefit payment strategies using the Legendre transform-dual technique. Finally, numerical analysis is presented to illustrate the sensitivity of the optimal strategies to model parameters. [ABSTRACT FROM AUTHOR]

The collocation method is one of the most powerful and effective techniques to solve nonlinear differential equations. This method reduces the original problem to a system of algebraic equations. Awareness of the nature of the problem and the analytical behavior of the solution has an important influence on the choice of type and number of basis functions required by the collocation method for solving the different classes of differential equations. In general, the polynomial basis functions such as the Legendre polynomials are not always an appropriate choice to approximate the functions that can be expanded based on monomial basis functions with real powers. In such cases, a combination of polynomial basis functions with some maps to replace the natural powers with arbitrary real powers can be a suitable alternative. This paper presents a collocation method based on the shifted fractional-order Legendre functions to solve the linear and nonlinear Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems. The convergence analysis of the proposed method has also been investigated. Moreover, the method is used to solve some practical problems with different solution structures such as resource extraction and Merton's portfolio selection models. Numerical results for four different examples indicated that the collocation method based on the fractional-order Legendre functions are provided. Due to the non differentiability of the solution of HJB PDEs at x = 0 for resource extraction and Merton's portfolio selection models, the Legendre collocation method cannot produce solutions with the desired accuracy. In fact, it is observed that the collocation method based on the fractional-order Legendre functions can be more efficient than the standard Legendre collocation method. Also, the sample paths of the control and state processes are obtained together with the estimate given by the Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

The present article deals with the application of refined plate theory and surface effects to investigate the nonlocal free vibration behavior of a functionally graded nano-plate composed of two piezoelectric face sheets resting on Orthotropic Visco-Pasternak. Surface and nonlocal effects are applied using the surface piezoelectricity and stress-strain gradient theories. Surface effects, refined quasi-3D higher-order theory, neutral surface position and in-plane mechanical preloads are considered to study piezoelectric sandwich nano-plate with a functionally graded core in this study for the first time, simultaneously. The size-dependent governing equations are derived based on 2D and 3D refined theory using Hamilton's principle and different boundary conditions are studied and investigated according to the semi-analytical solution method. Comparing the results of the vibration behavior of 2D and quasi-3D models, local and non-local models, and also, examining different foundations simultaneously can be said as one of the innovations of the current research. Finally, a parametric study is presented to examine the effects of different parameters such as surface effects, small scale parameters, mechanical preloads, thickness stretching effects, neutral surface position, non-homogeneous coefficient, applied voltage, different foundation constants, boundary conditions, geometrical ratios and different theories in details. It is indicated that the inclusion of surface effects and thickness stretching effects leads to a significant influence on the vibration behavior of three-layered nano-scale plates. The presented results in this article may provide useful guidance for the design and development of the next generation of nano-devices. [ABSTRACT FROM AUTHOR]

We deal, for the classical N -body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unified manner. Our approach is based on the minimization of a renormalized Lagrangian action on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions. [ABSTRACT FROM AUTHOR]

In this article, a distributed optimal control approach is proposed for a class of affine nonlinear multi‐agent systems (MASs) with unknown nonlinear dynamics. The game theory is used to formulate the distributed optimal control problem into a differential graphical game problem with synchronized updates of all nodes. A data‐based integral reinforcement learning (IRL) algorithm is used to learn the solution of the coupled Hamilton–Jacobi (HJ) equation without prior knowledge of the drift dynamics, and the actor‐critic neural networks (A‐C NNs) are used to approximate the control law and the cost function, respectively. To update the parameters synchronously, the gradient descent algorithm is used to design the weight update laws of the A‐C NNs. Combining the IRL and the A‐C NNs, a distributed consensus optimal control method is designed. By using the Lyapunov stability theory, the developed optimal control method can show that all signals in the considered system are uniformly ultimately bounded (UUB), and the systems can achieve Nash equilibrium when all agents update their controllers simultaneously. Finally, simulation results are given to illustrate the effectiveness of the developed optimal control approach. [ABSTRACT FROM AUTHOR]

By exploiting the contact Hamiltonian dynamics (T* M × ℝ, Φt) around the Aubry set of contact Hamiltonian systems, we provide a relation among the Mather set, the Φt-recurrent set, the strongly static set, the Aubry set, the Mañé set, and the Φt-non-wandering set. Moreover, we consider the strongly static set, as a new flow-invariant set between the Mather set and the Aubry set in the strictly increasing case. We show that this set plays an essential role in the representation of certain minimal forward weak Kolmogorov-Arnold-Moser (KAM) solutions and the existence of transitive orbits around the Aubry set. [ABSTRACT FROM AUTHOR]

ABSTRACT This article investigates a multiplayer nonzero‐sum differential game (MNDG) with switching topology and input delay. The Hamilton–Jacobi–Isaacs (HJI) equation is used to derive the optimal control strategies. To solve the issue of switching topology, the control strategies are decoupled with the communication topology. Then, the stability of the system can be achieved regardless of changes in communication topology with the designed control strategies. This does not require a restriction on the dwell time, making the system stability conditions more relaxed. The Lyapunov–Krasovskii functional (LKF) is constructed to prove the stability of the system in the presence of time‐varying input delay. To cope with the real‐world situation, the input delay is considered to be disturbed by a time‐varying perturbation in this article. With these treatments, the system is more in line with real‐world applications. Simulation examples are given to validate the efficiency of the presented methods. [ABSTRACT FROM AUTHOR]

For a continuous Hamiltonian H : (x , p , u) ∈ T ⁎ R n × R → R , we consider the asymptotic behavior of associated Hamilton–Jacobi equations with state-constraint { H (x , D u , λ u) ≤ C λ , x ∈ Ω λ ⊂ R n , H (x , D u , λ u) ≥ C λ , x ∈ Ω ‾ λ ⊂ R n , as λ → 0 +. When H satisfies certain convex, coercive and monotone conditions, the domain Ω λ : = (1 + r (λ)) Ω keeps bounded, star-shaped for all λ > 0 with lim λ → 0 + ⁡ r (λ) = 0 , and lim λ → 0 + ⁡ C λ = c (H) equals the ergodic constant of H (⋅ , ⋅ , 0) , we prove the convergence of solutions u λ to a specific solution of the critical equation { H (x , D u , 0) ≤ c (H) , x ∈ Ω ⊂ R n , H (x , D u , 0) ≥ c (H) , x ∈ Ω ‾ ⊂ R n. We also discuss the generalization of such a convergence for equations with more general C λ and Ω λ. [ABSTRACT FROM AUTHOR]

In this article, the multiplayer hierarchical decision‐making problem for discrete‐time nonlinear networks of service is studied from the perspective of Nash–Stackelberg–Nash games. A novel two‐level value iteration adaptive dynamic programming algorithm is developed to solve the coupled nonlinear Hamilton–Jacobi–Bellman equations associated with the game problem, which neither requires the system drift dynamics to be known as a priori nor requests the initial strategies to be admissible. Moreover, both the value function sequences and the strategy sequences generated by the algorithm converge to their theoretical optimality. An implementation framework for the algorithm is constructed by leveraging the regularized least‐squares method, and a convergence criterion that guarantees the admissibility of the approximated strategies is also proposed. The effectiveness of the proposed algorithm and implementation framework are finally demonstrated through two numerical simulation examples. [ABSTRACT FROM AUTHOR]

In the paper we prove the convergence of viscosity solutions u λ as λ → 0 + for the parametrized degenerate viscous Hamilton–Jacobi equation H ⁢ (x , d x ⁢ u , λ ⁢ u) = α ⁢ (x) ⁢ Δ ⁢ u , α ⁢ (x) ≥ 0 , x ∈ 핋 n under suitable convex and monotonic conditions on H : T * ⁢ M × ℝ → ℝ . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation H ⁢ (x , d x ⁢ u , 0) = α ⁢ (x) ⁢ Δ ⁢ u. [ABSTRACT FROM AUTHOR]

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$ , with $s\in (1/2,\,1)$ , and a coercive gradient term with subcritical power $0. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0 , involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1. [ABSTRACT FROM AUTHOR]

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing. [ABSTRACT FROM AUTHOR]

We consider the Hamilton-Jacobi equation H (x , D u) + λ (x) u = c , x ∈ M , where M is a connected, closed and smooth Riemannian manifold. The functions H (x , p) and λ (x) are continuous. H (x , p) is convex, coercive with respect to p , and λ (x) changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao [11] under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem. To the best of our knowledge, it is the first detailed description of the large time behavior of the HJ equations with non-monotone dependence on the unknown function. [ABSTRACT FROM AUTHOR]

This article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy. [ABSTRACT FROM AUTHOR]

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

This paper investigates a parameter-free H∞ differential game approach for nonlinear active vehicle suspensions. The study accounts for the geometric nonlinearity of the half-car active suspension and the cubic nonlinearity of the damping elements. The nonlinear H∞ control problem is reformulated as a zero-sum game between two players, leading to the establishment of the Hamilton–Jacobi–Isaacs (HJI) equation with a Nash equilibrium solution. To minimize reliance on model parameters during the solution process, an actor–critic framework employing neural networks is utilized to approximate the control policy and value function. An off-policy reinforcement learning method is implemented to iteratively solve the HJI equation. In this approach, the disturbance policy is derived directly from the value function, requiring only a limited amount of driving data to approximate the HJI equation's solution. The primary innovation of this method lies in its capacity to effectively address system nonlinearities without the need for model parameters, making it particularly advantageous for practical engineering applications. Numerical simulations confirm the method's effectiveness and applicable range. The off-policy reinforcement learning approach ensures the safety of the design process. For low-frequency road disturbances, the designed H∞ control policy enhances both ride comfort and stability. [ABSTRACT FROM AUTHOR]

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients. [ABSTRACT FROM AUTHOR]

The given text discusses different approaches to finding equations of motion for physical systems, such as the Newtonian, Lagrangian, and Hamiltonian approaches. It presents a supplementary tool using the total differential of the mechanical energy function to obtain equations of motion for certain systems. The text also explores the relationship between this approach and Hamilton's equations of motion. Additionally, the text provides concise and factual information about different types of motion and their corresponding equations, making it a useful resource for library patrons conducting research on these topics. [Extracted from the article]

In this article, we study the optimal control of stochastic differential equations with random impulses. We optimize the performance index and add the influence of random impulses to the performance index with a random compensation function. Using the idea of stochastic analysis and dynamic programming principle, a new Hamilton–Jacobi–Bellman (HJB) equation is obtained, and the existence and uniqueness of its viscosity solution are proved. [ABSTRACT FROM AUTHOR]

In this article, a distributed controller based on adaptive dynamic programming is proposed to solve the minimum loss problem of flywheel energy storage systems (FESS). We first formulate a performance function aiming to reduce total losses of FESS in power distribution applications. Then we use the Hamilton–Jacobi–Bellman (HJB) equation to solve this optimal control problem. The solution of the HJB equation is approximated by neural networks. To achieve distributed control, we estimate the global variables in the HJB equation by using the dynamic average consensus algorithm. A barrier Lyapunov function and a saturation function are introduced to handle the issue of state and input constraints, respectively. Then the stability of the system is proved through the Lyapunov stability analysis. Finally the effectiveness of the proposed strategy is verified by simulations. Simulation results show that FESS can track the power command while minimizing total power losses by interacting with neighbors. The proposed algorithm leads to a loss reduction of 1.13%$$ 1.13\% $$ compared to the equal power distribution strategy. [ABSTRACT FROM AUTHOR]

In this manuscript, given a metric tensor on the probability simplex, we define differential operators on the Wasserstein space of probability measures on a graph. This allows us to propose a notion of graph individual noise operator and investigate Hamilton–Jacobi equations on this Wasserstein space. We prove comparison principles for viscosity solutions of such Hamilton–Jacobi equations and show existence of viscosity solutions by Perron's method. We also discuss a model optimal control problem and show that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation. [ABSTRACT FROM AUTHOR]

In this paper we study Monge solutions to stationary Hamilton–Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result. [ABSTRACT FROM AUTHOR]

In this article, a finite‐horizon adaptive Pontryagin's maximum principle is presented for nonlinear systems with state inequality constraints. Concurrent learning (CL) technique is adopted to identify the unknown parameters of the dynamic systems. Based on the identification model, a novel adaptive iterative algorithm under the Pontryagin's framework is introduced to learn the finite‐horizon optimal control solution. Convergence analysis of the algorithm is provided by showing that the cost function sequence is monotonically decreasing. Furthermore, we extend the adaptive iterative algorithm to time‐varying nonlinear systems. The new algorithm overcomes the technical obstacles of the existing adaptive/approximate dynamic programming (ADP) approaches to deal with the time‐varying characteristic of Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE), especially when state constraints exist. Simulation examples are carried out to validate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the link between the solution to an Hamilton-Jacobi (HJ) equation and the solution to a Scalar Conservation Law (SCL) on a special network. When the equations are posed on the real axis, it is well known that the space derivative of the solution to the Hamilton-Jacobi equation is the solution to the corresponding scalar conservation law. On networks, the situation is more complicated and we show that this result still holds true in the convex case on a 1:1 junction. The correspondence between solutions to HJ equations and SCL on a 1:1 junction is done showing the convergence of associated numerical schemes. A second direct proof using semi-algebraic approximations is also given.Here a 1:1 junction is a simple network composed of two edges and one vertex. In the case of three edges or more, we show that the associated HJ germ is not a $ L^1 $-dissipative germ, while it is the case for only two edges.As an important byproduct of our numerical approach, we get a new result on the convergence of numerical schemes for scalar conservation laws on a junction. For a general desired flux condition which is discretized, we show that the numerical solution with the general flux condition converges to the solution of a SCL problem with an effective flux condition at the junction. Up to our knowledge, in previous works the effective condition was directly implemented in the numerical scheme. In general the effective flux condition differs from the desired one, and is its relaxation, which is very natural from the point of view of Hamilton-Jacobi equations. Here for SCL, this effective flux condition is encoded in a germ that we characterize at the junction. [ABSTRACT FROM AUTHOR]

This research is dedicated to developing a min–max robust control strategy for a dynamic game involving pursuers, evaders, and defenders in a multiple-missile scenario. The approach employs neural dynamic programming, utilizing multiple continuous differential neural networks (DNNs). The competitive controller devised addresses the robust optimization of a joint cost function that relies on the trajectories of the pursuer–evader–defender system, accommodating an uncertain mathematical model while adhering to control restrictions. The dynamic programming min–max formulation facilitates robust control by accounting for bounded modeling uncertainties and external disturbances for each game component. The value function of the Hamilton–Jacobi–Bellman (HJB) equation is approximated by a DNN, enabling the estimation of the closed-loop formulation for the joint dynamic game with state restrictions. The controller’s design is grounded in estimating the state trajectory under the worst possible uncertainties and perturbations, providing a robustness factor through the robust neural controller. The learning law class for the time-varying weights in the DNN is generated by studying the HJB partial differential equation for the missile motion for each player in the dynamic game. The controller incorporates the solution of the obtained learning laws and a time-varying Riccati equation, offering an online solution to the control implementation. A recurrent algorithm, based on the Kiefer–Wolfowitz method, adjusts the initial conditions for the weights to satisfy the final condition of the given cost function for the dynamic game. A numerical example is presented to validate the proposed robust control methodology, confirming the optimization solution based on the DNN approximation for Bellman’s value function. [ABSTRACT FROM AUTHOR]

This paper employs an integral reinforcement learning (IRL) method to investigate the optimal tracking control problem (OTCP) for nonlinear nonzero-sum (NZS) differential game systems with unknown drift dynamics. Unlike existing methods, which can only bound the tracking error, the proposed approach ensures that the tracking error asymptotically converges to zero. This study begins by constructing an augmented system using the tracking error and reference signal, transforming the original OTCP into solving the coupled Hamilton–Jacobi (HJ) equation of the augmented system. Because the HJ equation contains unknown drift dynamics and cannot be directly solved, the IRL method is utilized to convert the HJ equation into an equivalent equation without unknown drift dynamics. To solve this equation, a critic neural network (NN) is employed to approximate the complex value function based on the tracking error and reference information data. For the unknown NN weights, the least squares (LS) method is used to design an estimation law, and the convergence of the weight estimation error is subsequently proven. The approximate solution of optimal control converges to the Nash equilibrium, and the tracking error asymptotically converges to zero in the closed system. Finally, we validate the effectiveness of the proposed method in this paper based on MATLAB using the ode45 method and least squares method to execute Algorithm 2. [ABSTRACT FROM AUTHOR]

This article investigates the issue of data‐based distributed consensus optimal control for a class of affine nonlinear multi‐agent systems (MASs) under switching topology with external disturbances. With the help of the game theory, the distributed adaptive optimal consensus control issue can be formulated into a zero‐sum (ZM) game problem. In control design, a data‐based integral reinforcement learning (IRL) algorithm is used to solve the coupled Hamilton–Jacobi–Isaac (HJI) equation with unknown drift dynamics. Meanwhile, to relax the persistent excitation (PE) condition in the traditional optimal control design, the experience replay (ER) technique is introduced. Combining IRL algorithm and single critic neural network (NN), a distributed adaptive optimal consensus control approach is designed. The stability of the closed‐loop system is proved by combining the Lyapunov stability theory and the average dwell time method. Finally, a simulation example is given to illustrate the effectiveness of the developed optimal consensus control approach. [ABSTRACT FROM AUTHOR]

Given a smooth potential W : T n → R on the torus, the Quantum Guerra–Morato action functional is given by I (ψ) = ∫ ( D v D v * 2 (x) − W (x)) a (x) 2 d x , where ψ is described by ψ = a e i u ℏ , u = v + v * 2 , a = e v * − v 2 ℏ , v, v* are real functions, ∫a2(x)dx = 1, and D is the derivative on x ∈ Tn. It is natural to consider the constraint div(a2Du) = 0, which means flux zero. The a and u obtained from a critical solution (under variations τ) for such action functional, fulfilling such constraints, satisfy the Hamilton-Jacobi equation with a quantum potential. Denote ′ = d d τ . We show that the expression for the second variation of a critical solution is given by ∫a2D[v′] D[(v*)′] dx. Introducing the constraint ∫a2Du dx = V, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation. [ABSTRACT FROM AUTHOR]

This paper studies a maximal L q -regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a L p duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold q = d (γ − 1) / γ , d being the dimension of the ambient space and γ > 1 the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range q > 1. [ABSTRACT FROM AUTHOR]

It is clear that there is a relationship between time and space in nanostructures that are always attacked by waves with wavelengths within the intrinsic characteristic length of the nanostructure. Also, the viscoelastic behavior of nanostructures is affected by this relationship between nonlocal time and nonlocal space. It is worth noting that the conventional tempo-spatially decoupled nonlocal viscoelasticity does not give accurate results regarding the viscoelastic response of these structures. Hence, herein and for the first time, a novel fractional nonlocal time–space strain gradient viscoelasticity is implemented to be able to give a correct answer in addressing the dispersion of elastic waves inside functionally graded material (FGM) viscoelastic nanoplates. The constructive equations of this theory are based on the theory of nonlocal strain gradient elasticity. The properties of the FGM studied in this research are obtained by the power-law model, and the kinematic relations are extracted based on the refined higher-order plate theory. Subsequently, motion equations are written using Hamilton's principle and solved analytically. The results of this work reveal that an increase in the nonlocal parameter can reduce the loss factor in a remarkable way due to the coupling between the nonlocalities in the spatial and temporal domains. [ABSTRACT FROM AUTHOR]

This research aims to study the non-linear parametric vibration of laminated composite shallow (LCS) shells with the optimal fiber angles exposed to external and parametric excitations, including primary and 1:2 internal resonances. In this regard, optimal fiber angles are found with implementations of the P-T method for the objective functions and utilization of the particle swarm optimization (PSO). Also, the non-linear model of the shallow shells is established based on the stress function and the first-order shear deformation theory (FSDT). According to FSDT, Hooke's law, von-Kármán equation, Hamilton's principle, and Galerkin method, two-degree-of-freedom non-linear ordinary differential governing equations are discretized. [ABSTRACT FROM AUTHOR]

The vibration responses of a sandwich beam with partially covered by shear thickening fluid (STF) layer under an impact load are investigated. The nonlinear governing equations of the flexural vibration are derived based on extended Hamilton's principle and are solved by the finite difference method. The model is then validated and used to develop a complete parametric study of partially covered beams with the STF-filled core to properly design and place the STF patch. It is found that, for the first vibration mode, maximum damping, and the smallest change in the natural frequency are achieved when the coverage length of the partial STF patch exceeds 50% and the center of the patch is positioned at 56.25% from the left edge. For the second vibration mode, the coverage length is 37.5% and the center of the patch is located at 75% from the left edge of the beam. Additionally, it has been observed that maintaining a thickness ratio of 0.75 between the constraining layer and the base beam leads to increased damping, while simultaneously minimizing alterations in the natural frequency of the original beam. The results can be used for the structural design of sandwich beams partially covered by STF. [ABSTRACT FROM AUTHOR]

This study explores the nonlinear free vibrations of anisogrid lattice cylindrical panels under clamped circumstances for the first time. The lattice panel is made of repeating circumferential hexagonal unit cells with helical and circumferential isotropic ribs. The panel's stiffness and mass are calculated using a continuous model based on orthotropic deep panel theory. The strain–displacement relationships are based on von-Kármán's geometrically nonlinear theory. Using the Hamilton principle, nonlinear two-dimensional (2D) motion equations are extracted in their strong form. A 2D collocation approach based on the Sinc functions with double exponential (DE) transformation with a high convergence rate is used to solve the governing motion equations. The time variable is then eliminated from the equations using the weighted residual Galerkin procedure. Finally, the system's nonlinear natural frequencies are computed using an iterative algorithm based on the bilateral displacement control. The influence of unit cell features, such as the angle of helical ribs, the number and thickness of ribs, and the panel dimensions, is examined in detail on the panel's linear and nonlinear vibration behavior. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

Under arbitrary boundary conditions, a Jacobi–Ritz time domain semi-analytical method is proposed to study the vibration behaviors of uniform and stepped conical shells. Based on the idea of the differential element method and the first-order shear deformation theory (FSDT), the vibration analysis models of uniform and stepped conical shell structures are established. The axial and circumferential displacement tolerance functions are expressed by the Jacobi polynomial and the Fourier series. The complex boundary conditions are simulated by using artificial spring technology. In light of the Ritz method and the Newmark-β integral method, the free and forced vibration results of the structure can be figured out in the time domain. The numerical results demonstrate that the proposed method has excellent accuracy and remarkable dependability when compared to the finite element method (FEM). Numerical examples are used to examine the forced and free vibration behaviors under various boundary conditions, semi-vertex angles, thicknesses, and load characteristics. [ABSTRACT FROM AUTHOR]

This paper is the first to develop a meshless thermodynamic model with circumferentially variable dimensions to analyze the thermal-vibration mechanisms of typical functionally graded (FG) panel-type structures, mainly including conical, cylindrical, and annular panels. The longitudinal variation of the panel-type structure’s circumferential dimensions is taken into account according to specific rules. The Thin Plate Inverse Mapping (TPIM) meshless shape functions are introduced to characterize the displacement components associated with each substructure. Thermodynamic equations are formulated based on FSDT thermo-elastic theory as well as Hamilton’s principle. Rectangular and exponential pulse loads are taken into consideration for forced vibration analysis in this formulation. Numerical convergence and comparative cases are provided. Solutions from the literature and finite element simulation results are utilized for comparison, revealing that the percentage differences observed are all less than 0.025%. This confirms the reliability and accuracy of the developed analysis model. In conclusion, a detailed investigation is conducted on the influences of geometrical dimensions, boundary conditions, as well as thermal and external loads on the thermodynamic behaviors of FG panel-type structures with circumferentially variable dimensions. [ABSTRACT FROM AUTHOR]

AbstractFoldability capacity is now introduced as a novel nanofiller reinforcement production procedure using some operation to control the mechanical, thermal and electrical properties in the sport equipment. Application of this type of nanofillers in the curved structures like pole vault shell leads to a novel engineering and sport shell shape structures. This article is organized to suggest a vibration-based formulation for analysis of folded reinforced curved shell sport structure subjected to thermal and mechanical loading. Using computation of kinetic, strain and external energies, one can arrive the motion’s equations using the minimization of total energy and Hamilton’s principle. Using solution of the motion’s equations through an analytical approach, the parametric analysis is presented. The verified test is presented for confirmation of the solution and trend of results. [ABSTRACT FROM AUTHOR]

We prove homogenization for viscous Hamilton-Jacobi equations with a Hamiltonian of the form G (p) + V (x , ω) for a wide class of stationary ergodic random media in one space dimension. The momentum part G(p) of the Hamiltonian is a general (nonconvex) continuous function with superlinear growth at infinity, and the potential V (x , ω) is bounded and Lipschitz continuous. The class of random media we consider is defined by an explicit hill and valley condition on the diffusivity-potential pair which is fulfilled as long as the environment is not "rigid". [ABSTRACT FROM AUTHOR]

We study the well-posedness of an infinite-dimensional Hamilton--Jacobi equation posed on the set of nonnegative measures and with a monotonic nonlinearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton--Jacobi equations and use the monotonicity of the nonlinearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton--Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex nonlinearity, we also provide a Hopf--Lax variational representation of the solution. [ABSTRACT FROM AUTHOR]

We consider a singularly perturbed system of stochastic differential equations proposed by Chaudhari et al. (Res. Math. Sci. 2018) to approximate the entropic gradient descent in the optimization of deep neural networks via homogenization. We embed it in a much larger class of two-scale stochastic control problems and rely on convergence results for Hamilton--Jacobi--Bellman equations with unbounded data proved recently by ourselves (ESAIM Control Optim. Calc. Var. 2023). We show that the limit of the value functions is itself the value function of an effective control problem with extended controls and that the trajectories of the perturbed system converge in a suitable sense to the trajectories of the limiting effective control system. These rigorous results improve the understanding of the convergence of the algorithms used by Chaudhari et al., as well as of their possible extensions where some tuning parameters are modeled as dynamic controls. [ABSTRACT FROM AUTHOR]

The size-dependent nonlinear vibration model of the magneto-electro-elastic (MEE) sandwich nanobeam on Winkler-Pasternak foundation is built based on nonlocal elasticity theory and Reddy's third-order shear deformation theory. Employing the Hamilton variational principle and von Karman's geometrically nonlinear theory, the nonlinear governing equation system of MEE sandwich nanobeam is deduced. The expressions of nonlinear frequency ratio and critical external electric and magnetic potential are derived with the perturbation method. Several cases are studied in order to show the influence of nonlocal parameter, Winkler and Pasternak elastic foundation, stacking sequence, external electric and magnetic potential on size-dependent nonlinear vibration behaviors of MEE simply supported sandwich nanobeam. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation of initial‐boundary‐value problems of a three‐parameter family of Bona–Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted L2$L^{2}$ inner product. Well‐posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then, the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results. [ABSTRACT FROM AUTHOR]

Based on the nonlocal theory, a layered theoretical model is presented for vibration analysis of piezoelectric–piezomagnetic bimorph nanobeams. The layer-wise expressions of displacement and potential fields across the thickness are provided through ensuring the continuity of displacements and shear stresses at the interface as well as the shear stress free at the surface, and then the nonlocal governing equations and consistent boundary conditions are derived from the Hamilton's principle. For typical support conditions at beam ends, the differential quadrature method is developed to obtain the numerical results which is verified by comparing with analytical solutions for the simply-supported beam. A detailed parametric study is conducted to analyze the influence of nonlocal parameter, structure size, external load and boundary condition on the natural frequency and mode shape. It indicates the nonlocal small-scale effect should be considered to accurately predict the vibration properties of magnetoelectric laminated nanostructures. [ABSTRACT FROM AUTHOR]

The present work aims to investigate the dynamic behavior of a bidirectional functionally graded (BDFG) beam lying on nonlinear elastic substrate tacking into account the impact of the material distribution. For the first time, this effect will be studied. The beam characteristics are considered to change in the axial and thickness directions as a function of a specific function with four distinct distribution models. The BDFG beam is supposed to be in contact with a nonlinear three-parameter substrate with general boundary conditions. Hamilton's principle is used to determine the governing equations based on a quasi-3D solution. An analytical solution is then is applied to obtain the free and forced vibration parameters of the BDFG beam. Furthermore, the validation results show excellent agreement between the proposed theory and those given in the literature. On the other hand, a set of numerical results is exposed and discussed to assess the influence of several geometric and mechanical parameters like distribution patterns of material, beam geometry, foundation's parameters and external force on the forced and the free vibration of a BDFG beam sustained by a nonlinear elastic foundation. The results show that these parameters have a major influence on the dynamic response of BDFG beams supported by nonlinear foundation. These results can be used as reference solutions for future investigations. [ABSTRACT FROM AUTHOR]

A precise interpretation of the universe wave function is forbidden in the spirit of the Copenhagen School since a precise notion of measure operation cannot be satisfactorily defined. Here, we propose a Bohmian interpretation of the isotropic universe quantum dynamics, in which the Hamilton–Jacobi equation is restated by including quantum corrections, which lead to a classical trajectory containing effects of order ℏ 2 . This solution is then used to determine the spectrum of gauge-invariant quantum fluctuations living on the obtained background model. The analysis is performed adopting the wave function approach to describe the fluctuation dynamics, which gives a time-dependent harmonic oscillator for each Fourier mode and whose frequency is affected by the ℏ 2 corrections. The properties of the emerging spectrum are discussed, outlining the modification induced with respect to the scale-invariant result, and the hierarchy of the spectral index running is discussed. [ABSTRACT FROM AUTHOR]

The Langevin transducer is a widely used power ultrasonic device across both medical and industrial applications, from orthopaedic surgery to drilling and welding. It is a sandwich-type device which typically consists of piezoelectric ceramic rings between two metallic end-masses. The transducer is commonly operated at a particular resonant mode to deliver ultrasonic vibrations to a target structure of interest, and is generally modelled using approaches including the transfer matrix method and electromechanical equivalent circuit methods. Here, we propose a variational framework to study the dynamics of the Langevin transducer based on the Timoshenko-Ehrenfest beam theory and Hamilton's principle. The variational equation derived from this model is then discretized by the standard finite element method with spectral elements. To verify the proposed one-dimensional electromechanical model, the computed resonance frequencies, or natural frequencies, of the one-dimensional model are compared to those of a three-dimensional finite element model with respect to varying geometry parameters characterizing the transducer. The results of the reduced one-dimensional and full three-dimensional models are then compared to those measured through an experimental investigation consisting of laser Doppler vibrometry. This is undertaken for the first ten eigenfrequencies, including longitudinal and bending modes, where normalized amplitudes and vibration mode shapes are reported. The close correlation between modelling and experiment demonstrates that the proposed one-dimensional electromechanical model can deliver results consistent with the full three-dimensional model and from experiment, thus verifying a rapid and reliable method for studying the free vibrations and dynamics of the Langevin transducer which accounts for the axial vibrations of the piezoelectric ceramic stack in all cases. • Created a variational framework based on the Timoshenko-Ehrenfest beam theory. • Proposed one-dimensional electromechanical model. • The model was solved by the spectral element method. • Verified the robustness of 1D model by comparing it to 3D models and experiments. [ABSTRACT FROM AUTHOR]