Initial Value Function Research Articles

Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials

We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical diffusion, multilayer plates and envelopes, and others). From the viewpoint of the pure theory, they are important due to crucially new effects not arising in the case of differential equations and due to the fact that a number of classical methods, tools, and approaches turn out to be inapplicable in the nonlocal theory. The qualitative novelty of our investigation is that the initial-value function is assumed to be summable. Earlier, only the case of bounded (essentially bounded) initial-value functions was investigated. For the prototype problem (the spatial variable is single and the nonlocal term of the equation is single), we construct the integral representation of a solution and show its smoothness in the open half-plane. Further, we find a condition binding the coefficient at the nonlocal potential and the length of its translation such that this condition guarantees the uniform decay (weighted decay) of the constructed solution under the unbounded growth of time. The rate of this decay (weighted decay) is estimated as well.

Reform and Innovation of Civic and Political Education Based on Deep Enhanced Learning Methods

Abstract The use of an initial state value function and an optimal strategy are used in this paper to solve educational problems based on deep reinforcement learning. Deep reinforcement learning’s approximate function is defined, and the matrix model is created by training tuning using learning methods like gradient descent. To analyze the modeling process of reinforcement learning, reward values are added to the Markov decision transfer matrix and the expected value of cumulative returns is calculated. The weights are trained using the Bellman equation to enhance the algorithm’s stability. In evaluating the effect of reform and innovation in Civic Education, the teacher education concept is rated as 10 points. The reform and innovation of civic education, combined with deep reinforcement learning, can promote the reform of education and teaching modes, improving the efficiency and quality of education.

The Iterative Properties for Positive Solutions of a Tempered Fractional Equation

In this article, we investigate the iterative properties of positive solutions for a tempered fractional equation under the case where the boundary conditions and nonlinearity all involve tempered fractional derivatives of unknown functions. By weakening a basic growth condition, some new and complete results on the iterative properties of the positive solutions to the equation are established, which include the uniqueness and existence of positive solutions, the iterative sequence converging to the unique solution, the error estimate of the solution and convergence rate as well as the asymptotic behavior of the solution. In particular, the iterative process is easy to implement as it can start from a known initial value function.

Galerkin Method for a Backward Problem of Time-Space Fractional Symmetric Diffusion Equation

We investigate a backward problem of the time-space fractional symmetric diffusion equation with a source term, wherein the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric elliptic operators. The problem is ill-posed because the solution does not depend continuously on the measured data. In this paper, the existence and uniqueness of the solution and the conditional stability for the inverse problem are given and proven. Based on the least squares technique, we construct a Galerkin regularization method to overcome the ill-posedness of the considered problem. Under a priori and a posteriori selection rules for the regularization parameter, the Hölder-type convergence results of optimal order for the proposed method are derived. Meanwhile, we verify the regularized effect of our method by carrying out some numerical experiments where the initial value function is a smooth function or a non-smooth one. Numerical results show that this method works well in dealing with the backward problem of the time-space fractional parabolic equation.

Strong Left-Invertibility and Strong Input-Observability of Nonlinear Time-Delay Systems

In this letter, we study the problem of unknown inputs reconstruction for nonlinear time-delay systems. First we define two notions called strong left-invertibility and strong input-observability and the word “strong” is to address the causality properties of those two notions. Then necessary and sufficient conditions for the strong left-invertibility and the strong input-observability are given under the algebraic framework proposed in (Xia et al., 2002). We find that a sequence of inputs submodules plays an important role for the strong left-invertibility of time-delay systems. A structure algorithm is provided to construct that sequence and to formulate an input reconstructor. At last, several examples are given to illustrate how to check the strong left-invertibility and the strong input-observability by applying the proposed structure algorithm, and to show how to recover the inputs via causal outputs and the initial value functions of states (strong left-invertibility) or only via causal outputs (strong input-observability).

Periodic motions in a class of discontinuous delayed systems with a parabolic boundary.

Various periodic motions are investigated for a mass-spring-damper oscillator described by second-order discontinuous delayed systems with a parabolic boundary. The G-function related to a delayed variable is first developed to judge the crossing, sliding, and grazing motions at the boundary points. Then, basic and local mappings, together with their motion equations, are classified by current and delayed states of a local flow. It is shown that periodic motions can be predicted analytically in terms of the return mapping and the initial value function provided posteriorly. Finally, numerical simulations are given to illustrate the existence of slowly oscillating periodic solutions without or with sliding portions as well as a crossing periodic solution not slowly oscillating. These results not only enrich and extend switchability theory of discontinuous delayed systems, but also reveal some complicated periodic behaviors which cannot be found in continuous delayed systems and discontinuous systems without time delay.

Comparison principle for positive time-delay systems: An extension and its application

This paper proposes an extended comparison principle for continuous-time linear positive time-delay systems. Unlike the existing comparison principle, which uses a constant initial function for the comparison system, instead, we propose a time-varying initial function in order to derive a more general solution comparison. Based on this extended comparison principle, we develop a novel computational method, which exploits more effectively the information of the initial value function, to derive tighter exponential estimates for the state vector of positive time-delay systems. The effectiveness of our developed method is illustrated through two numerical examples.

Optimal control of port-Hamiltonian systems: A continuous-time learning approach

In this paper, we propose a continuous-time adaptive feedback controller for the optimal control of input-state-output port-Hamiltonian systems with respect to general Lagrangian performance indices. The proposed control law implements an online learning procedure which uses the Hamiltonian of the system as an initial value function candidate. The continuous-time learning of the value function is achieved by means of a certain Lagrange multiplier that allows to evaluate the optimality of the current solution. In particular, constructive conditions for stabilizing initial value function candidates are stated and asymptotic stability of the closed-loop equilibrium is proven. Simulations of an exemplary nonlinear optimal control problem demonstrate the performance of the controller resulting from the proposed online learning procedure.

The Convergence of a Cooperation Markov Decision Process System.

In a general Markov decision progress system, only one agent’s learning evolution is considered. However, considering the learning evolution of a single agent in many problems has some limitations, more and more applications involve multi-agent. There are two types of cooperation, game environment among multi-agent. Therefore, this paper introduces a Cooperation Markov Decision Process system with two agents, which is suitable for the learning evolution of cooperative decision between two agents. It is further found that the value function in the system also converges in the end, and the convergence value is independent of the choice of the value of the initial value function. This paper presents an algorithm for finding the optimal strategy pair in the system, whose fundamental task is to find an optimal strategy pair and form an evolutionary system . Finally, an example is given to support the theoretical results.

PH, Nanosheet Concentration, and Antioxidant Affect the Oxidation of Ti3C2Tx and Ti2CTx MXene Dispersions

AbstractThe chemical stability of 2D MXene nanosheets in aqueous dispersions must be maintained to foster their widespread application. MXene nanosheets react with water, which results in the degradation of their 2D structure into oxides and carbon residues. The latter detrimentally restricts the shelf life of MXene dispersions and devices. However, the mechanism of MXene degradation in aqueous environment has yet to be fully understood. In this work, the oxidation kinetics is investigated of Ti3C2Tx and Ti2CTx in aqueous media as a function of initial pH values, ionic strengths, and nanosheet concentrations. The pH value of the dispersion is found to change with time as a result of MXene oxidation. Specifically, MXene oxidation is accelerated in basic media by their reaction with hydroxyl anions. It is also demonstrated that oxidation kinetics are strongly dependent on nanosheet dispersion concentration, in which oxidation is accelerated for lower MXene concentrations. Ionic strength does not strongly affect MXene oxidation. The authors also report that citric acid acts as an effective antioxidant and mitigates the oxidation of both Ti3C2Tx and Ti2CTx MXenes. Reactive molecular dynamic simulations suggest that citric acid associates with the nanosheet edge to hinder the initiation of oxidation.

Improved value iteration for neural-network-based stochastic optimal control design

In this paper, a novel value iteration adaptive dynamic programming (ADP) algorithm is presented, which is called an improved value iteration ADP algorithm, to obtain the optimal policy for discrete stochastic processes. In the improved value iteration ADP algorithm, for the first time we propose a new criteria to verify whether the obtained policy is stable or not for stochastic processes. By analyzing the convergence properties of the proposed algorithm, it is shown that the iterative value functions can converge to the optimum. In addition, our algorithm allows the initial value function to be an arbitrary positive semi-definite function. Finally, two simulation examples are presented to validate the effectiveness of the developed method.

The Truncation Regularization Method for Identifying the Initial Value on Non-Homogeneous Time-Fractional Diffusion-Wave Equations

In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value functions. The convergence estimations are given in a priori regularization parameter choice regulations and a posteriori regularization parameter choice regulations. Numerical examples are given to demonstrate this is effective and practicable.

Backward difference formulae and spectral Galerkin methods for the Riesz space fractional diffusion equation

Approximating Riesz space fractional diffusion equation in time by k-step backward difference formula and in space by spectral Galerkin method, we establish a fully discrete scheme with high order both in time and in space. For k≤5, we prove the stability of full discretization and obtain the error estimate with order O(τk+Nα2−m), which depends only on the regularity of initial value and right-hand function. Moreover, we extend the proposed method to two dimensional case and derive similar results. Finally, we illustrate the theoretical estimates by numerical examples.

New Mechanistic Phosphorus Fertilizer Recommendation Model Based on Two Intrinsic Phosphorus Pools in Tropical Soils

No mechanistic model for phosphorus (P) fertilizer recommendation is available for nutrient management in tropical soils. Soil intrinsic P pools, including a weakly and a tightly adsorbed P pool, were proposed to quantify actual soil available P. The goal of this study was to develop a new mechanistic P fertilizer recommendation model based on the two pools. The public data set of a corn (Zea mays L.) field experiment was used to estimate the daily corn P uptake. Seven soils were used to demonstrate the effects of P‐sorption site density (PSSD) and desorption rate of the tightly adsorbed P pool on the P fertilizer recommendation. Dynamics of soil P pools in a corn–soil system were modeled based on the transformation between the pools, the daily crop P uptake, and P fertilizer added at the beginning of planting. The weakly adsorbed P pool was required to be positive to represent no constraint of crop P nutrition. Results showed that the derived P recommendation model was a linear model of the total crop P uptake minus the two pools with three different coefficients. The weakly adsorbed P pool was readily available for crop uptake. The coefficients of the total crop P uptake and the tightly adsorbed P pool were >1.0 and <1.0, respectively, and they changed slightly with PSSD but were dependent on the desorption rate. The coefficient of the total crop P uptake showed a maximum 1.679 at the desorption rate 0.0118 d−1.Core Ideas A P recommendation model was developed based on dynamic soil surface reaction and crop uptake. Intrinsic weakly and tightly adsorbed P pools quantify dynamics of soil P supply. P requirement was a linear function of crop uptake and initial values of P pools. Model parameters were determined by soil P‐sorption site density and desorption rate.

Downhole Microseismic Source Location Based on a Multi‐Dimensional DIRECT Algorithm for Unconventional Oil and Gas Reservoir Exploration

Downhole microseismic data has the significant advantages of high signal‐to‐noise ratio and well‐developed P and S waves and the core component of microseismic monitoring is microseismic event location associated with hydraulic fracturing in a relatively high confidence level and accuracy. In this study, we present a multidimensional DIRECT inversion method for microseismic locations and applicability tests over modeling data based on a downhole microseismic monitoring system. Synthetic tests inidcate that the objective function of locations can be defined as a multi‐dimensional matrix space by employing the global optimization DIRECT algorithm, because it can be run without the initial value and objective function derivation, and the discretely scattered objective points lead to an expeditious contraction of objective functions in each dimension. This study shows that the DIRECT algorithm can be extensively applied in real downhole microseismic monitoring data from hydraulic fracturing completions. Therefore, the methodology, based on a multidimensional DIRECT algorithm, can provide significant high accuracy and convergent efficiency as well as robust computation for interpretable spatiotemporal microseismic evolution, which is more suitable for real‐time processing of a large amount of downhole microseismic monitoring data.

On Large-Time Behavior of Solutions of Parabolic Nondivergent Equations with Increasing Principal Coefficients

We investigate sufficient conditions of the stabilization to zero for solutions of the Cauchy problem for linear parabolic second-order equations with increasing principal coefficients and initial-value functions growing at infinity as a power function.

Discrete-Time Local Value Iteration Adaptive Dynamic Programming: Convergence Analysis

In this paper, convergence properties are established for the newly developed discrete-time local value iteration adaptive dynamic programming (ADP) algorithm. The present local iterative ADP algorithm permits an arbitrary positive semidefinite function to initialize the algorithm. Employing a state-dependent learning rate function, for the first time, the iterative value function and iterative control law can be updated in a subset of the state space instead of the whole state space, which effectively relaxes the computational burden. A new analysis method for the convergence property is developed to prove that the iterative value functions will converge to the optimum under some mild constraints. Monotonicity of the local value iteration ADP algorithm is presented, which shows that under some special conditions of the initial value function and the learning rate function, the iterative value function can monotonically converge to the optimum. Finally, three simulation examples and comparisons are given to illustrate the performance of the developed algorithm.

General value iteration based single network approach for constrained optimal controller design of partially-unknown continuous-time nonlinear systems

In this paper, a novel iterative approximate dynamic programming scheme is proposed by introducing the learning mechanism of value iteration (VI) to solve the constrained optimal control problem for CT affine nonlinear systems with utilizing only one neural network. The idea is to show the feasibility of introducing the VI learning mechanism to solve for the constrained optimal control problem from a theoretical point of view, and thus the initial admissible control can be avoided compared with most existing works based on policy iteration (PI). Meanwhile, the initial condition of the proposed VI based method can be more general than the traditional VI method which requires the initial value function to be a zero function. A general analytical method is proposed to demonstrate the convergence property. To simplify the architecture, only one critic neural network is adopted to approximate the iterative value function while implementing the proposed method. At last, two simulation examples are proposed to validate the theoretical results.

Study of Time-Free Transfers into Libration Point Orbits with Multiple Constraints

Preliminary designs of time-free transfers connecting a low Earth orbit with a halo orbit around the sun–Earth point are developed with impulse maneuver and multiple orbital element constraints. As a practical application, the constraints, including orbital altitude, inclination, right ascension of ascending node, and flight-path angle, are considered to describe the circular Earth parking orbit. The problem is then solved directly by a differential correction, using the constraint partial derivatives relative to state vector and Julian day. To reduce the sensitivity of the initial condition for differential correction, initial value functions are constructed based on the relationship of free variables and constraints. Numerical results show that for halo orbit insertion ranges from a few meters per second to hundreds of meters per second, depending on the specific inclination and right ascension of ascending node of Earth parking orbit, and demonstrate that the method for designing transfer trajectories with multiple constraints is feasible.

An inverse backward problem for degenerate parabolic equations

This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017