Maximum Principle Research Articles

Convex hull property for elliptic and parabolic systems of PDE

We study the convex hull property for systems of partial differential equations. This is a generalization of the maximum principle for a single equation. We show that the convex hull property holds for a class of elliptic and parabolic systems of non-linear partial differential equations. In particular, this includes the case of the parabolic p-Laplace system. The coupling conditions for coefficients are demonstrated to be optimal by means of respective counterexamples.

Periodic on-off operations of the self-cycling ethanol fermentation process: Stability analysis and optimal operation strategies

Fuel ethanol is highlighted for being renewable and carbon neutral, but a multi-scale challenge exists that hampers its role as a future energy substitute, and with the breakthrough of cellulose pretreatment and enzyme hydrolysis technology, fermentation itself has become one of the main limiting factors for process enhancement. To tackle the problem, this paper studies periodic operation strategies to improve process performance and avoid possible instabilities. Firstly, superiority by periodic operation over the corresponding steady-state operation is verified. Then, to settle the problem of possible instabilities under impulsive control, stability of state-impulsive control is conducted and for the studied case, any self-cycling fermentation starts from batch mode with 50 % discharge-and-refill portion exhibits superior repeatability property, which allows continuous fermentation to operate under very slow overall dilution rate to cope with the extreme slow fermentation rate. Finally, on-off operation problem is formulated that is governed by a multidimensional nonlinear system, affine to the manipulated variable that is under the isoperimetric constraint; with the use of Pontryagin maximum principle, the optimal synthesis of the optimal strategies under integral constraint is computed out, and a practically plausible on-off operation strategy for ethanol fermentation is formulated with guaranteed periodic stability and optimality.

Global attractivity for reaction–diffusion equations with periodic coefficients and time delays

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson’s equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction.

Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul’yev methods

Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul’yev methods

Role Theory, Non-Coercive Influence, and the Agency of Target States: The Case of Kazakhstan’s Ambassadorial Corps and the Russian Diplomatic Academy

Abstract This article contributes to foreign policy analysis by applying role theory to the study of soft power, evaluating the relationship between a state’s chosen international role(s) and susceptibility to non-coercive influence. I argue that in countries where said role involves committed engagement with varied power poles, elites derive benefits from their various partners and thus abide soft power bases such as the Russian Diplomatic Academy (RDA) without violating the principle of autonomy maximization. I utilize a mixed methods approach, using logistic regression as part of a qualitative case study of Kazakhstan. Drawing on an original dataset of Kazakhstani ambassadors since 1992, I find that foreign policy role better explains the continued presence of RDA-trained diplomats within the Kazakhstani ambassadorial corps than traditional security arguments. This outcome demonstrates the promise of examining soft power through the lens of role theory, with an emphasis on the agency of potential targets of influence.

Phase stability and mechanical properties of the six-principal element TiVNbCrCoNi alloys

This study designed a series of six-principal element TiVNbCrCoNi alloys according to the maximum entropy principle under the constraint of valence electron concentration (VEC). The phase stability and mechanical properties were investigated. It was showed that at VEC levels below 5.2, the body-centered cubic (BCC)-structured solid-solution phase remained stable at room temperature. As VEC exceeded 8.0, the face-centered cubic (FCC)-structured solid-solution phase stabilized. In the VEC range of 5.2–8.0, (Co, Ni)Ti and Ni3Nb intermetallic compounds formed combining with the solid-solution phases. Both yield strength and ductility increased with increase of VEC in the BCC-structured solid-solution alloys. The phase stability was elucidated through First-principle and thermodynamic calculations, while the evolution of mechanical properties of solid-solution alloys was evaluated in terms of multiple parameters. The thermodynamic calculation achieved satisfactory accuracy after being adjusted to utilize atomic bond enthalpy instead of atomic electronegativity difference.

Dynamics of a discrete one‐predator two‐prey system with Michaelis–Menten‐type prey harvesting and prey refuge

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system.

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

Optimal control of an infected prey–predator model with fear effect

In this paper, we propose and analyze a prey–predator model with the functional response of Beddington–DeAngelis and the fear effect that have infection only in prey populations. We determine existence criteria of several equilibria, and the stability at different equilibria are presented. We exert pesticide control over prey and additional food control over predators, the optimal control is obtained by the Pontryagin maximum principle. We confirm that adding controls to the predator and prey yields better results. Further we enrich our analysis with the inclusion of the existence and uniqueness of the optimal control. Finally, some numerical results to illustrate our analysis are presented.

Analysis of the operation of the microprocessor control system of a bridge crane using optimum control laws

This work presents the results of studies on the influence of the error of the main parameters of the bridge crane on the quality of optimal control. The quality of control refers to the exact positioning of the load at the stopping point and the magnitude of the final fluctuations of the load after the crane stops. The concept of optimal control is based on L.S. Pontryagin's maximum principle, which is when the goal is achieved by switching the acceleration-braking control of the load suspension point according to a specific algorithm. To achieve the maximum speed of operation of the crane, certain restrictions and conditions have been removed, namely, the refusal to eliminate load fluctuations during crane acceleration, which simplifies the subsequent elimination of load fluctuations during crane braking. However, this is an unnecessary waste of time, and therefore, acceleration is carried out at the maximum possible acceleration without eliminating load fluctuations. Of course, at the same time, significant fluctuations in the cargo occur, and they continue throughout the entire time the crane is moving. Moreover, the final goal is achieved only at the end of the cycle by switching the acceleration-deceleration modes - the cargo is delivered precisely to the specified point without fluctuations after the crane stops. A PC program that generates optimal control laws for a specific crane has been developed. This program can be used for any lifting machine where the suspension point moves in a straight line, and flexible load suspension is used, such as overhead or gantry cranes, hoists, etc. The results of the research were tested on several objects: a 0.5t crane, a 5t bridge grab crane, and models of bridge cranes. In each case, a good coincidence of theoretical and experimental results was observed, resulting in effective management of the object. The original microprocessor system implemented the optimal control laws, built first based on the MKP-1 microcontroller and then based on Arduino. However, a good result is guaranteed only if the fundamental parameters of the crane coincide with the theoretical ones. This is not easy to achieve during the actual operation of the crane. They vary within a specific range. The article examines how the error in each of the main parameters of the crane affects the result and how the control system can be improved to reduce the negative impact to a minimum. Keywords: optimal control, elimination of load fluctuations, accurate positioning, crane, microprocessor, reloading cycle

Convergence of a TPFA finite volume scheme for nonisothermal immiscible compressible two-phase flow in porous media

This paper presents and analyzes a finite volume scheme for a fully degenerate parabolic system modeling the compressible two-phase flow, such as water-gas, problem in porous media considering the effects of temperature. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. We use a first-order time implicit Euler scheme. The spatial discretization uses a fully coupled fully implicit cell-centered two-point flux approximation scheme where upwinding of the flux across an interface between two control volumes is performed for each fluid phase. Under physically relevant assumptions on the data, we prove the maximum principle for both saturation and temperature. We obtain a priori estimates and we show that the discrete problem is well posed. Also, we exploit various a priori estimates as well as compactness arguments to establish the convergence of the numerical scheme toward the weak solution. The open-source platform DuMuX was employed for the implementation of the developed algorithm based on the above scheme. Numerical result are presented to demonstrate the efficiency of this scheme. The test addresses the evolution in 2D of CO2 injection into a layered aquifer. The algorithm's convergence with first order is validated by our numerical results, which support the theoretical results. To our best knowledge, this is the first convergence result of a finite volume scheme in the case of nonisothermal immiscible compressible two-phase flow in porous media.

Convergence analysis of the maximum principle preserving BDF2 scheme with variable time-steps for the space fractional Allen–Cahn equation

In this work, a fully implicit discrete scheme preserving the maximum principle is analyzed for solving the space fractional Allen–Cahn equation with variable step BDF2 in time. The convergence analysis in L∞ norm is rigorously proved under the same requirement of time-step ratios for the maximum principle by means of the kernels recombination technique. Furthermore, the unique solvability of the proposed scheme is theoretically guaranteed depending on the property of convex functions. The energy decay law with the consideration of Riesz fractional derivative is also proved under sufficient restrictions on time-steps and time-step ratios. Numerical experiments are given to validate the second order accuracy in time, the maximum principle as well as the energy decay law of the proposed numerical scheme. In particular, comparisons of errors and time levels for the adaptive time-stepping strategy and uniform time steps are conducted to demonstrate the efficiency of adaptive time-stepping strategy.

Analysis of the hate speech and racism co-existence dissemination model with optimal control strategies

Hate speech, racism, and their co-existence are the human mind infections that are major factors that affect the living conditions of people, societal well-being, and political stability, economic and social disturbance throughout the world especially in underdeveloped nations of the world. The main objective of this study is to formulate and analyze the hate-speech and racism co-existence model with optimal control strategies and investigate the optimal effects of protection and improvement (rehabilitation) strategies to minimize and tackle the hate speech and racism co-existence dissemination in the community. In this study, we have computed the hate speech sub-model, the racism sub-model, and the hate speech and racism co-existence model equilibrium points and analyzed their stabilities, using the next generation operator approach all the models' effective reproduction numbers are computed, examined the phenomenon of backward bifurcation whenever the model effective reproduction number is less than one where at which both the positive co-existence free equilibrium point and the positive co-existence dominance equilibrium point exist simultaneously. To achieve the objective minimizing the hate speech and racism co-existence dissemination dynamics by implementing the efforts towards the protection and improvement (rehabilitation) strategies, the optimal control problem is re-formulated, and optimal control analysis is performed for the co-existence model using Pontryagin's maximum principle. Numerical simulations using MATLAB ode45 solver with fourth-order Runge-Kutta numerical methods to investigate the behavior of the co-existence model solutions and to explore the optimal effects of the protections and improvements (rehabilitation) strategies for the hate speech and racism co-existence dynamics are then performed. From the results, we observed that implementing the protection and improvement (rehabilitation) strategies simultaneously is the most effective approach to minimize and tackle the speech and racism co-existence propagation in the community.

Maximum entropy determination of mammalian proteome dynamics

Full understanding of proteostasis and energy utilization in cells will require knowledge of the fraction of cell proteins being degraded with different half-lives and their rates of synthesis. We therefore developed a method to determine such information that combines mathematical analysis of protein degradation kinetics obtained in pulse-chase experiments with Bayesian data fitting using the maximum entropy principle. This approach will enable rapid analyses of whole-cell protein dynamics in different cell types, physiological states, and neurodegenerative disease. Using it, we obtained surprising insights about protein stabilities in cultured cells normally and upon activation of proteolysis by mTOR inhibition and increasing cAMP or cGMP. It revealed that >90% of protein content in dividing mammalian cell lines is long-lived, with half-lives of 24 to 200 h, and therefore comprises much of the proteins in daughter cells. The well-studied short-lived proteins (half-lives < 10 h) together comprise <2% of cell protein mass, but surprisingly account for 10 to 20% of measurable newly synthesized protein mass. Evolution thus appears to have minimized intracellular proteolysis except to rapidly eliminate misfolded and regulatory proteins.

CONTROL THEORY APPROACHES TO OPTIMIZE INFORMATION FLOW IN THE FACE OF UNCERTAINTY AND CONFLICTING DATA

The rapid spread of inaccurate information through social media has become a major concern in recent years. This paper presents a mathematical model for analyzing the impact of inaccurate information on the spread of accurate information in a network. We aim to evaluate the effectiveness of three different strategies of control: media literacy programs and fact-checking and a combination of both. The model is analyzed using the maximum principle of Pontryagin to derive the optimal control strategies for minimizing the spread of inaccurate information while maximizing the spread of accurate information. A numerical simulation is presented to illustrate the effectiveness of the control strategies, and statistical analysis is performed to compare the impact of the different control strategies on the spread of information. The results demonstrate that the combination of media literacy programs and fact-checking is the most effective strategy for increasing the spread of accurate information and reducing the spread of inaccurate information. These findings have important implications for the design of effective strategies to combat the spread of misinformation and promote the spread of accurate information in a network.

Revisiting Gradient Bach Solitons via Maximum Principles

Supposing that the Ricci curvature has an appropriate lower bound and applying suitable maximum principles, we establish triviality results which guarantee that a gradient Bach soliton must be trivial and Bach-flat. Our approach is based on three main cores: convergence to zero at infinity, polynomial volume growth (both related to complete noncompact Riemannian manifolds) and stochastic completeness.

A mathematical analysis of the corruption dynamics model with optimal control strategy

Corruption is a global problem that affects many countries by destroying economic, social, and political development. Therefore, we have formulated and analyzed a mathematical model to understand better control measures that reduce corruption transmission with optimal control strategies. To verify the validity of this model, we examined a model analysis showing that the solution of the model is positive and bounded. The basic reproduction number R0 was computed by using the next-generation matrix. The formulated model was studied analytically and numerically in the context of corruption dynamics. The stability analysis of the formulated model showed that the corruption-free equilibrium is locally and globally asymptotically stable for R0 &amp;lt; 1, but the corruption-endemic equilibrium is globally asymptotically stable for R0 &amp;gt; 1. Furthermore, the optimal control strategy was expressed through the Pontryagin Maximum Principle by incorporating two control variables. Finally, numerical simulations for the optimal control model were performed using the Runge-Kutta fourth order forward and backward methods. This study showed that applying both mass education and law enforcement is the most efficient strategy to reduce the spread of corruption.

A Probabilistic Structural Damage Identification Method with a Generic Non-Convex Penalty

Due to the advantage that the non-convex penalty accurately characterizes the sparsity of structural damage, various models based on non-convex penalties have been effectively utilized to the field of structural damage identification. However, these models generally ignore the influence of the uncertainty on the damage identification, which inevitably reduces the accuracy of damage identification. To improve the damage identification accuracy, a probabilistic structural damage identification method with a generic non-convex penalty is proposed, where the uncertainty corresponding to each mode is quantified using the separate Gaussian distribution. The proposed model is estimated via the iteratively reweighted least squares optimization algorithm according to the maximum likelihood principle. The numerical and experimental results illustrate that the proposed method improves the damage identification accuracy by 3.98% and 7.25% compared to the original model, respectively.

Evolution of tensile strength and ductility of the non-equimolar TiVNbMoCr alloys with maximum entropy under the VEC constraint

In this work, a series of non-equimolar TiVNbMoCr alloys were designed according to the maximum entropy principle under the valence electron concentration (VEC) constraint. The effect of the physical parameters on tensile strength and elongation was experimentally investigated and comprehensively discussed. It was found that the lattice constant a and the average grain size d vary with relative changes in the content of each component, and exhibit negative correlations to VEC. Whereas, the self-interstitial formation energy Ei, the atomic radius mismatch δ, the shear modulus mismatch η and the shear modulus G exhibit positive correlations to VEC. These parameters have a significant impact on tensile properties of the alloys. Except for Burgers vector magnitude, the critical physical parameters in the Senkov and Labusch-Maresca-Curtin's strengthening models are positively correlated to the yield strength. The ductility of the solid-solution TiVNbMoCr alloys is related to VEC, a, d and the grain boundary characteristics in strain accommodation, which respectively belong to intrinsic, crystallography structural and microstructural types of parameters.

Mass lumping of stable generalized finite element methods for parabolic interface problems

Stable generalized finite element method (SGFEM) for interface problem is based on simple mesh independent of interface curves and possesses merits of optimal convergence rates, conditioning of same order as that of finite element method (FEM), robustness, and freedom from any penalty parameters or stability schemes. This paper investigates mass lumping of SGFEM for parabolic interface problems. The mass lumping is important for computational reduction and maximum principle preserving. The difficulty in the mass lumping of conventional generalized FEM (GFEM) and SGFEM consists in their incorporation of extra enriched functions. The existing mass lumping methods of GFEM cannot work properly due to features of SGFEM that enriched shape functions vanish on mesh nodes, and the nodes are enriched partially rather than totally. In this study, we propose two mass lumping schemes for the SGFEM. The first one is to only lump the FEM part, and enriched parts are kept fixed. This scheme enables us to execute a rigorous optimal convergence analysis. It is noted that a theoretical analysis on the mass lumping of GFEM has not been available yet in the GFEM field. The second scheme is to further compress the enriched part and intersection part of mass matrix separately. The lumped mass matrix is almost diagonal, and there is only a line of nonzero entries in the intersection part of mass matrix. Numerical experiments indicate that two schemes both produce the optimal convergence rates; their errors are almost the same as those of SGFEM without using the mass lumping.