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Well-posedness of the shooting algorithm for control-affine problems with a scalar state constraint

COMPUTATIONAL & APPLIED MATHEMATICS 2023年第5期42卷 1-22页

作者： Aronna, M. Soledad Bonnans, J. Frederic San Goh, Bean FGV EMAp Escola Matemat Aplicada Praia Botafogo 190 BR-22250900 Rio De Janeiro Brazil Cent Supelec INRIA Saclay L2S F-91190 Gif Sur Yvette France Curtin Univ Sch Elect Engn Comp & Math Sci Perth 6845 Australia

We deal with a control-affine problem with scalar control subject to bounds, a scalar state constraint and endpoint constraints of equality type. For the numerical solution of this problem, we propose a shooting algorithm and provide a sufficient condition for its local convergence. We exhibit an example that illustrates the theory.

关键词： Optimal control shooting algorithm Singular arc Second order optimality conditions

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A global shooting algorithm for the facility location and capacity acquisition problem on a line with dense demand

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COMPUTERS & OPERATIONS RESEARCH 2016年 71卷 1-15页

作者： Murat, Alper Laporte, Gilbert Verter, Vedat Wayne State Univ Dept Ind & Syst Engn Detroit MI 48202 USA HEC Montreal Distribut Management 3000 Chemin Cote Ste Catherine Montreal PQ H3T 2A7 Canada McGill Univ Desautels Fac Management 1001 Sherbrooke St West Montreal PQ H3A 1G5 Canada

This paper describes the development of an exact allocation-based solution algorithm for the facility location and capacity acquisition problem (LCAP) on a line with dense demand data. Initially, the n-facility problem on a line is studied and formulated as a dynamic programming model in the allocation decision space. Next, we cast this dynamic programming formulation as a two-point boundary value problem and provide conditions for the existence and uniqueness of solutions. We derive sufficient conditions for non-empty service regions and necessary conditions for interior facility locations. We develop an efficient exact shooting algorithm to solve the problem as an initial value problem and illustrate on an example. A computational study is conducted to study the effect of demand density and other problem parameters on the solutions. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Location-allocation problem Capacity acquisition Dynamic programming shooting algorithm

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Buoyancy force and Arrhenius energy impacts on Buongiorno electromagnetic nanofluid flow containing gyrotactic microorganism

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PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART C-JOURNAL OF MECHANICAL ENGINEERING SCIENCE 2022年第17期236卷 9459-9471页

作者： Safdar, Rabia Gulzar, Iqra Jawad, Muhammad Jamshed, Wasim Shahzad, Faisal Eid, Mohamed R. Lahore Coll Women Univ Dept Math Lahore Pakistan Univ Faisalabad Sch Management Studies Faisalabad Pakistan Capital Univ Sci & Technol CUST Dept Math Islamabad Pakistan New Valley Univ Fac Sci Dept Math Al Kharga 72511 Al Wadi Al Gadi Egypt Northern Border Univ Fac Sci Dept Math Ar Ar 1321 Saudi Arabia

In this study, an incompressible, steady, and magnetohydrodynamic flow of Buongiorno nanofluid through a stretchable surface has been analyzed by adopting a theoretical and numerical approach. This paper involves the impacts of buoyancy forces, thermal conductivity, chemically diffusion, and Arrhenius activation energy. Moreover, the influence of the suspension of gyrotactic microorganisms in the nanofluids is also a part of this study. It is assumed that the behavior of viscosity varies as a function of time and the suspension of microorganisms remain consistent throughout the study. A system of PDEs is reduced to a solvable system of ODEs by applying a suitable similarity transformation. For the sake of numerical solutions, the shooting method has been employed. Wolfram Mathematica has been used to deal with the BVP. In addition, the behaviors of different emerging parameters comprising velocity outline, temperature outline, concentration distribution, the density outline of gyrotactic microorganisms have also been demonstrated by graphical illustrations. From the extracted results, it has been observed that the rising values of viscosity of fluid produce a decline in the velocity parameter. Also, an increment has been noticed in the temperature profile for the growing behavior of the mixed convective factor.

关键词： gyrotactic microorganisms Arrhenius activation energy chemical diffusion magnetohydrodynamics shooting algorithm Buongiorno nanofluid

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Numerical Simulation for Bioconvection of Unsteady Stagnation Point Flow of Oldroyd-B Nanofluid with Activation Energy and Temperature-Based Thermal Conductivity Past a Stretching Disk

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Computer Modeling in Engineering & Sciences 2022年第1期130卷 233-254页

作者： Muhammad Sami Rashad Haihu Liu Shan Ali Khan School of Energy and Power Engineering Xi’an Jiaotong UniversityXi’an710049China Department of Mathematics Government College University FaisalabadLayyah Campus31200Pakistan

A mathematical modeling is explored to scrutinize the unsteady stagnation point flow of Oldroyd-B nanofluid under the thermal conductivity and solutal diffusivity with bioconvection *** of Joule heating and Arrhenius activation energy including convective boundary conditions are studied,and the specified surface temperature and constant temperature of wall(CTW)are *** consequences of thermal conductivity and diffusivity are also taken into *** flow is generated through stretchable disk geometry,and the behavior of non-linear thermal radiation is incorporated in energy *** partial differential equations governing the fluid flow in the structure is reduced into dimensionless nonlinear ODEs by applying suitable similarity *** obtained system of non-dimensional nonlinear ODEs is treated numerically with the help of bvp4c solver in Matlab under shooting *** impact of various prominent parameters on velocity profile,thermal profile,volumetric nanoparticle concentration and microorganism distribution is depicted in graphical *** numerical outcomes for skin friction coefficient,heat transfer rate,Sherwood number as well as microorganism density number versus various parameters are listed in the *** results show that fluid velocity is reduced by increasing buoyancy ratio parameter,while the fluid flow increases with mixed convective *** temperature profile is enhanced with the amount of nonlinear thermal radiation and temperature dependent thermal ***,concentration profiles of nanoparticles have opposite behavior for Brownian motion coefficient and thermophoresis diffusion parameter,and it is noticed that by varying Peclet number the microorganisms profile is *** proposed study is useful to control and optimize heat transfer in industrial applications.

关键词： Oldroyd-B nanofluid thermal conductivity solutal diffusivity conductivity bioconvective microorganisms shooting algorithm

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Numerical Search for a Global Solution in a Two-Mode Economy Model with an Exhaustible Resource of Hydrocarbons

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Mathematical Models and Computer Simulations 2022年第2期14卷 213-223页

作者： Leonov, E.A. Polbin, A.V. Institute of Applied Economic Research Russian Presidential Academy of National Economy and Public Administration Moscow Russian Federation Gaidar Institute for Economic Policy Moscow Russian Federation

Abstract: This study is devoted to the numerical search for the optimal trajectory in the model of economic growth with two sectors: production and energy. In the energy sector, there is an opportunity to use energy resources from limited reserves. Switching the economy to a mode of operation without a resource, the moment of which is not fixed and depends on the behavior of agents, introduces features into the phase portrait and necessitates a numerical search for a global solution, which is what this study is aimed at. To solve this multidimensional problem, it is proposed to use a shooting algorithm using a modification of Newton’s method to update the starting point and also discuss the nuances of applying this approach to economic problems of this type. © 2022, Pleiades Publishing, Ltd.

关键词： dynamic general equilibrium model exhaustible resources global solution Newton’s method oil prices shooting algorithm

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Effects of nonlinear thermal radiation and activation energy on modified second-grade nanofluid with Cattaneo-Christov expressions

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JOURNAL OF THERMAL ANALYSIS AND CALORIMETRY 2021年第2期143卷 1175-1186页

作者： Khan, Sami Ullah Tlili, Iskander Waqas, Hassan Imran, Muhammad COMSATS Univ Islamabad Dept Math Sahiwal 57000 Pakistan Ton Duc Thang Univ Dept Management Sci & Technol Dev Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Sci Appl Ho Chi Minh City Vietnam Govt Coll Univ Dept Math Faisalabad Pakistan

The aim of the current contribution is to exploit the effects of nonlinear thermal radiation and activation energy on the flow of modified second-grade fluid with the utilization of nanoparticles. The induced flow has been considered by linear movement of stretched surface which is considered to be porous. Famous modified second-grade fluid enables one to capture the important rheological features of shear thinning and thickening. Following the modern aspects of heat and mass transportations, we have employed the theories of Cattaneo-Christov heat flux and generalized Fick's relations. The additional features of thermal radiation are also utilized in the energy equations with nonlinear expressions. Further, the impact of activation energy is also considered in the current continuation which makes the study quite versatile. While operating the suitable variables, the constituted problem has been distracted in a dimensionless form. The solution procedure has been followed with the implementation of the famous shooting technique with desirable accuracy. The involved engineering parameters are explained graphically with interesting physical consequences. It is noted that velocity distribution declined with a stretching ratio constant and combined parameter. An improved nanoparticles temperature distribution is observed with increases in stretching parameter and Brownian motion constant. The study also reports that the presence of activation energy can be more useful to enhance reaction processes. Based on the obtained scientific computations, it is claimed that the reported results can play a useful role in manufacturing processes and improvement in energy and heat resources.

关键词： Modified second-grade nanofluid Nonlinear thermal radiation Cattaneo-Christov heat flux Activation energy shooting algorithm

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Direct marketing of an event under hazards of customer saturation and forgetting

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ANNALS OF OPERATIONS RESEARCH 2020年第1期295卷 207-227页

作者： Kogan, Konstantin Herbon, Avi Venturi, Beatrice Bar Ilan Univ Dept Management IL-52900 Ramat Gan Israel Univ Cagliari Dept Econ & Business Sci Cagliari Italy

We address the problem of optimising the intensity of online advertising. In contrast to the classical literature, we tackle the advertising interaction between the firm and the potential customer, for the sale of a onetime event, in a very limited time horizon. The problem is intrinsically dynamic due to two conflicting situations: the first arises when the customer is subjected to intense advertising pressure, which may lead to customer saturation and even irritation, while the second is the tendency for customers to forget if they are not reminded systematically through advertising. In order to determine an optimal event-advertising policy and develop an efficient enumerative shooting algorithm to solve the problem, we suggest a hazard rate-based approach to modelling the conflicting factors. Our analysis shows that the initial level of customer interest in the event has a non-trivial effect on the dynamics of the optimal advertising policy. In particular, this advertising policy consists of a monotonic increase over time prior to the event in the case of high initial interest and a concave, peak-wise form in the case of low initial interest.

关键词： The maximum principle Direct marketing shooting algorithm Onetime event advertising

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Nonlinear thermally radiative heat transport for brinkman type micropolar nano-material over an inclined surface with motile microorganisms and exponential heat source

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INTERNATIONAL COMMUNICATIONS IN HEAT AND MASS TRANSFER 2021年 126卷 105351-105351页

作者： Song, Ying-Qing Waqas, Hassan Khan, Shan Ali Khan, Sami Ullah Khan, M. Ijaz Chu, Yu-Ming Qayyum, Sumaira Hunan City Univ Coll Sci Yiyang 413000 Peoples R China Univ Faisalabad Govt Coll Dept Math Layyah Campus Faisalabad 31200 Pakistan COMSATS Univ Islamabad Dept Math Sahiwal 57000 Pakistan Riphah Int Univ I 14 Dept Math & Stat Islamabad 44000 Pakistan Huzhou Univ Dept Math Huzhou 313000 Peoples R China Changsha Univ Sci Technol Hunan Prov Key Lab Math Modeling & Anal Engn Changsha 410114 Peoples R China Quaid I Azam Univ Dept Math Islamabad 44000 Pakistan

The novel aim of current analysis is to identify the characteristics of Arrhenius activation energy and nonlinear thermal radiation in bio-convective brinkman type micropolar nanofluid flow with gyrotactic motile microorganisms and melting phenomena through a stretched surface. The temperature dependent heat source/sink and exponential space-based thermal source are considered. Further, concentration of nano-materials and thermal fields are considered in the occurrence of Brownian motion and thermophoresis diffusion. The narrating boundary layer flow equations of brinkman type micropolar fluid model are reframed on the form of ordinary ones with the help of appropriate similarity transformation. The bvp4c collocation technique is used in the transformed model with shooting scheme to obtain a numerical solution of the original problem. The numerical data for the velocity, thermal and solutal fields, and microorganism's field is graphically designed and impacts of the prominent parameters are described in detail. It is notice from the current study velocity of fluid has diverse nature against magnetic parameter. Growing behavior of velocity field is depicted for higher estimation of mixed convection parameter. The microorganism field is boosted up by increasing melting parameter. The current model is applicable in nanotechnology, biotechnology, drug delivery, protein chips, DNA chips, food processing, cancer treatment, smart sensors and in many more fields

关键词： Micropolar nanofluid Microorganism Brinkman effect Temperature dependent heat source sink Exponential space-based heat source shooting algorithm

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Cattaneo-Christov double diffusions theories with bio-convection in nanofluid flow to enhance the efficiency of nanoparticles diffusion

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CASE STUDIES IN THERMAL ENGINEERING 2021年 26卷

作者： Khan, Shan Ali Waqas, Hassan Naqvi, Syed Muhammad Raza Shah Alghamdi, Metib Al-Mdallal, Qasem Govt Coll Univ Faisalabad Dept Math Layyah Campus Faisalabad 31200 Pakistan Peking Univ Dept Mech & Engn Sci Coll Engn Beijing 100871 Peoples R China King Khalid Univ Coll Sci Dept Math Abha 61413 Saudi Arabia King Abdulaziz Univ Dept Math Math Modelling & Appl Computat Res Grp MMAC POB 80203 Jeddah 21589 Saudi Arabia United Arab Emirates Univ Dept Math Sci POB 15551 Abu Dhabi U Arab Emirates

Purpose: The current mathematical model is developed to scrutinize the consequence of bioconvective cross diffusion flow of magnetized viscous nanofluid past multiple geometries (cone, wedge and plate) with convective boundary conditions. Together the nanoparticles and motile microorganism are incorporated into the dimensionless nonlinear differential expressions. The behavior of Cattaneo-Christov heat and mass flux is accounted for energy and concentration expressions. The influence of activation energy and thermal radiation are considered. The mathematical model is reduced into an ordinary one by using adequate similarity transformation. Buongiorno model is utilized for nanofluid (nanoliquids) analysis. Methodology/approach: The renovated dimensionless self-similarity systems are then solved numerically by utilizing shooting technique built-in function bvp4c solver with the help of commercial software Matlab. The obtained results are verified and an outstanding agreement has been found. Engineering quantities of interest are observed physically. Findings: The features of various emerging parameters against velocity distribution, thermal distribution, and solutal field of species, microorganism concentration as well as skin friction coefficient, gradient of temperature, local Sherwood number and density number of motile microorganisms are interpreted and deliberated in tabulated and graphical form. Results: The results indicate that velocity field is raises via larger Grashof number. The resultant velocity is decline via larger magnetic parameter. Larger estimation of thermal Biot number increases the heat transfer. Larger thermal relaxation parameter reduces the temperature of fluid. The concentration of nanoparticles is declines via concentration relaxation parameter. The microorganism's field is declines by varying the variations of Peclet number.

关键词： Cattaneo-Christov heat flux Bioconvection Nanoparticles Matlab shooting algorithm

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A shooting algorithm for Optimal Control Problems with Singular Arcs

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2013年第2期158卷 419-459页

作者： Aronna, M. Soledad Bonnans, J. Frederic Martinon, Pierre Univ Padua ITN Marie Curie Network SADCO Padua Italy INRIA Saclay Palaiseau France CMAP Ecole Polytech Palaiseau France

In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss-Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method.

关键词： Optimal control Singular arc Bang-singular control shooting algorithm Second order optimality condition Gauss-Newton method Stability analysis

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