Based on the elastic theory of porous media,embedded discrete fracture model and finite volume method,and considering the micro-seepage mechanism of shale gas,a fully coupled seepage-geomechanical model suitable for f...Based on the elastic theory of porous media,embedded discrete fracture model and finite volume method,and considering the micro-seepage mechanism of shale gas,a fully coupled seepage-geomechanical model suitable for fractured shale gas reservoirs is established,the optimization method of refracturing timing is proposed,and the influencing factors of refracturing timing are analyzed based on the data from shale gas well in Fuling of Sichuan Basin.The results show that due to the depletion of formation pressure,the percentage of the maximum horizontal principal stress reversal area in the total area increases and then decreases with time.The closer the area is to the hydraulic fracture,the shorter the time for the peak of the stress reversal area percentage curve to appear,and the shorter the time for the final zero return(to the initial state).The optimum time of refracturing is affected by matrix permeability,initial stress difference and natural fracture approach angle.The larger the matrix permeability and initial stress difference is,the shorter the time for stress reversal area percentage curve to reach peak and return to the initial state,and the earlier the time to take refracturing measures.The larger the natural fracture approach angle is,the more difficult it is for stress reversal to occur near the fracture,and the earlier the optimum refracturing time is.The more likely the stress reversal occurs at the far end of the artificial fracture,the later the optimal time of refracturing is.Reservoirs with low matrix permeability have a rapid decrease in single well productivity.To ensure economic efficiency,measures such as shut-in or gas injection can be taken to restore the stress,and refracturing can be implemented in advance.展开更多
The bipolar Navier-Stokes-Poisson system （BNSP） has been used to simulate the transport of charged particles （ions and electrons for instance） under the influence of electrostatic force governed by the self-consis...The bipolar Navier-Stokes-Poisson system （BNSP） has been used to simulate the transport of charged particles （ions and electrons for instance） under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L^2 time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate （1 ＋ t）^-3/4 in L^2-norm, while the individual momentum of the charged particles converges at the optimal rate （1 ＋ t）^-1/4 which is slower than the rate （1 ＋ t）^-3/4 for the compressible Navier-Stokes equations （NS）. In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that almost counteracts the influence of the electric field so that the total density and momentum of the two carriers converges at a faster rate （1 ＋ t）^-3/4＋ε for any small constant ε 〉 0. The above estimates reveal the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.展开更多
The isentropic bipolar compressible Navier-Stokes-Poisson （BNSP） system is investigated in R3 in the present paper. The optimal time decay rate of global strong solution is established. When the regular initial data...The isentropic bipolar compressible Navier-Stokes-Poisson （BNSP） system is investigated in R3 in the present paper. The optimal time decay rate of global strong solution is established. When the regular initial data belong to the Sobolev space H l（R3） ∩ B˙ s 1,1 （R3） with l ≥ 4 and s ∈ （0, 1], it is shown that the momenta of the charged particles decay at the optimal rate （1＋t） 1 4 s 2 in L2 -norm, which is slower than the rate （1＋t） 3 4 s 2 for the compressible Navier-Stokes （NS） equations [14]. In particular, a new phenomenon on the charge transport is observed. The time decay rate of total density and momentum was both （1 ＋ t） 3 4 due to the cancellation effect from the interplay interaction of the charged particles.展开更多
We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin＇s maximum principle. In particular, we discus...We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin＇s maximum principle. In particular, we discuss optimality （or nonoptimality） of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.展开更多
In this paper, we study the non-isentropic compressible magnetohydrodynamic system with a time periodic external force in R^n. Under the condition that the optimal time decay rates are obtained by spectral analysis, w...In this paper, we study the non-isentropic compressible magnetohydrodynamic system with a time periodic external force in R^n. Under the condition that the optimal time decay rates are obtained by spectral analysis, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension n 〉 5. Our proof is based on a combination of the energy method and the contraction mapping theorem.展开更多
Significant optical engineering advances at the University of Arizona are being made for design, fabrication, and construction of next generation astronomical telescopes. This summary review paper focuses on the techn...Significant optical engineering advances at the University of Arizona are being made for design, fabrication, and construction of next generation astronomical telescopes. This summary review paper focuses on the technological advances in three key areas. First is the optical fabrication technique used for constructing next-generation telescope mirrors. Advances in ground-based telescope control and instrumentation comprise the second area of development. This includes active alignment of the laser truss-based Large Binocular Telescope(LBT) prime focus camera, the new MOBIUS modular cross-dispersion spectroscopy unit used at the prime focal plane of the LBT, and topological pupil segment optimization. Lastly, future space telescope concepts and enabling technologies are discussed. Among these, the Nautilus space observatory requires challenging alignment of segmented multi-order diffractive elements. The OASIS terahertz space telescope presents unique challenges for characterizing the inflatable primary mirror, and the Hyperion space telescope pushes the limits of high spectral resolution, far-UV spectroscopy. The Coronagraphic Debris and Exoplanet Exploring Pioneer(CDEEP) is a Small Satellite(Small Sat) mission concept for high-contrast imaging of circumstellar disks and exoplanets using vector vortex coronagraph. These advances in optical engineering technologies will help mankind to probe, explore, and understand the scientific beauty of our universe.展开更多
Polymer flooding has been proven to effectively improve oil recovery in the Bohai Oil Field. However, due to high oil viscosity and significant formation heterogeneity, it is necessary to further improve the displacem...Polymer flooding has been proven to effectively improve oil recovery in the Bohai Oil Field. However, due to high oil viscosity and significant formation heterogeneity, it is necessary to further improve the displacement effectiveness of polymer flooding in heavy oil reservoirs in the service life of offshore platforms. In this paper, the effects of the water/oil mobility ratio in heavy oil reservoirs and the dimensionless oil productivity index on polymer flooding effectiveness were studied utilizing rel- ative permeability curves. The results showed that when the water saturation was less than the value, where the water/oil mobility ratio was equal to 1, polymer flooding could effectively control the increase of fractional water flow, which meant that the upper limit of water/oil ratio suitable for polymer flooding should be the value when the water/oil mobility ratio was equal to 1. Mean while, by injecting a certain volume of water to create water channels in the reservoir, the polymer flooding would be the most effective in improving sweep efficiency, and lower the fractional flow of water to the value corresponding to △Jmax. Considering the service life of the platform and the polymer mobility control capacity, the best polymer injection timing for heavy oil reservoirs was optimized. It has been tested for reservoirs with crude oil viscosity of 123 and 70 mPa s, the optimum polymer flooding effec- tiveness could be obtained when the polymer floods were initiated at the time when the fractional flow of water were 10 % and 25 %, respectively. The injection timing range for polymer flooding was also theoretically analyzed for the Bohai Oil Field utilizing which provided methods for effectiveness. relative permeability curves, improving polymer flooding展开更多
This is a survey paper on the study of compressible Navier-Stokes-Poisson equations. The emphasis is on the long time behavior of global solutions to multi-dimensional compressible Navier-Stokes-Poisson equations, and...This is a survey paper on the study of compressible Navier-Stokes-Poisson equations. The emphasis is on the long time behavior of global solutions to multi-dimensional compressible Navier-Stokes-Poisson equations, and the optimal decay rates for both unipolar and bipolar compressible Navier-Stokes-Poisson equations are discussed.展开更多
Energy transmission efficiency in the magnetic pulse generators varies with saturated time of magnetic switch. An optimal matching time exists and depends on the compression ratio, under which, the energy transmission...Energy transmission efficiency in the magnetic pulse generators varies with saturated time of magnetic switch. An optimal matching time exists and depends on the compression ratio, under which, the energy transmission efficiency can reach approximate 100%. The equation of required magnetic core volume is obtained by taken into account the optimal matching mode. It indicates that a great reduction on the volume is feasible under the optimal matching mode. The circuit simulation code-PSPICE is also introduced to simulate a 3-stage magnetic pulse compressor, and the results are in accordance with those of equivalent circuit analyses.展开更多
Shale samples of Silurian Longmaxi Formation in the Changning area of the Sichuan Basin, SW China, were selected to carry out scanning electron microscopy, CT imaging, high-pressure mercury injection, low-temperature ...Shale samples of Silurian Longmaxi Formation in the Changning area of the Sichuan Basin, SW China, were selected to carry out scanning electron microscopy, CT imaging, high-pressure mercury injection, low-temperature nitrogen adsorption and imbibition experiments to compare the hydration characteristics of montmorillonite and illite, analyze the main factors affecting the water block removal of shale, and reveal the mechanisms of pore structure evolution during shale hydration. The hydration characteristics of shale are closely related to the composition of clay minerals, the shale with high illite content is not susceptible to hydration and thus has limited room for pore structure improvement;the shale with high montmorillonite is susceptible to hydration expansion and thus has higher potential of pore structure improvement by stimulation;the shale with high illite content has stronger imbibition in the initial stage, but insufficient diffusion ability, and thus is likely to have water block;the shale with high montmorillonite content has weaker imbibition in the initial stage but better water diffusion, so water blocking in this kind of shale can be removed to some degree;the shale reservoir has an optimal hydration time, when it is best in physical properties, but hydration time too long would cause damage to the reservoir, and the shale with high illite content has a shorter optimal hydration time;inorganic cations can inhibit the hydration of clay minerals and have stronger inhibition to illite expansion, especially K^(+);for the reservoir with high content of montmorillonite, the cation content of fracturing fluid can be lowered to promote the shale hydration;fracturing fluid with high K^(+) content can be injected into reservoirs with high illite content to suppress hydration.展开更多
The compressible non-isentropic bipolar Navier-Stokes-Poisson （BNSP） sys- tem is investigated in R3 in the present paper, and the optimal time decay rates of global strong solution are shown. For initial data being ...The compressible non-isentropic bipolar Navier-Stokes-Poisson （BNSP） sys- tem is investigated in R3 in the present paper, and the optimal time decay rates of global strong solution are shown. For initial data being a perturbation of equilibrium state in Hl（R3） （R3） for 1 〉 4 and s E （0, 1], it is shown that the density and temperature for each charged particle （like electron or ion） decay at the same optimal rate （1 ＋ t）-3/4, but the momentum for each particle decays at the optimal rate （1 ＋ t）-1/4-3/2 which is slower than the rate （1 ＋ t）-3/4-3/2 for the compressible Navier-Stokes （NS） equations [19] for same initial data. However, the total momentum tends to the constant state at the rate （1 ＋t）-3/4 as well, due to the interplay interaction of charge particles which counteracts the influence of electric field.展开更多
The 7-DOF model of a full vehicle with an active suspension is developed in this paper.The model is written into the state equation style.Actuator forces are treated as inputs in the state equations.Based on the basic...The 7-DOF model of a full vehicle with an active suspension is developed in this paper.The model is written into the state equation style.Actuator forces are treated as inputs in the state equations.Based on the basic optimal control theory,the optimal gains for the control system are figured out.So an optimal controller is developed and implemented using Matlab/Simulink,where the Riccati equation with coupling terms is deduced using the Hamilton equation.The all state feedback is chosen for the controller.The gains for all vehicle variables are traded off so that majority of indexes were up to optimal.The active suspension with optimal control is simulated in frequency domain and time domain separately,and compared with a passive suspension.Throughout all the simulation results,the optimal controller developed in this paper works well in the majority of instances.In all,the comfort and ride performance of the vehicle are improved under the active suspension with optimal control.展开更多
In this paper, we consider the Vlasov-Maxwell-Fokker-Planck system with relativistic transport in the whole space. The global solutions to this system near the relativistic Maxwellian are constructed and the optimal t...In this paper, we consider the Vlasov-Maxwell-Fokker-Planck system with relativistic transport in the whole space. The global solutions to this system near the relativistic Maxwellian are constructed and the optimal time decay rate of global solutions are also obtained by an approach by combining the compensating function and energy method.展开更多
This article presents an equivalence theorem for three different kinds of optimal control problems, which are optimal target control problems, optimal norm control problems, and optimal time control problems. Controll...This article presents an equivalence theorem for three different kinds of optimal control problems, which are optimal target control problems, optimal norm control problems, and optimal time control problems. Controlled systems in this study are internally controlled Stokes equations.展开更多
基金Supported by National Natural Science Foundation Joint Fund Project(U21B2071)National Natural Science Foundation of China(52174033)National Natural Science Youth Foundation of China(52304041).
文摘Based on the elastic theory of porous media,embedded discrete fracture model and finite volume method,and considering the micro-seepage mechanism of shale gas,a fully coupled seepage-geomechanical model suitable for fractured shale gas reservoirs is established,the optimization method of refracturing timing is proposed,and the influencing factors of refracturing timing are analyzed based on the data from shale gas well in Fuling of Sichuan Basin.The results show that due to the depletion of formation pressure,the percentage of the maximum horizontal principal stress reversal area in the total area increases and then decreases with time.The closer the area is to the hydraulic fracture,the shorter the time for the peak of the stress reversal area percentage curve to appear,and the shorter the time for the final zero return(to the initial state).The optimum time of refracturing is affected by matrix permeability,initial stress difference and natural fracture approach angle.The larger the matrix permeability and initial stress difference is,the shorter the time for stress reversal area percentage curve to reach peak and return to the initial state,and the earlier the time to take refracturing measures.The larger the natural fracture approach angle is,the more difficult it is for stress reversal to occur near the fracture,and the earlier the optimum refracturing time is.The more likely the stress reversal occurs at the far end of the artificial fracture,the later the optimal time of refracturing is.Reservoirs with low matrix permeability have a rapid decrease in single well productivity.To ensure economic efficiency,measures such as shut-in or gas injection can be taken to restore the stress,and refracturing can be implemented in advance.
基金The research of the first author was partially supported by the NNSFC No.10871134the NCET support of the Ministry of Education of China+4 种基金the Huo Ying Dong Fund No.111033the Chuang Xin Ren Cai Project of Beijing Municipal Commission of Education #PHR201006107the Instituteof Mathematics and Interdisciplinary Science at CNUThe research of the second author was supported by the General Research Fund of Hong Kong (CityU 103109)the National Natural Science Foundation of China,10871082
文摘The bipolar Navier-Stokes-Poisson system （BNSP） has been used to simulate the transport of charged particles （ions and electrons for instance） under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L^2 time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate （1 ＋ t）^-3/4 in L^2-norm, while the individual momentum of the charged particles converges at the optimal rate （1 ＋ t）^-1/4 which is slower than the rate （1 ＋ t）^-3/4 for the compressible Navier-Stokes equations （NS）. In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that almost counteracts the influence of the electric field so that the total density and momentum of the two carriers converges at a faster rate （1 ＋ t）^-3/4＋ε for any small constant ε 〉 0. The above estimates reveal the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.
基金supported by NSFC (10872004)National Basic Research Program of China (2010CB731500)the China Ministry of Education (200800010013)
文摘The isentropic bipolar compressible Navier-Stokes-Poisson （BNSP） system is investigated in R3 in the present paper. The optimal time decay rate of global strong solution is established. When the regular initial data belong to the Sobolev space H l（R3） ∩ B˙ s 1,1 （R3） with l ≥ 4 and s ∈ （0, 1], it is shown that the momenta of the charged particles decay at the optimal rate （1＋t） 1 4 s 2 in L2 -norm, which is slower than the rate （1＋t） 3 4 s 2 for the compressible Navier-Stokes （NS） equations [14]. In particular, a new phenomenon on the charge transport is observed. The time decay rate of total density and momentum was both （1 ＋ t） 3 4 due to the cancellation effect from the interplay interaction of the charged particles.
文摘We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin＇s maximum principle. In particular, we discuss optimality （or nonoptimality） of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.
基金Supported by National Natural Science Foundation of China(11271305)
文摘In this paper, we study the non-isentropic compressible magnetohydrodynamic system with a time periodic external force in R^n. Under the condition that the optimal time decay rates are obtained by spectral analysis, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension n 〉 5. Our proof is based on a combination of the energy method and the contraction mapping theorem.
基金the Gordon and Betty Moore Foundation for their financial support of the development of the MODElens and its enabling alignment technologiesthe II-VI Foundation Block-Gift,Technology Research Initiative Fund Optics/Imaging Program。
文摘Significant optical engineering advances at the University of Arizona are being made for design, fabrication, and construction of next generation astronomical telescopes. This summary review paper focuses on the technological advances in three key areas. First is the optical fabrication technique used for constructing next-generation telescope mirrors. Advances in ground-based telescope control and instrumentation comprise the second area of development. This includes active alignment of the laser truss-based Large Binocular Telescope(LBT) prime focus camera, the new MOBIUS modular cross-dispersion spectroscopy unit used at the prime focal plane of the LBT, and topological pupil segment optimization. Lastly, future space telescope concepts and enabling technologies are discussed. Among these, the Nautilus space observatory requires challenging alignment of segmented multi-order diffractive elements. The OASIS terahertz space telescope presents unique challenges for characterizing the inflatable primary mirror, and the Hyperion space telescope pushes the limits of high spectral resolution, far-UV spectroscopy. The Coronagraphic Debris and Exoplanet Exploring Pioneer(CDEEP) is a Small Satellite(Small Sat) mission concept for high-contrast imaging of circumstellar disks and exoplanets using vector vortex coronagraph. These advances in optical engineering technologies will help mankind to probe, explore, and understand the scientific beauty of our universe.
基金supported by Open Fund (CRI2012RCPS0152CN) of State Key Laboratory of Offshore Oil Exploitationthe National Science and Technology Major Project (2011ZX05024-004-01)
文摘Polymer flooding has been proven to effectively improve oil recovery in the Bohai Oil Field. However, due to high oil viscosity and significant formation heterogeneity, it is necessary to further improve the displacement effectiveness of polymer flooding in heavy oil reservoirs in the service life of offshore platforms. In this paper, the effects of the water/oil mobility ratio in heavy oil reservoirs and the dimensionless oil productivity index on polymer flooding effectiveness were studied utilizing rel- ative permeability curves. The results showed that when the water saturation was less than the value, where the water/oil mobility ratio was equal to 1, polymer flooding could effectively control the increase of fractional water flow, which meant that the upper limit of water/oil ratio suitable for polymer flooding should be the value when the water/oil mobility ratio was equal to 1. Mean while, by injecting a certain volume of water to create water channels in the reservoir, the polymer flooding would be the most effective in improving sweep efficiency, and lower the fractional flow of water to the value corresponding to △Jmax. Considering the service life of the platform and the polymer mobility control capacity, the best polymer injection timing for heavy oil reservoirs was optimized. It has been tested for reservoirs with crude oil viscosity of 123 and 70 mPa s, the optimum polymer flooding effec- tiveness could be obtained when the polymer floods were initiated at the time when the fractional flow of water were 10 % and 25 %, respectively. The injection timing range for polymer flooding was also theoretically analyzed for the Bohai Oil Field utilizing which provided methods for effectiveness. relative permeability curves, improving polymer flooding
基金supported by the NSFC (10871134),supported by the NSFC (10871134, 10771008)the NCET support of the Ministry of Education of China+1 种基金the Huo Ying Dong Fund (111033)the funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201006107)
文摘This is a survey paper on the study of compressible Navier-Stokes-Poisson equations. The emphasis is on the long time behavior of global solutions to multi-dimensional compressible Navier-Stokes-Poisson equations, and the optimal decay rates for both unipolar and bipolar compressible Navier-Stokes-Poisson equations are discussed.
基金supported by the High Technology Resesarch Development Project of China (863)
文摘Energy transmission efficiency in the magnetic pulse generators varies with saturated time of magnetic switch. An optimal matching time exists and depends on the compression ratio, under which, the energy transmission efficiency can reach approximate 100%. The equation of required magnetic core volume is obtained by taken into account the optimal matching mode. It indicates that a great reduction on the volume is feasible under the optimal matching mode. The circuit simulation code-PSPICE is also introduced to simulate a 3-stage magnetic pulse compressor, and the results are in accordance with those of equivalent circuit analyses.
基金Supported by the Science and Technology Planning Project of Sichuan Province,China(2020YJ0135)National Natural Science Foundation of China(51874250)。
文摘Shale samples of Silurian Longmaxi Formation in the Changning area of the Sichuan Basin, SW China, were selected to carry out scanning electron microscopy, CT imaging, high-pressure mercury injection, low-temperature nitrogen adsorption and imbibition experiments to compare the hydration characteristics of montmorillonite and illite, analyze the main factors affecting the water block removal of shale, and reveal the mechanisms of pore structure evolution during shale hydration. The hydration characteristics of shale are closely related to the composition of clay minerals, the shale with high illite content is not susceptible to hydration and thus has limited room for pore structure improvement;the shale with high montmorillonite is susceptible to hydration expansion and thus has higher potential of pore structure improvement by stimulation;the shale with high illite content has stronger imbibition in the initial stage, but insufficient diffusion ability, and thus is likely to have water block;the shale with high montmorillonite content has weaker imbibition in the initial stage but better water diffusion, so water blocking in this kind of shale can be removed to some degree;the shale reservoir has an optimal hydration time, when it is best in physical properties, but hydration time too long would cause damage to the reservoir, and the shale with high illite content has a shorter optimal hydration time;inorganic cations can inhibit the hydration of clay minerals and have stronger inhibition to illite expansion, especially K^(+);for the reservoir with high content of montmorillonite, the cation content of fracturing fluid can be lowered to promote the shale hydration;fracturing fluid with high K^(+) content can be injected into reservoirs with high illite content to suppress hydration.
基金supported by the NSFC (10871134)supported by the NSFC (10871134,10910401059)+1 种基金the funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201006107)supported by the General Research Fund of Hong Kong,City Univ.103108
文摘The compressible non-isentropic bipolar Navier-Stokes-Poisson （BNSP） sys- tem is investigated in R3 in the present paper, and the optimal time decay rates of global strong solution are shown. For initial data being a perturbation of equilibrium state in Hl（R3） （R3） for 1 〉 4 and s E （0, 1], it is shown that the density and temperature for each charged particle （like electron or ion） decay at the same optimal rate （1 ＋ t）-3/4, but the momentum for each particle decays at the optimal rate （1 ＋ t）-1/4-3/2 which is slower than the rate （1 ＋ t）-3/4-3/2 for the compressible Navier-Stokes （NS） equations [19] for same initial data. However, the total momentum tends to the constant state at the rate （1 ＋t）-3/4 as well, due to the interplay interaction of charge particles which counteracts the influence of electric field.
文摘The 7-DOF model of a full vehicle with an active suspension is developed in this paper.The model is written into the state equation style.Actuator forces are treated as inputs in the state equations.Based on the basic optimal control theory,the optimal gains for the control system are figured out.So an optimal controller is developed and implemented using Matlab/Simulink,where the Riccati equation with coupling terms is deduced using the Hamilton equation.The all state feedback is chosen for the controller.The gains for all vehicle variables are traded off so that majority of indexes were up to optimal.The active suspension with optimal control is simulated in frequency domain and time domain separately,and compared with a passive suspension.Throughout all the simulation results,the optimal controller developed in this paper works well in the majority of instances.In all,the comfort and ride performance of the vehicle are improved under the active suspension with optimal control.
基金supported partially by the NNSFC Grant(11371151)the Scientific Research Foundation of Graduate School of South China Normal University
文摘In this paper, we consider the Vlasov-Maxwell-Fokker-Planck system with relativistic transport in the whole space. The global solutions to this system near the relativistic Maxwellian are constructed and the optimal time decay rate of global solutions are also obtained by an approach by combining the compensating function and energy method.
基金partially supported by the National Natural Science Foundation of China under grants 11371285 and 91130022
文摘This article presents an equivalence theorem for three different kinds of optimal control problems, which are optimal target control problems, optimal norm control problems, and optimal time control problems. Controlled systems in this study are internally controlled Stokes equations.
