In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and &l...In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and <em>L</em>1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.展开更多
The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem,due to the global properties of fractional...The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem,due to the global properties of fractional differential operators.In this paper,we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable.By the proposed first-order optimality condition consisting of a Lagrange multiplier,we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations.Furthermore,a priori error estimates for state,adjoint state and control variables are discussed in details.Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.展开更多
The main aim of this study is to prove that a class of fractional Burgers’equations has a unique solution under some special conditions.Then it is demonstrated that an optimal control problem for this class of fracti...The main aim of this study is to prove that a class of fractional Burgers’equations has a unique solution under some special conditions.Then it is demonstrated that an optimal control problem for this class of fractional Burgers’equations has at least one optimal solution.展开更多
In this paper, optimal control for a novel West Nile virus （WNV） model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations （FDEs）, where the...In this paper, optimal control for a novel West Nile virus （WNV） model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations （FDEs）, where the fractional derivative is defined in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pon- tryagin maximum principle. Two numerical methods are used to study the fractional- order optimal control problem. The methods are, the iterative optimal control method （OCM） and the generalized Euler method （GEM）. Positivity, boundedness and conver- gence of the IOCM are studied. Comparative studies between the proposed methods are implemented, it is found that the IOCM is better than the GEM.展开更多
Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different ...Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.展开更多
In that paper,we new study has been carried out on previous studies of one of the most important mathematical models that describe the global economic movement,and that is described as a non-linear fractional financia...In that paper,we new study has been carried out on previous studies of one of the most important mathematical models that describe the global economic movement,and that is described as a non-linear fractional financial model of awareness,where the studies are represented at the steps following:One:The schematic of the model is suggested.Two:The disease-free equilibrium point(DFE)and the stability of the equilibrium point are discussed.Three:The stability of the model is fulfilled by drawing the Lyapunov exponents and Poincare map.Fourth:The existence of uniformly stable solutions have discussed.Five:The Caputo is described as the fractional derivative.Six:Fractional optimal control for NFFMA is discussed by clarifying the fractional optimal control through drawing before and after control.Seven:Reduced differential transform method(RDTM)and Sumudu Decomposition Method(SDM)are used to take the resolution of an NFFMA.Finally,we display that SDM and RDTM are highly identical.展开更多
In this paper,we propose a fractional-order and two-patch model of tuberculosis(TB)epidemic,in which susceptible,slow latent,fast latent and infectious individuals can travel freely between the patches,but not under t...In this paper,we propose a fractional-order and two-patch model of tuberculosis(TB)epidemic,in which susceptible,slow latent,fast latent and infectious individuals can travel freely between the patches,but not under treatment infected individuals,due to medical reasons.We obtain the basic reproduction number Ro for the model and extend the classical LaSalle's invariance principle for fractional differential equations.We show that if R0<1,the disease-free equilibrium(DFE)is locally and globally asymptotically stable.If Ro>l,we obtain sufficient conditions under which the endernic equilibrium is unique and globally asymptotically stable.We extend the model by inclusion the time-dependent controls(effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches),and formulate a fractional optimal control problem to reduce the spread of the disease.The numerical results show that the use of all controls has the most impact on disease control,and decreases the size of all infected compartments,but increases the size of susceptible compartment in both patches.We,also,investigate the impact of the fractional derivative order a on the values of the controls(0.7≤α≤1).The results show that the maximum levels of effective treatment controls in both patches increase when a is reduced from l,while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when o limits to 1.展开更多
文摘In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and <em>L</em>1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.
基金This work was partly supported by National Natural Science Foundation of China(Grant Nos.:12101283,12271233 and 12171287)Natural Science Foundation of Shandong Province(Grant Nos.:ZR2019YQ05,2019KJI003,and ZR2016JL004).
文摘The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem,due to the global properties of fractional differential operators.In this paper,we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable.By the proposed first-order optimality condition consisting of a Lagrange multiplier,we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations.Furthermore,a priori error estimates for state,adjoint state and control variables are discussed in details.Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.
文摘The main aim of this study is to prove that a class of fractional Burgers’equations has a unique solution under some special conditions.Then it is demonstrated that an optimal control problem for this class of fractional Burgers’equations has at least one optimal solution.
文摘In this paper, optimal control for a novel West Nile virus （WNV） model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations （FDEs）, where the fractional derivative is defined in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pon- tryagin maximum principle. Two numerical methods are used to study the fractional- order optimal control problem. The methods are, the iterative optimal control method （OCM） and the generalized Euler method （GEM）. Positivity, boundedness and conver- gence of the IOCM are studied. Comparative studies between the proposed methods are implemented, it is found that the IOCM is better than the GEM.
基金supported by CNPq and CAPES(Brazilian research funding agencies)Portuguese funds through the Center for Research and Development in Mathematics and Applications(CIDMA)the Portuguese Foundation for Science and Technology(FCT),within project UID/MAT/04106/2013
文摘Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.
文摘In that paper,we new study has been carried out on previous studies of one of the most important mathematical models that describe the global economic movement,and that is described as a non-linear fractional financial model of awareness,where the studies are represented at the steps following:One:The schematic of the model is suggested.Two:The disease-free equilibrium point(DFE)and the stability of the equilibrium point are discussed.Three:The stability of the model is fulfilled by drawing the Lyapunov exponents and Poincare map.Fourth:The existence of uniformly stable solutions have discussed.Five:The Caputo is described as the fractional derivative.Six:Fractional optimal control for NFFMA is discussed by clarifying the fractional optimal control through drawing before and after control.Seven:Reduced differential transform method(RDTM)and Sumudu Decomposition Method(SDM)are used to take the resolution of an NFFMA.Finally,we display that SDM and RDTM are highly identical.
文摘In this paper,we propose a fractional-order and two-patch model of tuberculosis(TB)epidemic,in which susceptible,slow latent,fast latent and infectious individuals can travel freely between the patches,but not under treatment infected individuals,due to medical reasons.We obtain the basic reproduction number Ro for the model and extend the classical LaSalle's invariance principle for fractional differential equations.We show that if R0<1,the disease-free equilibrium(DFE)is locally and globally asymptotically stable.If Ro>l,we obtain sufficient conditions under which the endernic equilibrium is unique and globally asymptotically stable.We extend the model by inclusion the time-dependent controls(effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches),and formulate a fractional optimal control problem to reduce the spread of the disease.The numerical results show that the use of all controls has the most impact on disease control,and decreases the size of all infected compartments,but increases the size of susceptible compartment in both patches.We,also,investigate the impact of the fractional derivative order a on the values of the controls(0.7≤α≤1).The results show that the maximum levels of effective treatment controls in both patches increase when a is reduced from l,while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when o limits to 1.
