In this paper,we consider parabolic distributed control problems with cost functional of pointwise observation type either in space or in time.First,we show the well-posedness of the optimization problems and derive t...In this paper,we consider parabolic distributed control problems with cost functional of pointwise observation type either in space or in time.First,we show the well-posedness of the optimization problems and derive the first order optimality systems,where the adjoint state can be expressed as the linear combination of solutions to two backward parabolic equations that involve the Dirac delta distribution as source either in space or in time.Second,we use a space-time finite element method to discretize the control problems,where the state variable is approximated by piecewise constant functions in time and continuous piecewise linear polynomials in space,and the control variable is discretized by following the variational discretization concept.We obtain a priori error estimates for the control and state variables with order O(k 12+h)up to a logarithmic factor under the L 2-norm.Finally,we perform several numerical experiments to support our theoretical results.展开更多
This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems.Based on elliptic reconstruction approach introduced earlier by Makrid...This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems.Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto[25],a residual based a posteriori error estimators for the state,co-state and control variables are derived.The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements,whereas the piecewise constant functions are employed for the control variable.The temporal discretization is based on the backward Euler method.We derive a posteriori error estimates for the state,co-state and control variables in the L^(∞)(0,T;L^(2)(Ω))-norm.Finally,a numerical experiment is performed to illustrate the performance of the derived estimators.展开更多
In this paper,the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation.The optimal convergent order O(h^(2)+k^(2))is obtained...In this paper,the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation.The optimal convergent order O(h^(2)+k^(2))is obtained for the numerical solution in a discrete L^(2)-norm.A numerical experiment is presented to test the theoretical result.展开更多
Control constrained parabolic optimal control problems are generally challenging,from either theoretical analysis or algorithmic design perspectives.Conceptually,the well-known alternating direction method of multipli...Control constrained parabolic optimal control problems are generally challenging,from either theoretical analysis or algorithmic design perspectives.Conceptually,the well-known alternating direction method of multipliers(ADMM)can be directly applied to such problems.An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations.At each iteration of the ADMM,the main computation is for solving an unconstrained parabolic optimal control subproblem.Because of its inevitably high dimensionality after space-time discretization,the parabolicoptimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required.It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations,and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm.To implement the ADMM efficiently,we propose an inexactness criterion that is independent of the mesh size of the involved discretization,and that can be performed automatically with no need to set empirically perceived constant accuracy a priori.The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly,yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously.Efficiency of this ADMM implementation is promisingly validated by some numerical results.Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations,hyperbolic equations,convection-diffusion equations,and fractional parabolic equations.展开更多
This paper considers the variational discretization for the constrained optimal control problem governed by linear parabolic equations.The state and co-state are approximated by RaviartThomas mixed finite element spac...This paper considers the variational discretization for the constrained optimal control problem governed by linear parabolic equations.The state and co-state are approximated by RaviartThomas mixed finite element spaces,and the authors do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control.A priori error estimates are derived for the state,the co-state,and the control.Some numerical examples are presented to confirm the theoretical investigations.展开更多
基金supported in part by the Strategic Priority Research Program of Chi-nese Academy of Sciences(Grant No.XDB 41000000)the National Key Basic Research Program(Grant No.2018YFB0704304)+1 种基金the National Natural Science Foundation of China(Grants No.12071468,11671391)Xiaoping Xie was supported in part by the National Natural Science Foundation of China(Grants No.12171340,11771312).
文摘In this paper,we consider parabolic distributed control problems with cost functional of pointwise observation type either in space or in time.First,we show the well-posedness of the optimization problems and derive the first order optimality systems,where the adjoint state can be expressed as the linear combination of solutions to two backward parabolic equations that involve the Dirac delta distribution as source either in space or in time.Second,we use a space-time finite element method to discretize the control problems,where the state variable is approximated by piecewise constant functions in time and continuous piecewise linear polynomials in space,and the control variable is discretized by following the variational discretization concept.We obtain a priori error estimates for the control and state variables with order O(k 12+h)up to a logarithmic factor under the L 2-norm.Finally,we perform several numerical experiments to support our theoretical results.
文摘This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems.Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto[25],a residual based a posteriori error estimators for the state,co-state and control variables are derived.The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements,whereas the piecewise constant functions are employed for the control variable.The temporal discretization is based on the backward Euler method.We derive a posteriori error estimates for the state,co-state and control variables in the L^(∞)(0,T;L^(2)(Ω))-norm.Finally,a numerical experiment is performed to illustrate the performance of the derived estimators.
基金This work is supported by National Natural Science Foundation of China(Grant Nos.11271145 and 11031006)Foundation of Guizhou Science and Technology Department(Grant No.[2011]2098)+1 种基金Foundation for Talent Introduction of Guangdong Provincial University,Specialized Research Fund for the Doctoral Programof Higher Education(Grant No.20114407110009)the Project of Department of Education of Guangdong Province(Grant No.2012KJCX0036).
文摘In this paper,the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation.The optimal convergent order O(h^(2)+k^(2))is obtained for the numerical solution in a discrete L^(2)-norm.A numerical experiment is presented to test the theoretical result.
基金supported by the seed fund for basic research at The University of Hong Kong(project No.201807159005)a General Research Fund from Hong Kong Research Grants Council。
文摘Control constrained parabolic optimal control problems are generally challenging,from either theoretical analysis or algorithmic design perspectives.Conceptually,the well-known alternating direction method of multipliers(ADMM)can be directly applied to such problems.An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations.At each iteration of the ADMM,the main computation is for solving an unconstrained parabolic optimal control subproblem.Because of its inevitably high dimensionality after space-time discretization,the parabolicoptimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required.It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations,and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm.To implement the ADMM efficiently,we propose an inexactness criterion that is independent of the mesh size of the involved discretization,and that can be performed automatically with no need to set empirically perceived constant accuracy a priori.The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly,yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously.Efficiency of this ADMM implementation is promisingly validated by some numerical results.Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations,hyperbolic equations,convection-diffusion equations,and fractional parabolic equations.
基金supported by the National Natural Science Foundation of Chinaunder Grant No.11271145Foundation for Talent Introduction of Guangdong Provincial University+3 种基金Fund for the Doctoral Program of Higher Education under Grant No.20114407110009the Project of Department of Education of Guangdong Province under Grant No.2012KJCX0036supported by Hunan Education Department Key Project 10A117the National Natural Science Foundation of China under Grant Nos.11126304 and 11201397
文摘This paper considers the variational discretization for the constrained optimal control problem governed by linear parabolic equations.The state and co-state are approximated by RaviartThomas mixed finite element spaces,and the authors do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control.A priori error estimates are derived for the state,the co-state,and the control.Some numerical examples are presented to confirm the theoretical investigations.
