抛物系统时间最优控制问题有限维逼近的误差估计
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摘要
论文研究了一种抽象抛物系统时间最优控制问题的有限维逼近的误差估计.基于抽象空间到有限维空间的正交投影逼近,文章设计了有限维逼近问题.证明了逼近问题的最优时间和最优控制的收敛性,得到了最优时间的误差估计.最后给出了有限元逼近和谱逼近的应用例子.
This paper is devoted to the study of error estimates of finite dimensional approximations for the time optimal control problems of abstract parabolic systems.Firstly, based on the orthogonal projection approximation from abstract space to finite dimensional space, we design the finite dimensional approximation problem.Then we derive the convergence of optimal time and optimal control. Moreover, the error estimate for the optimal time is obtained. Finally, we give some application examples for the finite element approximation and spectral method approximation.
This paper is devoted to the study of error estimates of finite dimensional approximations for the time optimal control problems of abstract parabolic systems.Firstly, based on the orthogonal projection approximation from abstract space to finite dimensional space, we design the finite dimensional approximation problem.Then we derive the convergence of optimal time and optimal control. Moreover, the error estimate for the optimal time is obtained. Finally, we give some application examples for the finite element approximation and spectral method approximation.
引文
[1]Barbu V. Analysis and Control of Nonlinear Infinite Dimensional Systems. Boston:Academic Press, 1993.
[2]Fattorini H O. Infinite Dimensional Optimization and Control Theory. Cambridge:Cambridge University Press, 1999.
[3]Lasalle J P. The Time Optimal Control Problem, Contributions to the Theory of Nonlinear Oscillations. Princeton:Princeton University Press, 1960.
[4]Lions J L. Optimal Control of Systems Governed by Partial Differential Equations. Berlin:Springer-Verlag, 1971.
[5]Li X J, Yong J M. Optimal Control Theory for Infinite Dimensional Systems. Boston:Birkhauser Boston, 1995.
[6]Phung K D, Wang L J, Zhang C. Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. H. Poincare Anal. Non Lineaire, 2014, 31(3):477-499.
[7]Wang G S. L~∞null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim., 2008, 47(4):1701-1720.
[8]Wang G S, Wang L J. The bang-bang principle of time optimal controls for the heat equation with internal controls. Systems Control Lett., 2007, 56(11):709-713.
[9]Wang L J, Wang G S. The optimal time control of a phase-field system. SIAM J. Control Optim.,2003, 42(4):1483-1508.
[10]Wang G S, Zuazua E. On the equivalence of minimal time and minimal norm controls for internally controlled heat equations. SIAM J. Control Optim., 2012, 50(5):2938-2958.
[11]Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods in Fluid Dynamics. Berlin:Springer-Verlag, 1998.
[12]Ciarlet P G. The Finite Element Methods for Elliptic Problems. Amsterdam:North-Holland,1978.
[13]Thomee V. Galerkin Finite Element Methods for Parabolic Problems, Berlin:Springer-Verlag,2006.
[14]Liu W B, Yan N N. Adaptive Finite Element Methods for Optimal Control Governed by PDEs.Beijing:Science Press, 2008.
[15]Knowles G. Finite element approximation of parabolic time optimal control problems. SIAM J.Control Optim., 1982, 20(3):414-427.
[16]Lasiecka I. Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions. SIAM J. Control Optim., 1984, 22(3):477-500.
[17]Wang G S, Zheng G J. An approach to the optimal time for a time optimal control problem of an internally controlled heat equation. SIAM J. Control Optim., 2012, 50(2):601-628.
[18]Huang J F, Yu X, Liu K S. Semidiscrete finite element approximation of time optimal control problems for semilinear heat equations with nonsmooth initial data. Systems&Control Letters,2018, 116:32-40.
[19]Tucsnak M, Valein J, Wu C T. Numerical approximation of some time optimal control problems.Control Conference, IEEE, 2015.
[20]Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York:Springer-Verlag, 1983.
[21]Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations:Continuous and Approximation Theories, Abstract Parabolic Systems(Vol. 1). Cambridge:Cambridge University Press, 2000.
[22]Fujita H, Saito N, Suzuki T. Operator Theory and Numerical Methods. Amsterdam:NorthHolland Publishing Co., 2001.
[23]Lin F H. A uniqueness theorem for parabolic equation. Comm. Pure Appl. Math., 1990, 43:127-136.
[24]Phung K D, Wang G S. Quantitative unique continuation for the semilinear heat equation in a convex domain. J. Funct. Anal., 2010, 259:1230-1247.
[25]Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. New York:Springer-Verlag, 1994.
[26]Quarteroni A, Valli A. Numerical Approximation of Partial Differential Equations. Berlin:Springer-Verlag, 1994.
[2]Fattorini H O. Infinite Dimensional Optimization and Control Theory. Cambridge:Cambridge University Press, 1999.
[3]Lasalle J P. The Time Optimal Control Problem, Contributions to the Theory of Nonlinear Oscillations. Princeton:Princeton University Press, 1960.
[4]Lions J L. Optimal Control of Systems Governed by Partial Differential Equations. Berlin:Springer-Verlag, 1971.
[5]Li X J, Yong J M. Optimal Control Theory for Infinite Dimensional Systems. Boston:Birkhauser Boston, 1995.
[6]Phung K D, Wang L J, Zhang C. Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. H. Poincare Anal. Non Lineaire, 2014, 31(3):477-499.
[7]Wang G S. L~∞null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim., 2008, 47(4):1701-1720.
[8]Wang G S, Wang L J. The bang-bang principle of time optimal controls for the heat equation with internal controls. Systems Control Lett., 2007, 56(11):709-713.
[9]Wang L J, Wang G S. The optimal time control of a phase-field system. SIAM J. Control Optim.,2003, 42(4):1483-1508.
[10]Wang G S, Zuazua E. On the equivalence of minimal time and minimal norm controls for internally controlled heat equations. SIAM J. Control Optim., 2012, 50(5):2938-2958.
[11]Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods in Fluid Dynamics. Berlin:Springer-Verlag, 1998.
[12]Ciarlet P G. The Finite Element Methods for Elliptic Problems. Amsterdam:North-Holland,1978.
[13]Thomee V. Galerkin Finite Element Methods for Parabolic Problems, Berlin:Springer-Verlag,2006.
[14]Liu W B, Yan N N. Adaptive Finite Element Methods for Optimal Control Governed by PDEs.Beijing:Science Press, 2008.
[15]Knowles G. Finite element approximation of parabolic time optimal control problems. SIAM J.Control Optim., 1982, 20(3):414-427.
[16]Lasiecka I. Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions. SIAM J. Control Optim., 1984, 22(3):477-500.
[17]Wang G S, Zheng G J. An approach to the optimal time for a time optimal control problem of an internally controlled heat equation. SIAM J. Control Optim., 2012, 50(2):601-628.
[18]Huang J F, Yu X, Liu K S. Semidiscrete finite element approximation of time optimal control problems for semilinear heat equations with nonsmooth initial data. Systems&Control Letters,2018, 116:32-40.
[19]Tucsnak M, Valein J, Wu C T. Numerical approximation of some time optimal control problems.Control Conference, IEEE, 2015.
[20]Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York:Springer-Verlag, 1983.
[21]Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations:Continuous and Approximation Theories, Abstract Parabolic Systems(Vol. 1). Cambridge:Cambridge University Press, 2000.
[22]Fujita H, Saito N, Suzuki T. Operator Theory and Numerical Methods. Amsterdam:NorthHolland Publishing Co., 2001.
[23]Lin F H. A uniqueness theorem for parabolic equation. Comm. Pure Appl. Math., 1990, 43:127-136.
[24]Phung K D, Wang G S. Quantitative unique continuation for the semilinear heat equation in a convex domain. J. Funct. Anal., 2010, 259:1230-1247.
[25]Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. New York:Springer-Verlag, 1994.
[26]Quarteroni A, Valli A. Numerical Approximation of Partial Differential Equations. Berlin:Springer-Verlag, 1994.