Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

In this paper, we are concerned with the space-time spectral discretization of an optimal control problem governed by a fourth-order parabolic partial differential equations (PDEs) in three control constraint cases. The dual Petrov-Galerkin spectral method in time and the spectral method in space are adopted to discrete the continuous system. By means of the obtained optimality condition for the continuous system and that of its spectral discrete system, we establish a priori error estimate for the spectral approximation in details. Four numerical examples are, subsequently, executed to confirm the theoretical results. The experiment results show the high efficiency and a good precision of the space-time spectral method for this kind of problems.

[1]  Jun Zhang,et al.  Fully-discrete Spectral-Galerkin scheme with second-order time-accuracy and unconditionally energy stability for the volume-conserved phase-field lipid vesicle model , 2021, J. Comput. Appl. Math..

[2]  Bing Sun,et al.  Galerkin spectral method for a fourth-order optimal control problem with H1-norm state constraint , 2021, Comput. Math. Appl..

[3]  Yuzhu Han.  A class of fourth-order parabolic equation with arbitrary initial energy , 2018, Nonlinear Analysis: Real World Applications.

[4]  Zhong Zheng,et al.  Error estimates of the space-time spectral method for parabolic control problems , 2017, Comput. Math. Appl..

[5]  Yanping Chen,et al.  Spectral Method Approximation of Flow Optimal Control Problems withH1-Norm State Constraint , 2017 .

[6]  Juan Zhang,et al.  Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state , 2016, Comput. Math. Appl..

[7]  Yanping Chen,et al.  Galerkin Spectral Approximation of Elliptic Optimal Control Problems with $$H^1$$H1-Norm State Constraint , 2016, J. Sci. Comput..

[8]  Arnd Rösch,et al.  Finite Element Discretization of State-Constrained Elliptic Optimal Control Problems with Semilinear State Equation , 2015, SIAM J. Control. Optim..

[9]  Yanping Chen,et al.  Error estimates for spectral approximation of elliptic control problems with integral state and control constraints , 2014, Comput. Math. Appl..

[10]  Wei Gong,et al.  Error estimates for parabolic optimal control problems with control and state constraints , 2013, Comput. Optim. Appl..

[11]  Boris Vexler,et al.  A priori error estimates for space–time finite element discretization of semilinear parabolic optimal control problems , 2011, Numerische Mathematik.

[12]  Rolf Rannacher,et al.  A Priori Error Estimates for Finite Element Discretizations of Parabolic Optimization Problems with Pointwise State Constraints in Time , 2011, SIAM J. Control. Optim..

[13]  Danping Yang,et al.  International Journal of Computer Mathematics Spectral Mixed Galerkin Method for State Constrained Optimal Control Problem Governed by the First Bi- Harmonic Equation Spectral Mixed Galerkin Method for State Constrained Optimal Control Problem Governed by the First Bi-harmonic Equation , 2022 .

[14]  Yunqing Huang,et al.  A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations , 2008 .

[15]  Boris Vexler,et al.  A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems , 2019, Constrained Optimization and Optimal Control for Partial Differential Equations.

[16]  Changchun Liu.  A fourth order parabolic equation with nonlinear principal part , 2008 .

[17]  Jie Shen,et al.  Fourierization of the Legendre--Galerkin method and a new space--time spectral method , 2007 .

[18]  B. Guo,et al.  Spectral Methods and Their Applications , 1998 .

[19]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[20]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[21]  Fawang Liu,et al.  Spectral method for the two-dimensional time distributed-order diffusion-wave equation on a semi-infinite domain , 2022, J. Comput. Appl. Math..

[22]  Zuliang Lu,et al.  A posteriori error estimates of hp spectral element method for parabolic optimal control problems , 2022, AIMS Mathematics.

[23]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[24]  Nianyu Yi,et al.  A Legendre-Galerkin Spectral Method for Optimal Control Problems Governed by Elliptic Equations , 2008, SIAM J. Numer. Anal..

[25]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[26]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .