A numerical approach to an optimal boundary control of the viscous Burgers’ equation

The dynamics of the forced Burgers’ equation subject to Dirichlet boundary conditions using boundary control is analyzed with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient method is suggested to solve t...

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Vydáno v:Applied mathematics and computation Ročník 210; číslo 1; s. 126 - 135
Hlavní autoři: Kucuk, Ismail, Sadek, Ibrahim
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier Inc 01.04.2009
Elsevier
Témata:
Boundary control
Burgers’ equation
Calculus of variations and optimal control
Control parametrization
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematics
Modal expansion technique
Nonlinear partial differential equation
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in mathematical programming, optimization and calculus of variations
Optimal control
Partial differential equations
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Modal expansion technique
Burgers’ equation
Boundary control
Control parametrization
Nonlinear partial differential equation
Optimal control
Differential equation
Optimization method
Dynamical system
Boundary condition
Parameterization
Partial differential equation
Non linear equation
Mathematical programming
Initial value problem
Burgers equation
Runge Kutta method
Fourier series
Fourier coefficient
Extrapolation
Numerical analysis
Boundary value problem
Multistep method
Applied mathematics
Optimal solution
Numerical simulation
Burgers' equation
ISSN:0096-3003, 1873-5649
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Shrnutí:The dynamics of the forced Burgers’ equation subject to Dirichlet boundary conditions using boundary control is analyzed with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient method is suggested to solve the optimal boundary control of the Burgers’ equation. The solution method involves the transformation of the original problem into one with homogeneous boundary conditions. This modifies the problem from one in which there are boundary controls to one in which there are distributed controls. The Modal space technique is applied on the distributed controls of the forced Burgers’ equation to generate a low-dimensional dynamical systems. The time-variant controls are approximated by a finite term of the Fourier series whose coefficients and frequencies giving optimal solutions are to be determined, thereby converting the optimal control problem into mathematical programming problem. The approximate solution space based on the control parameterization is obtained by using the Runge–Kutta method. Numerical simulations for the boundary controls are presented for various target functions to assess the efficiency of the proposed method.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2008.12.056