In this paper, two approaches are used to solve a class of the distributed optimal control problems defined on rectangular domains. In the first approach, a meshless method for solving the distributed optimal control problems is proposed; this method is based on separable representation of state and control functions. The approximation process is done in two fundamental stages. First, the partial differential equation (PDE) constraint is transformed to an algebraic system by weighted residual method, and then, Bezier curves are used to approximate the action of control and state. In the second approach, the Bernstein polynomials together with Galerkin method are utilized to solve partial differential equation coupled system, which is a necessary and sufficient condition for the main problem. The proposed techniques are easy to implement, efficient, and yield accurate results. Numerical examples are provided to illustrate the flexibility and efficiency of the proposed method.
Issue Section:
Research Papers
References
1.
Rad
, J. A.
, Kazem
, S.
, and Parand
, K.
, 2014
, “Optimal Control of a Parabolic Distributed Parameter System Via Radial Basis Functions
,” Commun. Nonlinear Sci. Numer. Simul.
, 19
(8
), pp. 2559
–2567
.2.
Pearson
, J. W.
, 2013
, “A Radial Basis Function Method for Solving PDE-Constrained Optimization Problems
,” Numer. Algorithms
, 64
(3
), pp. 481
–506
.3.
Eppler
, K.
, Harbrecht
, H.
, and Mommer
, M. S.
, 2008
, “A New Fictitious Domain Method in Shape Optimization
,” Comput. Optim. Appl.
, 40
(2
), pp. 281
–298
.4.
Alavi
, S. A.
, Kamyad
, A. V.
, and Farahi
, M. H.
, 1997
, “The Optimal Control of an Inhomogeneous Wave Problem With Internal Control and Their Numerical Solution
,” Bull. Iran. Math. Soc.
, 23
(2
), pp. 9
–36
.5.
Borzabadi
, A. H.
, Kamyad
, A. V.
, and Farahi
, M. H.
, 2004
, “Optimal Control of the Heat Equation in an Inhomogeneous Body
,” J. Appl. Math. Comput.
, 15
(1–2
), pp. 127
–146
.6.
Farahi
, M. H.
, 2009
, “Control the Fiber Orientation Distribution at the Outlet of Contraction
,” Acta Appl. Math.
, 106
(2
), pp. 279
–292
.7.
Farahi
, M. H.
, Rubio
, J. E.
, and Wilson
, D. A.
, 1996
, “The Optimal Control of the Linear Wave Equation
,” Int. J. Control
, 63
(5
), pp. 833
–848
.8.
Farahi
, M. H.
, Rubio
, J. E.
, and Wilson
, D. A.
, 1996
, “The Global Control of a Nonlinear Wave Equation
,” Int. J. Control
, 65
(1
), pp. 1
–15
.9.
Ravindran
, S. S.
, 2000
, “A Reduced-Order Approach for Optimal Control of Fluids Using Proper Orthogonal Decomposition
,” Int. J. Numer. Method Fluids
, 34
(5
), pp. 425
–448
.10.
Weiser
, M.
, and Schiela
, A.
, 2004
, “Function Space Interior Point Methods for PDE Constrained Optimization
,” PAMM
, 4
(1
), pp. 43
–46
.11.
Laumen
, M.
, 2000
, “Newton's Method for a Class of Optimal Shape Design Problems
,” Siam J. Optim.
, 10
(2
), pp. 503
–533
.12.
Benamou
, J. D.
, 1999
, “Domain Decomposition, Optimal Control of Systems Governed by Partial Differential Equations, and Synthesis of Feedback Laws
,” J. Optim. Theory Appl.
, 102
(1
), pp. 15
–36
.13.
Akkouche
, A.
, Maidi
, A.
, and Aidene
, M.
, 2014
, “Optimal Control of Partial Differential Equations Based on the Variational Iteration Method
,” Comput. Math. Appl.
, 68
(5
), pp. 622
–631
.14.
Casas
, E.
, Troltzsch
, F.
, and Unger
, A.
, 2000
, “Second Order Sufficient Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations
,” SIAM J. Constraint. Optim.
, 38
(5
), pp. 1369
–1381
.15.
Casas
, E.
, and Raymond
, J. P.
, 2006
, “Error Estimates for the Numerical Approximation of Dirichlet Boundary Control for Semilinear Elliptic Equations
,” SIAM J. Control Optim.
, 45
(5
), pp. 1586
–1611
.16.
Griesse
, R.
, and Volkwein
, S.
, 2005
, “A Primal-Dual Active Set Strategy for Optimal Boundary Control of a Nonlinear Reaction-Diffusion System
,” SIAM J. Constraint Optim.
, 44
(2
), pp. 467
–494
.17.
Neittaanma
, P.
, and Tiba
, D.
, 1994
, “Optimal Control of Nonlinear Parabolic Systems
,” Theory, Algorithms and Applications
, Marcel Dekker
, New York
.18.
Rosch
, A.
, and Troltzsch
, F.
, 2007
, “On Regularity of Solutions and Lagrange Multipliers of Optimal Control Problems for Semilinear Equations With Mixed Pointwise Control-State Constraints
,” SIAM J. Constraint Optim.
, 46
(3
), pp. 1098
–1115
.19.
Serovaiskii
, S. Y.
, 2013
, “An Optimal Control Problem for a Nonlinear Elliptic Equation With a Phase Constraint and State Variation
,” Russ. Math.
, 57
(9
), pp. 67
–70
.20.
Solsvik
, J.
, Tangen
, S.
, and Jakobsen
, H. A.
, 2013
, “Evaluation of Weighted Residual Methods for the Solution of the Pellet Equations: The Orthogonal Collocation, Galerkin, Tau and Least-Squares Methods
,” Comput. Chem. Eng.
, 58
, pp. 223
–259
.21.
Harada
, K.
, and Nakamae
, E.
, 1982
, “Application of the Bezier Curve to Data Interpolation, Computer-Aided Design
,” Comput. Aided Des.
, 14
(1
), pp. 55
–59
.22.
Nurnberger
, G.
, and Zeilfelder
, F.
, 2000
, “Developments in Bivariate Spline Interpolation
,” J. Comput. Appl. Math.
, 121
(1–2
), pp. 125
–152
.23.
Zheng
, J.
, Sederberg
, T. W.
, and Johnson
, R. W.
, 2004
, “Least Squares Methods for Solving Differential Equations Using Bezier Control Points
,” Appl. Numer. Math.
, 48
(2
), pp. 237
–252
.24.
Evrenosoglu
, M.
, and Somali
, S.
, 2008
, “Least Squares Methods for Solving Singularity Perturbed Two-Points Boundary Value Problems Using Bezier Control Point
,” Appl. Math. Lett.
, 21
(10
), pp. 1029
–1032
.25.
Ghomanjani
, F.
, Farahi
, M. H.
, and Gachpazan
, M.
, 2014
, “Optimal Control of Time-Varying Linear Delay Systems Based on the Bezier Curves
,” Comput. Appl. Math.
, 33
(3
), pp. 687
–715
.26.
Ghomanjani
, F.
, and Farahi
, M. H.
, 2012
, “The Bezier Control Points Method for Solving Delay Differential Equation
,” Intell. Control Autom.
, 3
(2
), pp. 188
–196
.27.
Beltran
, J. V.
, and Monterde
, I.
, 2004
, “Bezier Solutions of the Wave Equation
,” Lect. Notes Comput. Sci.
, 2
, pp. 631
–640
.28.
Cholewa
, R.
, Nowak
, A. J.
, Bialecki
, R. A.
, and Wrobel
, L. C.
, 2002
, “Cubic Bezier Splines for BEM Heat Transfer Analysis of the 2-D Continuous Casting Problems
,” Comput. Mech.
, 28
(3), pp. 282
–290
.29.
Lang
, B.
, 2004
, “The Synthesis of Wave Forms Using Bezier Curves With Control Point Modulation
,” The Second CEMS Research Student Conference
, 1st ed., Morgan Kaufamann
, San Francisco, CA
.30.
Shi
, Y. Q.
, and Sun
, H.
, 2000
, Image and Video Compression for Multimedia Engineering
, CRC Press, Boca Raton, FL.31.
Chu
, C. H.
, Wang
, C. C. L.
, and Tsai
, C. R.
, 2008
, “Computer Aided Geometric Design of Strip Using Developable Bezier Patches
,” Comput. Ind.
, 59
(6
), pp. 601
–611
.32.
Farin
, G. E.
, 1998
, Curve and Surfaces for Computer Aided Geometric Design
, 1st ed., Academic Press
, New York
.33.
Gresho
, P. M.
, and Sani
, R. L.
, 1998
, Incompressible Flow and the Finite Element Method
, Wiley
, New York
.34.
Dehghan
, M.
, Yousefi
, S. A.
, and Rashedi
, K.
, 2013
, “Ritz–Galerkin Method for Solving an Inverse Heat Conduction Problem With a Nonlinear Source Term Via Bernstein Multi-Scaling Functions and Cubic B-Spline Functions
,” Inverse Probl. Sci. Eng.
, 21
(3
), pp. 500
–523
.35.
Watson
, G. A.
, 1980
, Approximation Theory and Numerical Methods
, Wiley-Interscience
.36.
Langtangen
, H. P.
, 1998
, Computational Partial Differential Equations Numerical Methods and Diffpack Programming
, Springer
, Berlin
.37.
Tröltzsch
, F.
, 2010
, Optimal Control of Partial Differential Equations
(Graduate Studies in Mathematics), American Mathematical Society
, Providence, RI
.38.
Lass
, O.
, Vallejos
, M.
, Borzi
, A.
, and Douglas
, C. C.
, 2009
, “Implementation and Analysis of Multigrid Schemes With Finite Elements for Elliptic Optimal Control Problems
,” Computing
, 84
(1–2
), pp. 27
–48
.39.
Rudin
, W.
, 1986
, Principles of Mathematical Analysis
, McGraw-Hill
, New York
.40.
Bellucci
, M. A.
, 2014
, “On the Explicit Representation of Orthonormal Bernstein Polynomials
,” arXiv preprint arXiv:1404.2293.Copyright © 2016 by ASME
You do not currently have access to this content.
