×

Adaptive algorithms for optimal control of time-dependent partial differential-algebraic equation systems. (English) Zbl 1062.49506

Summary: This paper describes an adaptive algorithm for optimal control of time-dependent partial differential-algebraic equation (PDAE) systems. A direct method based on a modified multiple shooting type technique and sequential quadratic programming (SQP) is used for solving the optimal control problem, while an adaptive mesh refinement (AMR) algorithm is employed to dynamically adapt the spatial integration mesh. Issues of coupling the AMR solver to the optimization algorithm are addressed. For time-dependent PDAEs which can benefit from the use of an adaptive mesh, the resulting method is shown to be highly efficient.

MSC:

49M25 Discrete approximations in optimal control
49M37 Numerical methods based on nonlinear programming
65L80 Numerical methods for differential-algebraic equations
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
90C55 Methods of successive quadratic programming type

Software:

ADIFOR; SNOPT; DASPK 3.0
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gill, Journal of Computational and Applied Mathematics 120 pp 197– (2000)
[2] COOPT?Control and optimization of dynamic systems?Users’ Guide. Technical Report UCSB-ME-99-1, Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA, 1999.
[3] Serban, Mathematics and Computers in Simulation 56 pp 187– (2001)
[4] Solution adapted mesh refinement and sensitivity analysis for parabolic partial differential equation systems. Lecture Notes in Computational Science and Engineering 30, (eds), Springer-Verlag: Heidelberg, to appear, 2003. · Zbl 1062.65100
[5] Li, Journal of Computational and Applied Mathematics 125 pp 131– (2000)
[6] SNOPT: An SQP algorithm for large-scale constrained optimization. Technical Report 97-2, Department of Mathematics, University of California, San Diego, La Jolla, CA, 1997.
[7] Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM: Philadelphia, PA, 1995. · doi:10.1137/1.9781611971224
[8] Design of new DASPK for sensitivity analysis. Technical Report, Department of Computer Science, University of California, Santa Barbara, CA, 1999.
[9] Maly, Applied Numerical Mathematics 20 pp 57– (1997)
[10] Optimal control and optimization of viscous, incompressible flows. In Incompressible Computational Fluid Dynamics, Gunzburger MD, Nicolaides RA (ed.). Cambridge. 1993; 109-150. · Zbl 1189.76447
[11] Ghattas, Journal of Computational Physics 136 pp 231– (1997)
[12] Applied Optimal Design. Wiley: New York, 1979.
[13] A parallel reduced Hessian SQP method for shape optimization. In Multidisciplinary Design Optimization: State of the Art, (ed.). SIAM: Philadelphia, PA, 1997; 133-152.
[14] Computational strategies for shape optimization of Navier-Stokes flows. Technical Report CMU-CML-97-102, Computational Mechanics Lab, Department of Civil and Environmental Engineering, Carnegie Mellon University, 1997.
[15] Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM: Philadelphia, PA, 1995. · doi:10.1137/1.9781611971231
[16] Bischof, Scientific Programming 1 pp 11– (1992) · doi:10.1155/1992/717832
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.