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Numerical Optimal Control of Parabolic PDES Using DASOPT

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Large-Scale Optimization with Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 93))

Abstract

This paper gives a preliminary description of DASOPT, a software system for the optimal control of processes described by time-dependent partial differential equations (PDEs). DASOPT combines the use of efficient numerical methods for solving differential-algebraic equations (DAEs) with a package for large-scale optimization based on sequential quadratic programming (SQP). DASOPT is intended for the computation of the optimal control of time-dependent nonlinear systems of PDEs in two (and eventually three) spatial dimensions, including possible inequality constraints on the state variables. By the use of either finite-difference or finite-element approximations to the spatial derivatives, the PDEs are converted into a large system of ODEs or DAEs. Special techniques are needed in order to solve this very large optimal control problem. The use of DASOPT is illustrated by its application to a nonlinear parabolic PDE boundary control problem in two spatial dimensions. Computational results with and without bounds on the state variables are presented.

This research was partially supported by National Science Foundation grants CCR95–27151 and DMI-9424639, National Institute of Standards and Technology contract 60 NANB2D 1272, Department of Energy grant FG02–92ER25130, Office of Naval Research grants N00014-90-J-1242 and N00014–96–1–0274, the Army High Performance Computing Research Center ARL Cooperative agreement DAAH04–95–2–0003 and contract DAAH04–95-C-0008, and the Minnesota Supercomputing Institute.

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References

  1. U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Applied Mathematics, 13, Society for Industrial and Applied Mathematics (SIAM) Publications, Philadelphia, PA, 1995. ISBN 0–89871–354–4.

    Book  MATH  Google Scholar 

  2. J. T. Betts, Issues in the direct transcription of optimal control problems to sparse nonlinear programs, in Control Applications of Optimization, R. Bulirsch and D. Kraft, eds., vol. 115 of Internat. Ser. Numer. Math., Basel, 1994, Birkhäuser, pp. 3–17.

    Google Scholar 

  3. J. T. Betts, and W. P. Huffman, The application of sparse nonlinear programming to trajectory optimization, J. of Guidance, Control, and Dynamics, 14 (1991), pp. 338–348.

    Article  Google Scholar 

  4. L. T. Biegler, J. Nocedal, and C. Schmid, A reduced Hessian method for large-scale constrained optimization, SIAM J. Optim., 5 (1995), pp. 314–347.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Bischof, A. Carle, G. Corliss, A. Griewank, and P. Hovland, ADIFOR—generating derivative codes from Fortran programs, Scientific Programming, 1 (1992), pp. 11–29.

    Google Scholar 

  6. K. E. Brenan, Differential-algebraic equations issues in the direct transcription of path constrained optimal control problems, Ann. Numer. Math., 1 (1994), pp. 247–263.

    MathSciNet  MATH  Google Scholar 

  7. K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial–Value Problems in Differential–Algebraic Equations, SIAM Publications, Philadelphia, second ed., 1995. ISBN 0–89871–353–6.

    Book  Google Scholar 

  8. A. Buckley, and A. Lenir, QN-like variable storage conjugate gradients, Math. Prog., 27 (1983), pp. 155–175.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Bulirsch, E. Nerz, H. J. Pesch, and O. Von Stryk, Combining direct and indirect methods in optimal control: Range maximization of a hang glider, in Calculus of Variations, Optimal Control Theory and Numerical Methods., R. Bulirsch, A. Miele, J. Stoer, and K. H. Well, eds., vol. 111 of Internat. Ser. Numer. Math., Basel, 1993, Birkhäuser, pp. 273–288.

    Google Scholar 

  10. M. Caracotsios, and W. E. Stewart, Sensitivity analysis of initial value problems with mixed ODEs and algebraic equations, Computers and Chemical Engineering, 9 (1985), pp. 359–365.

    Article  Google Scholar 

  11. E. D. Dickmanns, and K. H. Well, Approximate solution of optimal control problems using third-order Hermite polynomial functions, in Proc. 6th Technical Conference on Optimization Techniques, Berlin, Heidelberg, New York, London, Paris and Tokyo, 1975, Springer Verlag.

    Google Scholar 

  12. S. K. Eldersveld, Large-scale sequential quadratic programming algorithms, PhD thesis, Department of Operations Research, Stanford University, Stanford, CA, 1991.

    Google Scholar 

  13. J. C. Gilbert, and C. Lemaréchal, Some numerical experiments with variable-storage quasi-Newton algorithms, Math. Prog., (1989), pp. 407–435.

    Google Scholar 

  14. P. E. Gill, and W. Murray, Conjugate-gradient methods for large-scale nonlinear optimization, Report SOL 79–15, Department of Operations Research, Stanford University, Stanford, CA, 1979.

    Google Scholar 

  15. P. E. Gill, W. Murray, and M. A. Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization, Numerical Analysis Report 96–2, Department of Mathematics, University of California, San Diego, La Jolla, CA, 1996.

    Google Scholar 

  16. P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, Maintaining LU factors of a general sparse matrix, Linear Algebra and its Applications, 88 /89 (1987), pp. 239–270.

    Article  MathSciNet  Google Scholar 

  17. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization, Academic Press, London and New York, 1981. ISBN 0–12–283952–8.

    MATH  Google Scholar 

  18. D. M. Gritsis, C. C. Pantelides, and R. W. H. Sargent, Optimal control of systems described by index two differential-algebraic equations, SIAM J. Sci. Comput., 16 (1995), pp. 1349–1366.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. R. Hargraves, and S. W. Paris, Direct trajectory optimization using nonlinear programming and collocation, J. of Guidance, Control, and Dynamics, 10 (1987), pp. 338–348.

    Article  MATH  Google Scholar 

  20. D. Kraft, On converting optimal control problems into nonlinear programming problems, in Computational Mathematical Programming, K. Schittkowski, ed., NATO ASI Series F: Computer and Systems Sciences 15, Springer Verlag, Berlin and Heidelberg, 1985, pp. 261–280.

    Chapter  Google Scholar 

  21. R. Lamour, A well posed shooting method for transferable DAEs, Numer. Math., 59 (1991), pp. 815–829.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Leibfritz, and E. W. Sachs, Numerical solution of parabolic state constrained control problems using SQP and interior point methods, in Large Scale Optimization: the State of the Art, W. W. Hager, D. W. Hearn, and P. M. Pardalos, eds., Kluwer Academic Publishers, Dordrecht, 1994, pp. 245–258. ISBN 0–7923–2798–5.

    Chapter  Google Scholar 

  23. G. Leitmann, The Calculus of Variations and Optimal Control, Mathematical Concepts and Methods in Science and Engineering, 24, Plenum Press, New York and London, 1981. ISBN 0–306–40707–8.

    MATH  Google Scholar 

  24. T. Maly, and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems. Applied Numerical Mathematics, 20 (1996), pp. 57–79.

    Article  MathSciNet  MATH  Google Scholar 

  25. B. A. Murtagh, and M. A. Saunders, MINOS 5.4 User’s Guide, Report SOL 83–20R, Department of Operations Research, Stanford University, Stanford, CA, 1993.

    Google Scholar 

  26. J. Nocedal, Updating quasi-Newton matrices with limited storage, Math. Corn-put., 35 (1980), pp. 773–782.

    Article  MathSciNet  MATH  Google Scholar 

  27. C. C. Pantelides, R. W. H. Sargent, and V. S. Vassiliadis, Optimal control of multistage systems described by high-index differential-algebraic equations, Internat. Ser. Numer. Math., 115 (1994), pp. 177–191.

    MathSciNet  Google Scholar 

  28. H. J. Pesch, Real-time computation of feedback controls for constrained optimal control problems, Optimal Control Appl. Methods, 10 (1989), pp. 129–171.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. H. Schultz, Reduced SQP methods for large-scale optimal control problems in DAE with application to path planning problems for satellite mounted robots, PhD thesis, University of Heidelberg, 1996.

    Google Scholar 

  30. V. H. Schulz, H.-G. Bock, and M. C. Steinbach, Exploiting invariants in the numerical solution of multipoint boundary value problems for DAE. submitted to SIAM J. Sci. Comput., 1996.

    Google Scholar 

  31. M. C. Steinbach, A structured interior point SQP method for nonlinear optimal control problems, in Control Applications of Optimization, R. Bulirsch and D. Kraft, eds., vol. 115 of Internat. Ser. Numer. Math., Basel, 1994, Birkhäuser, pp. 213–222.

    Google Scholar 

  32. P. Tanartkit, and L. T. Biegler, Stable decomposition for dynamic optimization, Ind. Eng. Chem. Res., (1995), pp. 1253–1266.

    Google Scholar 

  33. I.-B. Tjoa, and L. T. Biegler, A reduced SQP strategy for errors-in-variables estimation, Comput. Chem. Eng., 16 (1992), pp. 523–533.

    Article  Google Scholar 

  34. O. Von Stryk, Numerical solution of optimal control problems by direct collocation, in Control Applications of Optimization, R. Bulirsch, A. Miele, J. Stoer, and K. H. Well, eds., vol. 111 of Internat. Ser. Numer. Math., Basel, 1993, Birkhäuser, pp. 129–143.

    Google Scholar 

  35. O. Von Stryk, and R. Bulirsch, Direct and indirect methods for trajectory optimization, Ann. Oper. Res., 37 (1992), pp. 357–373.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Minnesota, Minneapolis, Minnesota, 55455, USA

    Linda Petzold & J. Ben Rosen

  2. Department of Computer Science and Engineering, University of California, San Diego, La Jolla, California, 92093-0114, USA

    J. Ben Rosen

  3. Department of Mathematics, University of California, San Diego, La Jolla, California, 92093-0112, USA

    Philip E. Gill

  4. Department of Computer Science, University of Minnesota, Minneapolis, Minnesota, 55455, USA

    Laurent O. Jay

  5. School of Mechanical Engineering, Kookmin University, Seoul, Korea

    Kihong Park

Authors
  1. Linda Petzold
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  2. J. Ben Rosen
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  3. Philip E. Gill
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  4. Laurent O. Jay
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  5. Kihong Park
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Editor information

Editors and Affiliations

  1. Chemical Engineering Department, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Lorenz T. Biegler

  2. Computer Science Department, Cornell University, Ithaca, NA, 14853-0001, USA

    Thomas F. Coleman

  3. Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY, 10598, USA

    Andrew R. Conn

  4. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Fadil N. Santosa

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© 1997 Springer Science+Business Media New York

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Petzold, L., Rosen, J.B., Gill, P.E., Jay, L.O., Park, K. (1997). Numerical Optimal Control of Parabolic PDES Using DASOPT. In: Biegler, L.T., Coleman, T.F., Conn, A.R., Santosa, F.N. (eds) Large-Scale Optimization with Applications. The IMA Volumes in Mathematics and its Applications, vol 93. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1960-6_12

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  • DOI: https://doi.org/10.1007/978-1-4612-1960-6_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7356-1

  • Online ISBN: 978-1-4612-1960-6

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