April  2011, 29(2): 647-670. doi: 10.3934/dcds.2011.29.647

Numerical procedure for optimal control of higher index DAEs

1. 

Institute of Automatic Control and Robotics, Warsaw University of Technology, 02-525 Warsaw, Poland

Received  September 2009 Revised  March 2010 Published  October 2010

The paper deals with optimal control problems described by higher index DAEs. We introduce a numerical procedure for solving these problems. The procedure, based on the appropriately defined adjoint equations, refers to an implicit Runge--Kutta method for differential--algebraic equations. Assuming that higher index DAEs can be solved numerically the gradients of functionals defining the control problem are evaluated with the help of well--defined adjoint equations. The paper presents numerical examples related to index three DAEs showing the validity of the proposed approach.
Citation: Radoslaw Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 647-670. doi: 10.3934/dcds.2011.29.647
References:
[1]

K. Balla, Linear subspaces for linear DAE's of index 1,, Comput. Math. Appl., 32 (1996), 81. Google Scholar

[2]

K. Balla, Boundary conditions and their transfer for differential algebraic equations of index 1,, Comput. Math. Appl., 31 (1996), 1. Google Scholar

[3]

K. Balla and R. März, Transfer of boundary conditions for DAE's of index 1,, SIAM J. Numer. Anal., 33 (1996), 2318. Google Scholar

[4]

K. Balla and R. März, "An Unified Approach to Linear Differential Algebraic Equations and Their Adjoint Equations,", Institute of Mathematics Technical Report, (2000). Google Scholar

[5]

K. Balla, K. and R. März, Linear differential algebraic equations and their adjoint equations,, Results in Mathematica, 37 (2000), 13. Google Scholar

[6]

K. E. Brenan, S. L. Campbell and L. R. Petzold, "Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations,", North-Holland, (1989). Google Scholar

[7]

Y. Cao, S. Li, L. Petzold and R. Serben, Adjoint sensitivity analysis for differential-algebraic equations: Part I, The adjoint DAE system and its numerical solution,, SIAM J. Sci. Comp., 24 (2000), 1076. Google Scholar

[8]

Y. Cao, S. Li and L. Petzold, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software,, J. Comput. Appl. Math., 149 (2002), 171. Google Scholar

[9]

W. F. Frenery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems,, Appl. Numer. Math., 25 (1997), 41. Google Scholar

[10]

C. W. Gear, Differential-algebraic equation index transformations,, SIAM J. Sci. Stat. Comput., 9 (1988), 39. Google Scholar

[11]

C. W. Gear, Differential algebraic equations, indices and integral algebraic equations,, SIAM J. Numer. Anal., 27 (1990), 1527. Google Scholar

[12]

D. R. A. Giles, "A Comparison of Three Problem-Oriented Simulation Programs for Dynamic Mechanical Systems,", Thesis, (1978). Google Scholar

[13]

E. Hairer, Ch. Lubich and M. Roche, "The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods,", Lecture Notes in Mathematics, 1409 (1989). Google Scholar

[14]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II,", Springer-Verlag, (1996). Google Scholar

[15]

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers,", preprint, (2004). Google Scholar

[16]

L. Kershenbaum, D. Q. Mayne, R. Pytlak and R. B. Vinter, Receding horizon control,, in, (1993), 20. Google Scholar

[17]

S. Li and L. R. Petzold, Software and algorithms for sensitivity analysis of large-scale differential-algebraic systems,, J. Comput. Appl. Math., 125 (2000), 131. Google Scholar

[18]

S. Li, L. R. Petzold and W. Zhu, Sensitivity analysis of differential-algebraic equations: A comparison of methods on a special problem,, Appl. Numer. Math., 32 (2000), 161. Google Scholar

[19]

T. Maly, and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems,, Appl. Numer. Math., 20 (1997), 57. Google Scholar

[20]

D. W. Manning, "A Computer Technique for Simulating Dynamic Multibody Systems Based on Dynamic Formalizm,", Thesis, (1981). Google Scholar

[21]

R. März, Differential algebraic systems anew,, Applied Numetical Mathematics, 42 (2002), 315. Google Scholar

[22]

R. März, Characterizing differential algebraic equations without the use of derivative arrays,, Int. J. Comp. & Mathem. with Appl., 50 (2005), 1141. Google Scholar

[23]

M. D. R. de Pinho and R. Vinter, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations,, J. Math. Anal. Appl., 212 (1997), 493. Google Scholar

[24]

R. Pytlak, Optimal control of differential-algebraic equations,, Proceed. of the 33rd IEEE CDC, (1994), 951. Google Scholar

[25]

R. Pytlak, "Numerical Methods for Optimal Control Problem With State Constraints,", Lecture Notes in Mathematics, 1707 (1999). Google Scholar

[26]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 1: First order approximations,, J. Optimization Theory and Applications, 134 (2007), 61. Google Scholar

[27]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 2: Necessary optimality conditions,, J. Optimization Theory and Applications, 134 (2007), 77. Google Scholar

[28]

R. Pytlak and R. B. Vinter, A feasible directions algorithm for optimal control problems with state and control constraints: Convergence analysis,, SIAM J. Control and Optimization, 36 (1998), 1999. Google Scholar

[29]

R. Pytlak and R. B. Vinter, A feasible directions type algorithm for optimal control problems with state and control constraints: Implementation,, J. Optimization Theory and Applications, 101 (1999), 623. Google Scholar

[30]

W. Schiehlen (ed.), "Mulitbody Systems Handbook,", Springer-Verlag, (1990). Google Scholar

show all references

References:
[1]

K. Balla, Linear subspaces for linear DAE's of index 1,, Comput. Math. Appl., 32 (1996), 81. Google Scholar

[2]

K. Balla, Boundary conditions and their transfer for differential algebraic equations of index 1,, Comput. Math. Appl., 31 (1996), 1. Google Scholar

[3]

K. Balla and R. März, Transfer of boundary conditions for DAE's of index 1,, SIAM J. Numer. Anal., 33 (1996), 2318. Google Scholar

[4]

K. Balla and R. März, "An Unified Approach to Linear Differential Algebraic Equations and Their Adjoint Equations,", Institute of Mathematics Technical Report, (2000). Google Scholar

[5]

K. Balla, K. and R. März, Linear differential algebraic equations and their adjoint equations,, Results in Mathematica, 37 (2000), 13. Google Scholar

[6]

K. E. Brenan, S. L. Campbell and L. R. Petzold, "Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations,", North-Holland, (1989). Google Scholar

[7]

Y. Cao, S. Li, L. Petzold and R. Serben, Adjoint sensitivity analysis for differential-algebraic equations: Part I, The adjoint DAE system and its numerical solution,, SIAM J. Sci. Comp., 24 (2000), 1076. Google Scholar

[8]

Y. Cao, S. Li and L. Petzold, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software,, J. Comput. Appl. Math., 149 (2002), 171. Google Scholar

[9]

W. F. Frenery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems,, Appl. Numer. Math., 25 (1997), 41. Google Scholar

[10]

C. W. Gear, Differential-algebraic equation index transformations,, SIAM J. Sci. Stat. Comput., 9 (1988), 39. Google Scholar

[11]

C. W. Gear, Differential algebraic equations, indices and integral algebraic equations,, SIAM J. Numer. Anal., 27 (1990), 1527. Google Scholar

[12]

D. R. A. Giles, "A Comparison of Three Problem-Oriented Simulation Programs for Dynamic Mechanical Systems,", Thesis, (1978). Google Scholar

[13]

E. Hairer, Ch. Lubich and M. Roche, "The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods,", Lecture Notes in Mathematics, 1409 (1989). Google Scholar

[14]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II,", Springer-Verlag, (1996). Google Scholar

[15]

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers,", preprint, (2004). Google Scholar

[16]

L. Kershenbaum, D. Q. Mayne, R. Pytlak and R. B. Vinter, Receding horizon control,, in, (1993), 20. Google Scholar

[17]

S. Li and L. R. Petzold, Software and algorithms for sensitivity analysis of large-scale differential-algebraic systems,, J. Comput. Appl. Math., 125 (2000), 131. Google Scholar

[18]

S. Li, L. R. Petzold and W. Zhu, Sensitivity analysis of differential-algebraic equations: A comparison of methods on a special problem,, Appl. Numer. Math., 32 (2000), 161. Google Scholar

[19]

T. Maly, and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems,, Appl. Numer. Math., 20 (1997), 57. Google Scholar

[20]

D. W. Manning, "A Computer Technique for Simulating Dynamic Multibody Systems Based on Dynamic Formalizm,", Thesis, (1981). Google Scholar

[21]

R. März, Differential algebraic systems anew,, Applied Numetical Mathematics, 42 (2002), 315. Google Scholar

[22]

R. März, Characterizing differential algebraic equations without the use of derivative arrays,, Int. J. Comp. & Mathem. with Appl., 50 (2005), 1141. Google Scholar

[23]

M. D. R. de Pinho and R. Vinter, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations,, J. Math. Anal. Appl., 212 (1997), 493. Google Scholar

[24]

R. Pytlak, Optimal control of differential-algebraic equations,, Proceed. of the 33rd IEEE CDC, (1994), 951. Google Scholar

[25]

R. Pytlak, "Numerical Methods for Optimal Control Problem With State Constraints,", Lecture Notes in Mathematics, 1707 (1999). Google Scholar

[26]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 1: First order approximations,, J. Optimization Theory and Applications, 134 (2007), 61. Google Scholar

[27]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 2: Necessary optimality conditions,, J. Optimization Theory and Applications, 134 (2007), 77. Google Scholar

[28]

R. Pytlak and R. B. Vinter, A feasible directions algorithm for optimal control problems with state and control constraints: Convergence analysis,, SIAM J. Control and Optimization, 36 (1998), 1999. Google Scholar

[29]

R. Pytlak and R. B. Vinter, A feasible directions type algorithm for optimal control problems with state and control constraints: Implementation,, J. Optimization Theory and Applications, 101 (1999), 623. Google Scholar

[30]

W. Schiehlen (ed.), "Mulitbody Systems Handbook,", Springer-Verlag, (1990). Google Scholar

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