American Institute of Mathematical Sciences

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

Numerical procedure for optimal control of higher index DAEs

Radoslaw Pytlak 1,

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.
Keywords: Optimal control, adjoint equations., implicit systems.
Mathematics Subject Classification: Primary: 49K99, 34A09; Secondary: 49M0.
Citation: Radoslaw Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete and Continuous Dynamical Systems, 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), 81-86.

[2]

K. Balla, Boundary conditions and their transfer for differential algebraic equations of index 1, Comput. Math. Appl., 31 (1996), 1-5.

[3]

K. Balla and R. März, Transfer of boundary conditions for DAE's of index 1, SIAM J. Numer. Anal., 33 (1996), 2318-2332.

[4]

K. Balla and R. März, "An Unified Approach to Linear Differential Algebraic Equations and Their Adjoint Equations," Institute of Mathematics Technical Report, Humboldt University, Berlin, 2000.

[5]

K. Balla, K. and R. März, Linear differential algebraic equations and their adjoint equations, Results in Mathematica, 37 (2000), 13-35.

[6]

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

[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), 1076-1089.

[8]

Y. Cao, S. Li and L. Petzold, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software, J. Comput. Appl. Math., 149 (2002), 171-191.

[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), 41-54.

[10]

C. W. Gear, Differential-algebraic equation index transformations, SIAM J. Sci. Stat. Comput., 9 (1988), 39-47.

[11]

C. W. Gear, Differential algebraic equations, indices and integral algebraic equations, SIAM J. Numer. Anal., 27 (1990), 1527-1534.

[12]

D. R. A. Giles, "A Comparison of Three Problem-Oriented Simulation Programs for Dynamic Mechanical Systems," Thesis, Univ. Waterloo, Ontario, 1978.

[13]

E. Hairer, Ch. Lubich and M. Roche, "The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods," Lecture Notes in Mathematics, 1409, Springer-Verlag, Berlin, Heidelberg, 1989.

[14]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II," Springer-Verlag, Berlin Heidelberg New York, 1996.

[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, UCRL-JRNL-200037, 2004.

[16]

L. Kershenbaum, D. Q. Mayne, R. Pytlak and R. B. Vinter, Receding horizon control, in "Advances in Model-Based Predictive Control, Oxford, September 20-21, Conference Proceedings," 1993.

[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), 131-145.

[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), 161-174.

[19]

T. Maly, and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems, Appl. Numer. Math., 20 (1997), 57-79.

[20]

D. W. Manning, "A Computer Technique for Simulating Dynamic Multibody Systems Based on Dynamic Formalizm," Thesis, Univ. Waterloo, Ontario, 1981.

[21]

R. März, Differential algebraic systems anew, Applied Numetical Mathematics, 42 (2002), 315-335.

[22]

R. März, Characterizing differential algebraic equations without the use of derivative arrays, Int. J. Comp. & Mathem. with Appl., 50 (2005), 1141-1156.

[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), 493-516.

[24]

R. Pytlak, Optimal control of differential-algebraic equations, Proceed. of the 33rd IEEE CDC, Orlando, Florida, (1994), 951-956.

[25]

R. Pytlak, "Numerical Methods for Optimal Control Problem With State Constraints," Lecture Notes in Mathematics, 1707, Springer-Verlag, 1999.

[26]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 1: First order approximations, J. Optimization Theory and Applications, 134 (2007), 61-75.

[27]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 2: Necessary optimality conditions, J. Optimization Theory and Applications, 134 (2007), 77-90.

[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), 1999-2019.

[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), 623-649.

[30]

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

show all references

References:
[1]

K. Balla, Linear subspaces for linear DAE's of index 1, Comput. Math. Appl., 32 (1996), 81-86.

[2]

K. Balla, Boundary conditions and their transfer for differential algebraic equations of index 1, Comput. Math. Appl., 31 (1996), 1-5.

[3]

K. Balla and R. März, Transfer of boundary conditions for DAE's of index 1, SIAM J. Numer. Anal., 33 (1996), 2318-2332.

[4]

K. Balla and R. März, "An Unified Approach to Linear Differential Algebraic Equations and Their Adjoint Equations," Institute of Mathematics Technical Report, Humboldt University, Berlin, 2000.

[5]

K. Balla, K. and R. März, Linear differential algebraic equations and their adjoint equations, Results in Mathematica, 37 (2000), 13-35.

[6]

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

[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), 1076-1089.

[8]

Y. Cao, S. Li and L. Petzold, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software, J. Comput. Appl. Math., 149 (2002), 171-191.

[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), 41-54.

[10]

C. W. Gear, Differential-algebraic equation index transformations, SIAM J. Sci. Stat. Comput., 9 (1988), 39-47.

[11]

C. W. Gear, Differential algebraic equations, indices and integral algebraic equations, SIAM J. Numer. Anal., 27 (1990), 1527-1534.

[12]

D. R. A. Giles, "A Comparison of Three Problem-Oriented Simulation Programs for Dynamic Mechanical Systems," Thesis, Univ. Waterloo, Ontario, 1978.

[13]

E. Hairer, Ch. Lubich and M. Roche, "The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods," Lecture Notes in Mathematics, 1409, Springer-Verlag, Berlin, Heidelberg, 1989.

[14]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II," Springer-Verlag, Berlin Heidelberg New York, 1996.

[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, UCRL-JRNL-200037, 2004.

[16]

L. Kershenbaum, D. Q. Mayne, R. Pytlak and R. B. Vinter, Receding horizon control, in "Advances in Model-Based Predictive Control, Oxford, September 20-21, Conference Proceedings," 1993.

[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), 131-145.

[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), 161-174.

[19]

T. Maly, and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems, Appl. Numer. Math., 20 (1997), 57-79.

[20]

D. W. Manning, "A Computer Technique for Simulating Dynamic Multibody Systems Based on Dynamic Formalizm," Thesis, Univ. Waterloo, Ontario, 1981.

[21]

R. März, Differential algebraic systems anew, Applied Numetical Mathematics, 42 (2002), 315-335.

[22]

R. März, Characterizing differential algebraic equations without the use of derivative arrays, Int. J. Comp. & Mathem. with Appl., 50 (2005), 1141-1156.

[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), 493-516.

[24]

R. Pytlak, Optimal control of differential-algebraic equations, Proceed. of the 33rd IEEE CDC, Orlando, Florida, (1994), 951-956.

[25]

R. Pytlak, "Numerical Methods for Optimal Control Problem With State Constraints," Lecture Notes in Mathematics, 1707, Springer-Verlag, 1999.

[26]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 1: First order approximations, J. Optimization Theory and Applications, 134 (2007), 61-75.

[27]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 2: Necessary optimality conditions, J. Optimization Theory and Applications, 134 (2007), 77-90.

[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), 1999-2019.

[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), 623-649.

[30]

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

[1]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[2]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[3]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279
[4]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[5]

Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109
[6]

Dan Tiba. Implicit parametrizations and applications in optimization and control. Mathematical Control and Related Fields, 2020, 10 (3) : 455-470. doi: 10.3934/mcrf.2020006
[7]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks and Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007
[8]

Y. Gong, X. Xiang. A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. Journal of Industrial and Management Optimization, 2009, 5 (1) : 1-10. doi: 10.3934/jimo.2009.5.1
[9]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control and Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[10]

Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367
[11]

Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275
[12]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial and Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1
[13]

Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial and Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63
[14]

Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107
[15]

Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149
[16]

Luca Galbusera, Sara Pasquali, Gianni Gilioli. Stability and optimal control for some classes of tritrophic systems. Mathematical Biosciences & Engineering, 2014, 11 (2) : 257-283. doi: 10.3934/mbe.2014.11.257
[17]

Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics and Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187
[18]

Qiying Hu, Wuyi Yue. Optimal control for discrete event systems with arbitrary control pattern. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 535-558. doi: 10.3934/dcdsb.2006.6.535
[19]

Eduardo Casas, Konstantinos Chrysafinos. Analysis and optimal control of some quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 607-623. doi: 10.3934/mcrf.2018025
[20]

Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy