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2015, 2015(special): 515-524. doi: 10.3934/proc.2015.0515

Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin

Matthias Gerdts 1, and  Sven-Joachim Kimmerle 1,

1. 

Institut für Mathematik und Rechneranwendung (LRT-1), Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg/München, Germany, Germany

Received  September 2014 Revised  June 2015 Published  November 2015

We consider a numerical study of an optimal control problem for a truck with a fluid basin, which leads to an optimal control problem with a coupled system of partial differential equations (PDEs) and ordinary differential equations (ODEs). The motion of the fluid in the basin is modeled by the nonlinear hyperbolic Saint-Venant (shallow water) equations while the vehicle dynamics are described by the equations of motion of a mechanical multi-body system. These equations are fully coupled through boundary conditions and force terms. We pursue a first-discretize-then-optimize approach using a Lax-Friedrich scheme. To this end a reduced optimization problem is obtained by a direct shooting approach and solved by a sequential quadratic programming method. For the computation of gradients we employ an efficient adjoint scheme. Numerical case studies for optimal braking maneuvers of the truck and the basin filled with a fluid are presented.
Keywords: vehicle dynamics., ODE-PDE constrained optimization, shallow water equations, adjoint gradient computation, Saint-Venant equations, coupled ODE-PDE system, Lax-Friedrich scheme, hyperbolic conservation law.
Mathematics Subject Classification: Primary: 35Q35, 49J15, 49J20; Secondary: 35L04, 49N90, 76B1.
Citation: Matthias Gerdts, Sven-Joachim Kimmerle. Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin. Conference Publications, 2015, 2015 (special) : 515-524. doi: 10.3934/proc.2015.0515
References:
[1]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs 136, American Mathematical Society, Providence, RI, 2007.

[2]

F. Dubois, N. Petit and P. Rochon, Motion planning and nonlinear simulations for a tank containing a fluid,, in Proc. of the 5th European Control Conf. (ECC 99), (). 

[3]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 2010.

[4]

M. Gerdts, Optimal Control of ODEs and DAEs, de Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012.

[5]

M. Gerdts, OCPID-DAE1, Optimal Control and Parameter Identification with Differential-Algebraic Equations of Index 1. User Guide (Online Documentation), Universität der Bundeswehr München, Neubiberg/München, 2010.

[6]

M. Gugat and G. Leugering, Global boundary controllability of the De St. Venant equations between steady states, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 1-11.

[7]

M. Gugat and G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction, Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 257-270.

[8]

D. Kroener, Numerical Schemes for Conservation Laws, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Ltd., Chichester / B. G. Teubner, Stuttgart, 1997.

[9]

A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system, M2AN Math. Model. Numer. Anal., 36 (2002), 397-425.

[10]

P. D. Lax, Hyperbolic Partial Differential Equations, with an appendix by Cathleen S. Morawetz, Courant Lecture Notes in Mathematics 14, New York University, Courant Institute of Mathematical Sciences, New York / American Mathematical Society, Providence, RI, 2006.

[11]

C. B. Vreugdenhil, Numerical Methods for Shallow-Water Flow, reprinted edition, Kluwer Academic Publishers, Dordrecht , 1998.

show all references

References:
[1]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs 136, American Mathematical Society, Providence, RI, 2007.

[2]

F. Dubois, N. Petit and P. Rochon, Motion planning and nonlinear simulations for a tank containing a fluid,, in Proc. of the 5th European Control Conf. (ECC 99), (). 

[3]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 2010.

[4]

M. Gerdts, Optimal Control of ODEs and DAEs, de Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012.

[5]

M. Gerdts, OCPID-DAE1, Optimal Control and Parameter Identification with Differential-Algebraic Equations of Index 1. User Guide (Online Documentation), Universität der Bundeswehr München, Neubiberg/München, 2010.

[6]

M. Gugat and G. Leugering, Global boundary controllability of the De St. Venant equations between steady states, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 1-11.

[7]

M. Gugat and G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction, Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 257-270.

[8]

D. Kroener, Numerical Schemes for Conservation Laws, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Ltd., Chichester / B. G. Teubner, Stuttgart, 1997.

[9]

A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system, M2AN Math. Model. Numer. Anal., 36 (2002), 397-425.

[10]

P. D. Lax, Hyperbolic Partial Differential Equations, with an appendix by Cathleen S. Morawetz, Courant Lecture Notes in Mathematics 14, New York University, Courant Institute of Mathematical Sciences, New York / American Mathematical Society, Providence, RI, 2006.

[11]

C. B. Vreugdenhil, Numerical Methods for Shallow-Water Flow, reprinted edition, Kluwer Academic Publishers, Dordrecht , 1998.

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