# American Institute of Mathematical Sciences

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

 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.
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, (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, (2010). [4] M. Gerdts, Optimal Control of ODEs and DAEs,, de Gruyter Textbook, (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, (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. [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. [8] D. Kroener, Numerical Schemes for Conservation Laws,, Wiley-Teubner Series Advances in Numerical Mathematics, (1997). [9] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system,, M2AN Math. Model. Numer. Anal., 36 (2002), 397. [10] P. D. Lax, Hyperbolic Partial Differential Equations,, with an appendix by Cathleen S. Morawetz, (2006). [11] C. B. Vreugdenhil, Numerical Methods for Shallow-Water Flow,, reprinted edition, (1998).

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##### References:
 [1] J. M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs 136, (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, (2010). [4] M. Gerdts, Optimal Control of ODEs and DAEs,, de Gruyter Textbook, (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, (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. [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. [8] D. Kroener, Numerical Schemes for Conservation Laws,, Wiley-Teubner Series Advances in Numerical Mathematics, (1997). [9] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system,, M2AN Math. Model. Numer. Anal., 36 (2002), 397. [10] P. D. Lax, Hyperbolic Partial Differential Equations,, with an appendix by Cathleen S. Morawetz, (2006). [11] C. B. Vreugdenhil, Numerical Methods for Shallow-Water Flow,, reprinted edition, (1998).
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