Abstract
The objective of this work is to develop a mathematical framework for the modeling, control and optimization of dynamic control systems whose state variable is driven by interacting ODE’s (ordinary differential equations) and solutions of PDE’s (partial differential equations). The ultimate goal is to provide a sound basis for the design and control of new advanced engineering systems arising in many important classes of applications, some of which may encompass, for example, underwater gliders and mechanical fishes. For now, the research effort has been focused in gaining insight by applying necessary conditions of optimality for shear flow driven dynamic control systems which can be easily reduced to problems with ODE dynamics. In this article we present and discuss the problem of minimum time control of a particle advected in a Couette and Poiseuille flows, and solve it by using the maximum principle.
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References
Arutyunov, A., Karamzin, D., Lobo Pereira, F.: The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: revisited. J. Optim. Theory Appl. 149, 474–493 (2011)
Batchelor, G.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1992)
Clarke, F.: The maximum principle in optimal control, then and now. J. Control Cybern. 34, 709–722 (2005)
Clarke, F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Grilo, T., Lobo Pereira, F., Gama, S.: Optimal control of particle advection in Couette and Poiseuille flows. J. Conf. Papers Math. 2013, Art. ID 783510 (2013). http://dx.doi.org/10.1155/2013/783510
Lions, J.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)
Liu, J., Hu, H.: Biological inspiration: from carangiform fish to multi-joint robotic fish. J. Bionic Eng. 7, 35–48 (2010)
Mahmoudian, N., Woolsey, C.: Dynamics and control of underwater gliders II: motion planning and control. Virginia Center for Autonomous Systems, Technical Report No. VaCAS-2010-02 (2010)
Mahmoudian, N., Geisbert, J., Woolsey, C.: Dynamics and control of underwater gliders I: steady motions. Virginia Center for Autonomous Systems, Technical Report No. VaCAS-2007-01 (2009)
Pontryagin, L., Boltyanskiy, V., Gamkrelidze, R., Mishchenko, E.: Mathematical Theory of Optimal Processes. Interscience Publishers, New York (1962)
Acknowledgements
The first and the third authors gratefully acknowledge the financial support of the FCT funded the doctoral grant POPH/FSE/SFRH/BD/94131/2013, and the R&D projects PEST-OE/EEI/UI0147/2014 and SYSTEC R&D Unit ref. UID/EEA/00147/2013, respectively. The second author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
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Grilo, T., Gama, S.M.A., Pereira, F.L. (2015). On the Optimal Control of Flow Driven Dynamic Systems. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Mathematics of Energy and Climate Change. CIM Series in Mathematical Sciences, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-16121-1_7
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DOI: https://doi.org/10.1007/978-3-319-16121-1_7
Publisher Name: Springer, Cham
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