# American Institute of Mathematical Sciences

June  2016, 6(2): 293-334. doi: 10.3934/mcrf.2016005

## An optimal control approach to ciliary locomotion

 1 Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile 2 Inria, Villers-les-Nancy F-54600, France 3 Institut de Mathématiques de Bordeaux, Université de Bordeaux/CNRS/ Institut National Polytechnique de Bordeaux, 351 Cours de Libération, 33405 Talence Cedex, France

Received  February 2015 Revised  July 2015 Published  April 2016

We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.
Citation: Jorge San Martín, Takéo Takahashi, Marius Tucsnak. An optimal control approach to ciliary locomotion. Mathematical Control and Related Fields, 2016, 6 (2) : 293-334. doi: 10.3934/mcrf.2016005
##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. [2] F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers, Math. Models Methods Appl. Sci., 21 (2011), 361-387. doi: 10.1142/S0218202511005088. [3] F. Alouges, A. DeSimone and A. Lefebvre, Biological fluid dynamics, nonlinear partial differential equations, in Encyclopedia of Complexity and Systems Science, (2009), 548-554. [4] F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymetric microswimmers, Eur. Phys. J. E., 28 (2009), 279-284. [5] F. Alouges and L. Giraldi, Enhanced controllability of low Reynolds number swimmers in the presence of a wall, Acta Appl. Math., 128 (2013), 153-179. doi: 10.1007/s10440-013-9824-5. [6] J. R. Blake, A spherical envelope approch to ciliary propulsion, J. Fluid Mech., 46 (1971), 199-208. [7] T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid, J. Nonlinear Sci., 21 (2011), 325-385. doi: 10.1007/s00332-010-9084-8. [8] _________, Generic controllability of 3D swimmers in a perfect fluid, SIAM J. Control Optim., 50 (2012), 2814-2835. [9] S. Childress, Mechanics of Swimming and Flying, vol. 2 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1981. [10] J.-M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. [11] G. Dassios, M. Hadjinicolaou and A. C. Payatakes, Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates, Quart. Appl. Math., 52 (1994), 157-191. [12] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Monographs in Mathematics, Springer, New York, second ed., 2011. Steady-state problems. doi: 10.1007/978-0-387-09620-9. [13] D. Gérard-Varet and L. Giraldi, Rough wall effect on micro-swimmers, ESAIM Control Optim. Calc. Var., 21 (2015), 757-788. doi: 10.1051/cocv/2014046. [14] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Prentice-Hall Inc., Englewood Cliffs, N.J., 1965. [15] T. Ishikawa, M. P. Simmonds and T. J. Pedley, Hydrodynamic interaction of two swimming model micro-organisms, J. Fluid Mech., 568 (2006), 119-160. doi: 10.1017/S0022112006002631. [16] V. Jurdjevic, Geometric Control Theory, vol. 52 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1997. [17] E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Sci., 15 (2005), 255-289. doi: 10.1007/s00332-004-0650-9. [18] E. Lauga and T. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys., 72 (2009), 096601 (36pp). doi: 10.1088/0034-4885/72/9/096601. [19] J. Lohéac and A. Munnier, Controllability of 3D low Reynolds number swimmers, ESAIM Control Optim. Calc. Var., 20 (2014), 236-268. doi: 10.1051/cocv/2013063. [20] J. Lohéac and J.-F. Scheid, Time optimal control for a nonholonomic system with state constraint, Math. Control Relat. Fields, 3 (2013), 185-208. doi: 10.3934/mcrf.2013.3.185. [21] J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers, Acta Appl. Math., 123 (2013), 175-200. doi: 10.1007/s10440-012-9760-9. [22] S. Michelin and E. Lauga, Efficiency optimization and symmetry-breaking in a model of ciliary locomotion, Physics of Fluids, 22 (2010), p. 111901. doi: 10.1063/1.3507951. [23] A. Munnier, On the self-displacement of deformable bodies in a potential fluid flow, Math. Models Methods Appl. Sci., 18 (2008), 1945-1981. doi: 10.1142/S021820250800325X. [24] A. Munnier and B. Pinçon, Locomotion of articulated bodies in an ideal fluid: 2D model with buoyancy, circulation and collisions, Math. Models Methods Appl. Sci., 20 (2010), 1899-1940. doi: 10.1142/S0218202510004829. [25] J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math., 65 (2007), 405-424. doi: 10.1090/S0033-569X-07-01045-9. [26] A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number, J. Fluid. Mech., 198 (1989), 587-599. doi: 10.1017/S0022112089000261. [27] M. Sigalotti and J.-C. Vivalda, Controllability properties of a class of systems modeling swimming microscopic organisms, ESAIM Control Optim. Calc. Var., 16 (2010), 1053-1076. doi: 10.1051/cocv/2009034. [28] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. [29] G. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London. Ser. A., 209 (1951), 447-461. doi: 10.1098/rspa.1951.0218. [30] E. Trélat, Contrôle Optimal, Mathématiques Concrètes. [Concrete Mathematics], Vuibert, Paris, 2005. Théorie & applications. [Theory and applications]. [31] A. Wächter and L. T. Biegler, on the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-27. [32] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608759.

show all references

##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. [2] F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers, Math. Models Methods Appl. Sci., 21 (2011), 361-387. doi: 10.1142/S0218202511005088. [3] F. Alouges, A. DeSimone and A. Lefebvre, Biological fluid dynamics, nonlinear partial differential equations, in Encyclopedia of Complexity and Systems Science, (2009), 548-554. [4] F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymetric microswimmers, Eur. Phys. J. E., 28 (2009), 279-284. [5] F. Alouges and L. Giraldi, Enhanced controllability of low Reynolds number swimmers in the presence of a wall, Acta Appl. Math., 128 (2013), 153-179. doi: 10.1007/s10440-013-9824-5. [6] J. R. Blake, A spherical envelope approch to ciliary propulsion, J. Fluid Mech., 46 (1971), 199-208. [7] T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid, J. Nonlinear Sci., 21 (2011), 325-385. doi: 10.1007/s00332-010-9084-8. [8] _________, Generic controllability of 3D swimmers in a perfect fluid, SIAM J. Control Optim., 50 (2012), 2814-2835. [9] S. Childress, Mechanics of Swimming and Flying, vol. 2 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1981. [10] J.-M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. [11] G. Dassios, M. Hadjinicolaou and A. C. Payatakes, Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates, Quart. Appl. Math., 52 (1994), 157-191. [12] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Monographs in Mathematics, Springer, New York, second ed., 2011. Steady-state problems. doi: 10.1007/978-0-387-09620-9. [13] D. Gérard-Varet and L. Giraldi, Rough wall effect on micro-swimmers, ESAIM Control Optim. Calc. Var., 21 (2015), 757-788. doi: 10.1051/cocv/2014046. [14] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Prentice-Hall Inc., Englewood Cliffs, N.J., 1965. [15] T. Ishikawa, M. P. Simmonds and T. J. Pedley, Hydrodynamic interaction of two swimming model micro-organisms, J. Fluid Mech., 568 (2006), 119-160. doi: 10.1017/S0022112006002631. [16] V. Jurdjevic, Geometric Control Theory, vol. 52 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1997. [17] E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Sci., 15 (2005), 255-289. doi: 10.1007/s00332-004-0650-9. [18] E. Lauga and T. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys., 72 (2009), 096601 (36pp). doi: 10.1088/0034-4885/72/9/096601. [19] J. Lohéac and A. Munnier, Controllability of 3D low Reynolds number swimmers, ESAIM Control Optim. Calc. Var., 20 (2014), 236-268. doi: 10.1051/cocv/2013063. [20] J. Lohéac and J.-F. Scheid, Time optimal control for a nonholonomic system with state constraint, Math. Control Relat. Fields, 3 (2013), 185-208. doi: 10.3934/mcrf.2013.3.185. [21] J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers, Acta Appl. Math., 123 (2013), 175-200. doi: 10.1007/s10440-012-9760-9. [22] S. Michelin and E. Lauga, Efficiency optimization and symmetry-breaking in a model of ciliary locomotion, Physics of Fluids, 22 (2010), p. 111901. doi: 10.1063/1.3507951. [23] A. Munnier, On the self-displacement of deformable bodies in a potential fluid flow, Math. Models Methods Appl. Sci., 18 (2008), 1945-1981. doi: 10.1142/S021820250800325X. [24] A. Munnier and B. Pinçon, Locomotion of articulated bodies in an ideal fluid: 2D model with buoyancy, circulation and collisions, Math. Models Methods Appl. Sci., 20 (2010), 1899-1940. doi: 10.1142/S0218202510004829. [25] J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math., 65 (2007), 405-424. doi: 10.1090/S0033-569X-07-01045-9. [26] A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number, J. Fluid. Mech., 198 (1989), 587-599. doi: 10.1017/S0022112089000261. [27] M. Sigalotti and J.-C. Vivalda, Controllability properties of a class of systems modeling swimming microscopic organisms, ESAIM Control Optim. Calc. Var., 16 (2010), 1053-1076. doi: 10.1051/cocv/2009034. [28] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. [29] G. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London. Ser. A., 209 (1951), 447-461. doi: 10.1098/rspa.1951.0218. [30] E. Trélat, Contrôle Optimal, Mathématiques Concrètes. [Concrete Mathematics], Vuibert, Paris, 2005. Théorie & applications. [Theory and applications]. [31] A. Wächter and L. T. Biegler, on the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-27. [32] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608759.
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