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July  2018, 14(3): 1251-1269. doi: 10.3934/jimo.2018052

Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels

1. 

Faculty of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, China

2. 

State Key Laboratory of Industrial Control Technology, and Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou, Zhejiang 310027, China

3. 

School of Automation, Guangdong University of Technology, Guangzhou, Guangdong 510006, China

* Corresponding author: Tehuan Chen.

Received  August 2017 Revised  January 2018 Published  April 2018

Fund Project: This work was partially supported by the National Natural Science Foundation of China (61703217, 61703114, 61473253), the K. C. Wong Magna Fund in Ningbo University, the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (No. ICT170301, ICT170288) and the Medical Science and Technology Project, Zhejiang Province (2016154240, 2017192644).

Diffusiophoresis is a common phenomenon that occurs when colloids are placed in the non-uniform solute concentration. It generates solute gradients which force the colloids to transfer toward or away from the higher solute concentration side. In this paper, we consider the input sequence control of the colloid transport in a dead-end micro-channel with a boundary solute concentration being manipulated, which has a wide range of applications such as drug delivery, biology transport, oil recovery system and so on. We model this process by a coupled system, which involves the solute diffusion equation and the colloid transport model. Then an optimal control problem is formulated, in which the goal is to minimize colloid density distribution deviation between the computational one and the target at a pre-specified terminal time. To solve this partial differential equation (PDE) optimal control problem, we first apply the control parameterization method to discretize the boundary control and transfer it into an optimal parameter selection problem. Then, using the variational method, the gradient of the objective function with respect to the decision parameters can be derived, which depends on the solution of the coupled system and the costate system. Based on this, we propose an effective computational method and a gradient-based optimization algorithm to solve the optimal control problem numerically. Finally, we give the simulation results to demonstrate that the objective function based on the proposed method is less nearly two orders of magnitude than that of a constant value control strategy, which well illustrates the effectiveness of the proposed method.

Citation: Tehuan Chen, Chao Xu, Zhigang Ren. Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1251-1269. doi: 10.3934/jimo.2018052
References:
[1]

K. E. AbderrezzakR. Ata and F. Zaoui, One-dimensional numerical modelling of solute transport in streams: The role of longitudinal dispersion coefficient, Journal of Hydrology, 527 (2015), 978-989.  doi: 10.1016/j.jhydrol.2015.05.061.  Google Scholar

[2]

B. AbécassisC. Cottin-BizonneC. YbertA. Ajdari and L. Bocquet, Boosting migration of large particles by solute contrasts, Nature Materials, 7 (2008), 785-789.   Google Scholar

[3]

J. L. Anderson and D. C. Prieve, Diffusiophoresis: Migration of colloidal particles in gradients of solute concentration, Separation and Purification Methods, 13 (2012), 67-103.  doi: 10.1080/03602548408068407.  Google Scholar

[4]

A. M. AnnaswamyM. FleifilJ. W. RumseyR. PrasanthJ. P. Hathout and A. F. Ghoniem, Thermoacoustic instability: Model-based optimal control designs and experimental validation, IEEE Transactions on Control Systems and Technology, 8 (2000), 905-918.  doi: 10.1109/87.880593.  Google Scholar

[5]

E. E. L. BernalV. I. KovalchukE. K. Zholkovskiy and A. Yaroshchuk, Hydrodynamic dispersion in long microchannels under conditions of electroosmotic circulation. i. non-electrolytes, Microfluidics and Nanofluidics, 20 (2016), 1-19.   Google Scholar

[6]

M. A. BevanD. M. FordM. A. GroverB. ShapiroD. MaroudasY. YangR. ThyagarajanX. Tang and R. M. Sehgal, Controlling assembly of colloidal particles into structured objects: Basic strategy and a case study, Journal of Process Control, 27 (2015), 64-75.  doi: 10.1016/j.jprocont.2014.11.011.  Google Scholar

[7]

L. T. Biegler, Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, Society for Industrial and Applied Mathematics, 2010.  Google Scholar

[8]

A. M. BizerayS. ZhaoS. Duncan and D. A. Howey, Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter, Journal of Power Sources, 296 (2015), 400-412.  doi: 10.1016/j.jpowsour.2015.07.019.  Google Scholar

[9]

H. Bockelmann, V. Heuveline and D. P. Barz, Optimization of an electrokinetic mixer for microfluidic applications, Biomicrofluidics, 6 (2012), 024123 doi: 10.1063/1.4722000.  Google Scholar

[10]

J. Brodie and G. Jerauld, Impact of Salt Diffusion on Low-Salinity Enhanced Oil Recovery, SPE Improved Oil Recovery Symposium. Society of Petroleum Engineers, 2014. doi: 10.2118/169097-MS.  Google Scholar

[11]

R. L. Burden and J. D. Faires, Numerical Analysis, Cengage Learning, 1993. Google Scholar

[12]

M. CallewaertW. De MalscheH. OttevaereH. Thienpont and G. Desmet, Assessment and numerical search for minimal Taylor-Aris dispersion in micro-machined channels of nearly rectangular cross-section, Journal of Chromatography A, 1368 (2014), 70-81.  doi: 10.1016/j.chroma.2014.09.009.  Google Scholar

[13]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486.  doi: 10.3934/jimo.2013.9.471.  Google Scholar

[14]

T. Chen and C. Xu, Computational optimal control of the Saint-Venant PDE model using the time-scaling technique, Asia-Pacific Journal of Chemical Engineering, 11 (2016), 70-80.  doi: 10.1002/apj.1944.  Google Scholar

[15]

X. Chen, Topology optimization of microfluidics-a review, Microchemical Journal, 127 (2016), 52-61.  doi: 10.1016/j.microc.2016.02.005.  Google Scholar

[16]

T. ChenZ. RenC. Xu and R. Loxton, Optimal boundary control for water hammer suppression in fluid transmission pipelines, Computers & Mathematics With Applications, 69 (2015), 275-290.  doi: 10.1016/j.camwa.2014.11.008.  Google Scholar

[17]

N. M. Denovio and J. E. Saiers, Colloid movement in unsaturated porous media, Vadose Zone Journal, 3 (2004), 338-351.  doi: 10.2136/vzj2004.0338.  Google Scholar

[18]

A. Erbe, M. Zientara, L. Baraban, C. Kreidler and P. Leiderer, Various driving mechanisms for generating motion of colloidal particles, Journal of Physics: Condensed Matter, 20 (2008). doi: 10.1088/0953-8984/20/40/404215.  Google Scholar

[19]

A. E. Frankel and A. S. Khair, Dynamics of a self-diffusiophoretic particle in shear flow, Physical Review E, 90 (2014), 013030. doi: 10.1103/PhysRevE.90.013030.  Google Scholar

[20]

R. Golestanian, Anomalous diffusion of symmetric and asymmetric active colloids, Physical Review Letters, 102 (2009), 188305. doi: 10.1103/PhysRevLett.102.188305.  Google Scholar

[21]

R. Golestanian, T. B. Liverpool and A. Ajdari, Designing phoretic micro-and nano-swimmers, New Journal of Physics, 9 (2007), p126. doi: 10.1088/1367-2630/9/5/126.  Google Scholar

[22]

U. Hashim, T. Adam, P. N. A. Diyana and T. T. Seng, Computational micro fluid dynamics using COMSOL multiphysics for sample delivery in sensing domain, in IEEE International Conference on Biomedical Engineering and Sciences, Langkawi, UK, December 17th -19th, 2012. doi: 10.1109/IECBES.2012.6498208.  Google Scholar

[23]

D. Huang and J. Xu, Steady-state iterative learning control for a class of nonlinear PDE processes, Journal of Process Control, 21 (2011), 1155-1163.  doi: 10.1016/j.jprocont.2011.06.018.  Google Scholar

[24]

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Courier Corporation, 2009.  Google Scholar

[25]

F. Jülicher and J. Prost, Generic theory of colloidal transport, The European Physical Journal E, 29 (2009), 27-36.   Google Scholar

[26]

T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, Journal of Computational and Applied Mathematics, 215 (2008), 182-195.  doi: 10.1016/j.cam.2007.04.003.  Google Scholar

[27]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, 2008.  Google Scholar

[28]

J. Lee and W. F. Ramirez, Optimal fed-batch control of induced foreign protein production by recombinant bacteria, AIChE Journal, 40 (1994), 899-907.  doi: 10.1002/aic.690400516.  Google Scholar

[29]

D. Li, Electrokinetics in Microfluidics, Academic Press, 2004. Google Scholar

[30]

Q. LinR. C. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.   Google Scholar

[31]

C. LiuC. XueJ. Sun and G. Hu, A generalized formula for inertial lift on a sphere in microchannels, Lab on a Chip, 16 (2016), 884-892.  doi: 10.1039/C5LC01522G.  Google Scholar

[32]

H. Liu and Y. Zhang, Modelling thermocapillary migration of a microfluidic droplet on a solid surface, Journal of Computational Physics, 280 (2015), 37-53.  doi: 10.1016/j.jcp.2014.09.015.  Google Scholar

[33]

C. LiuR. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7.  Google Scholar

[34]

A. Logg, K. A. Mardal and G. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer Berlin Heidelberg, 2012. Google Scholar

[35]

D. Lombardi and P. S. Dittrich, Advances in microfluidics for drug discovery, Expert Opinion on Drug Discovery, 5 (2010), 1081-1094.   Google Scholar

[36]

R. LoxtonK. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[37]

Y. Z. Lu and T. J. Williams, Modelling, Estimation and Control of the Soaking Pit: An Example of the Development and Application of Some Modern Control Techniques to Industrial Processes, Purdue University, 1982. Google Scholar

[38]

D. Michler and R. Sprik, Directed vesicle transport by diffusio-osmosis, Europhysics Letters, 110 (2015), p28001. doi: 10.1209/0295-5075/110/28001.  Google Scholar

[39]

S. J. Moura and H. K. Fathy, Optimal boundary control & estimation of diffusion-reaction PDEs, in Proceedings of the 2011 American Control Conference, San Francisco, USA, June 29-July 1, 2011. doi: 10.1109/ACC.2011.5990900.  Google Scholar

[40]

V. T. NguyenD. Georges and G. Besancon, State and parameter estimation in 1-{D} hyperbolic {PDE}s based on an adjoint method, Automatica, 67 (2016), 185-191.  doi: 10.1016/j.automatica.2016.01.031.  Google Scholar

[41]

J. Palacci, B. Abecassis, C. Cottinbizonne, C. Ybert and L. Bocquet, Colloidal motility and pattern formation under rectified diffusiophoresis, Physical Review Letters, 104 (2010), 138302. doi: 10.1103/PhysRevLett.104.138302.  Google Scholar

[42]

R. Piazza, Thermophoresis: Moving particles with thermal gradients, Soft Matter, 4 (2008), 1740-1744.  doi: 10.1039/b805888c.  Google Scholar

[43]

O. Rosen and R. Luus, Evaluation of gradients for piecewise constant optimal control, Computers & Chemical Engineering, 15 (1991), 273-281.  doi: 10.1016/0098-1354(91)85013-K.  Google Scholar

[44]

B. Sabass and U. Seifert, Dynamics and efficiency of a self-propelled, diffusiophoretic swimmer, Journal of Chemical Physics, 136 (2011), 64508. Google Scholar

[45]

W. E. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations, Academic Press, Inc., San Diego, CA, 1991.  Google Scholar

[46]

S. ShinE. UmB. SabassJ. T. AultM. RahimiP. B. Warren and H. A. Stone, Size-dependent control of colloid transport via solute gradients in dead-end channels, Proceedings of the National Academy of Sciences, 113 (2016), 257-261.  doi: 10.1073/pnas.1511484112.  Google Scholar

[47]

K. L. TeoV. Rehbock and K. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

[48]

S. Torkzaban, Colloid transport and retention in unsaturated porous media: A review of interface-, collector-, and pore-scale processes and models, Vadose Zone Journal, 7 (2008), 667-681.   Google Scholar

[49]

S. Wang and X. Lou, An optimization approach to the estimation of effective drug diffusivity: From a planar disc into a finite external volume, Journal of Industrial and Management Optimization, 5 (2009), 127-140.   Google Scholar

[50]

Z. Wu and G. Chen, Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe, Journal of Fluid Mechanics, 740 (2014), 196-213.  doi: 10.1017/jfm.2013.648.  Google Scholar

[51]

C. YuK. L. TeoL. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

show all references

References:
[1]

K. E. AbderrezzakR. Ata and F. Zaoui, One-dimensional numerical modelling of solute transport in streams: The role of longitudinal dispersion coefficient, Journal of Hydrology, 527 (2015), 978-989.  doi: 10.1016/j.jhydrol.2015.05.061.  Google Scholar

[2]

B. AbécassisC. Cottin-BizonneC. YbertA. Ajdari and L. Bocquet, Boosting migration of large particles by solute contrasts, Nature Materials, 7 (2008), 785-789.   Google Scholar

[3]

J. L. Anderson and D. C. Prieve, Diffusiophoresis: Migration of colloidal particles in gradients of solute concentration, Separation and Purification Methods, 13 (2012), 67-103.  doi: 10.1080/03602548408068407.  Google Scholar

[4]

A. M. AnnaswamyM. FleifilJ. W. RumseyR. PrasanthJ. P. Hathout and A. F. Ghoniem, Thermoacoustic instability: Model-based optimal control designs and experimental validation, IEEE Transactions on Control Systems and Technology, 8 (2000), 905-918.  doi: 10.1109/87.880593.  Google Scholar

[5]

E. E. L. BernalV. I. KovalchukE. K. Zholkovskiy and A. Yaroshchuk, Hydrodynamic dispersion in long microchannels under conditions of electroosmotic circulation. i. non-electrolytes, Microfluidics and Nanofluidics, 20 (2016), 1-19.   Google Scholar

[6]

M. A. BevanD. M. FordM. A. GroverB. ShapiroD. MaroudasY. YangR. ThyagarajanX. Tang and R. M. Sehgal, Controlling assembly of colloidal particles into structured objects: Basic strategy and a case study, Journal of Process Control, 27 (2015), 64-75.  doi: 10.1016/j.jprocont.2014.11.011.  Google Scholar

[7]

L. T. Biegler, Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, Society for Industrial and Applied Mathematics, 2010.  Google Scholar

[8]

A. M. BizerayS. ZhaoS. Duncan and D. A. Howey, Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter, Journal of Power Sources, 296 (2015), 400-412.  doi: 10.1016/j.jpowsour.2015.07.019.  Google Scholar

[9]

H. Bockelmann, V. Heuveline and D. P. Barz, Optimization of an electrokinetic mixer for microfluidic applications, Biomicrofluidics, 6 (2012), 024123 doi: 10.1063/1.4722000.  Google Scholar

[10]

J. Brodie and G. Jerauld, Impact of Salt Diffusion on Low-Salinity Enhanced Oil Recovery, SPE Improved Oil Recovery Symposium. Society of Petroleum Engineers, 2014. doi: 10.2118/169097-MS.  Google Scholar

[11]

R. L. Burden and J. D. Faires, Numerical Analysis, Cengage Learning, 1993. Google Scholar

[12]

M. CallewaertW. De MalscheH. OttevaereH. Thienpont and G. Desmet, Assessment and numerical search for minimal Taylor-Aris dispersion in micro-machined channels of nearly rectangular cross-section, Journal of Chromatography A, 1368 (2014), 70-81.  doi: 10.1016/j.chroma.2014.09.009.  Google Scholar

[13]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486.  doi: 10.3934/jimo.2013.9.471.  Google Scholar

[14]

T. Chen and C. Xu, Computational optimal control of the Saint-Venant PDE model using the time-scaling technique, Asia-Pacific Journal of Chemical Engineering, 11 (2016), 70-80.  doi: 10.1002/apj.1944.  Google Scholar

[15]

X. Chen, Topology optimization of microfluidics-a review, Microchemical Journal, 127 (2016), 52-61.  doi: 10.1016/j.microc.2016.02.005.  Google Scholar

[16]

T. ChenZ. RenC. Xu and R. Loxton, Optimal boundary control for water hammer suppression in fluid transmission pipelines, Computers & Mathematics With Applications, 69 (2015), 275-290.  doi: 10.1016/j.camwa.2014.11.008.  Google Scholar

[17]

N. M. Denovio and J. E. Saiers, Colloid movement in unsaturated porous media, Vadose Zone Journal, 3 (2004), 338-351.  doi: 10.2136/vzj2004.0338.  Google Scholar

[18]

A. Erbe, M. Zientara, L. Baraban, C. Kreidler and P. Leiderer, Various driving mechanisms for generating motion of colloidal particles, Journal of Physics: Condensed Matter, 20 (2008). doi: 10.1088/0953-8984/20/40/404215.  Google Scholar

[19]

A. E. Frankel and A. S. Khair, Dynamics of a self-diffusiophoretic particle in shear flow, Physical Review E, 90 (2014), 013030. doi: 10.1103/PhysRevE.90.013030.  Google Scholar

[20]

R. Golestanian, Anomalous diffusion of symmetric and asymmetric active colloids, Physical Review Letters, 102 (2009), 188305. doi: 10.1103/PhysRevLett.102.188305.  Google Scholar

[21]

R. Golestanian, T. B. Liverpool and A. Ajdari, Designing phoretic micro-and nano-swimmers, New Journal of Physics, 9 (2007), p126. doi: 10.1088/1367-2630/9/5/126.  Google Scholar

[22]

U. Hashim, T. Adam, P. N. A. Diyana and T. T. Seng, Computational micro fluid dynamics using COMSOL multiphysics for sample delivery in sensing domain, in IEEE International Conference on Biomedical Engineering and Sciences, Langkawi, UK, December 17th -19th, 2012. doi: 10.1109/IECBES.2012.6498208.  Google Scholar

[23]

D. Huang and J. Xu, Steady-state iterative learning control for a class of nonlinear PDE processes, Journal of Process Control, 21 (2011), 1155-1163.  doi: 10.1016/j.jprocont.2011.06.018.  Google Scholar

[24]

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Courier Corporation, 2009.  Google Scholar

[25]

F. Jülicher and J. Prost, Generic theory of colloidal transport, The European Physical Journal E, 29 (2009), 27-36.   Google Scholar

[26]

T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, Journal of Computational and Applied Mathematics, 215 (2008), 182-195.  doi: 10.1016/j.cam.2007.04.003.  Google Scholar

[27]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, 2008.  Google Scholar

[28]

J. Lee and W. F. Ramirez, Optimal fed-batch control of induced foreign protein production by recombinant bacteria, AIChE Journal, 40 (1994), 899-907.  doi: 10.1002/aic.690400516.  Google Scholar

[29]

D. Li, Electrokinetics in Microfluidics, Academic Press, 2004. Google Scholar

[30]

Q. LinR. C. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.   Google Scholar

[31]

C. LiuC. XueJ. Sun and G. Hu, A generalized formula for inertial lift on a sphere in microchannels, Lab on a Chip, 16 (2016), 884-892.  doi: 10.1039/C5LC01522G.  Google Scholar

[32]

H. Liu and Y. Zhang, Modelling thermocapillary migration of a microfluidic droplet on a solid surface, Journal of Computational Physics, 280 (2015), 37-53.  doi: 10.1016/j.jcp.2014.09.015.  Google Scholar

[33]

C. LiuR. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7.  Google Scholar

[34]

A. Logg, K. A. Mardal and G. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer Berlin Heidelberg, 2012. Google Scholar

[35]

D. Lombardi and P. S. Dittrich, Advances in microfluidics for drug discovery, Expert Opinion on Drug Discovery, 5 (2010), 1081-1094.   Google Scholar

[36]

R. LoxtonK. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[37]

Y. Z. Lu and T. J. Williams, Modelling, Estimation and Control of the Soaking Pit: An Example of the Development and Application of Some Modern Control Techniques to Industrial Processes, Purdue University, 1982. Google Scholar

[38]

D. Michler and R. Sprik, Directed vesicle transport by diffusio-osmosis, Europhysics Letters, 110 (2015), p28001. doi: 10.1209/0295-5075/110/28001.  Google Scholar

[39]

S. J. Moura and H. K. Fathy, Optimal boundary control & estimation of diffusion-reaction PDEs, in Proceedings of the 2011 American Control Conference, San Francisco, USA, June 29-July 1, 2011. doi: 10.1109/ACC.2011.5990900.  Google Scholar

[40]

V. T. NguyenD. Georges and G. Besancon, State and parameter estimation in 1-{D} hyperbolic {PDE}s based on an adjoint method, Automatica, 67 (2016), 185-191.  doi: 10.1016/j.automatica.2016.01.031.  Google Scholar

[41]

J. Palacci, B. Abecassis, C. Cottinbizonne, C. Ybert and L. Bocquet, Colloidal motility and pattern formation under rectified diffusiophoresis, Physical Review Letters, 104 (2010), 138302. doi: 10.1103/PhysRevLett.104.138302.  Google Scholar

[42]

R. Piazza, Thermophoresis: Moving particles with thermal gradients, Soft Matter, 4 (2008), 1740-1744.  doi: 10.1039/b805888c.  Google Scholar

[43]

O. Rosen and R. Luus, Evaluation of gradients for piecewise constant optimal control, Computers & Chemical Engineering, 15 (1991), 273-281.  doi: 10.1016/0098-1354(91)85013-K.  Google Scholar

[44]

B. Sabass and U. Seifert, Dynamics and efficiency of a self-propelled, diffusiophoretic swimmer, Journal of Chemical Physics, 136 (2011), 64508. Google Scholar

[45]

W. E. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations, Academic Press, Inc., San Diego, CA, 1991.  Google Scholar

[46]

S. ShinE. UmB. SabassJ. T. AultM. RahimiP. B. Warren and H. A. Stone, Size-dependent control of colloid transport via solute gradients in dead-end channels, Proceedings of the National Academy of Sciences, 113 (2016), 257-261.  doi: 10.1073/pnas.1511484112.  Google Scholar

[47]

K. L. TeoV. Rehbock and K. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

[48]

S. Torkzaban, Colloid transport and retention in unsaturated porous media: A review of interface-, collector-, and pore-scale processes and models, Vadose Zone Journal, 7 (2008), 667-681.   Google Scholar

[49]

S. Wang and X. Lou, An optimization approach to the estimation of effective drug diffusivity: From a planar disc into a finite external volume, Journal of Industrial and Management Optimization, 5 (2009), 127-140.   Google Scholar

[50]

Z. Wu and G. Chen, Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe, Journal of Fluid Mechanics, 740 (2014), 196-213.  doi: 10.1017/jfm.2013.648.  Google Scholar

[51]

C. YuK. L. TeoL. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

Figure 1.  General layout of the main channel combined with a dead-end micro-channel
Figure 2.  Target distribution and optimal parameters
Figure 3.  Optimal colloid density distribution at fixed time point and numerical errors
Figure 4.  Optimal spatiotemporal evolution of colloid transport
Table 1.  Optimal control parameters $u(\tau) = {\sigma}^k, k = 4$
$p=4$ 1 2 3 4
${\sigma}^k$ $0.019997$ $0.010004$ $0.080603$ $0.029775$
$p=4$ 1 2 3 4
${\sigma}^k$ $0.019997$ $0.010004$ $0.080603$ $0.029775$
Table 2.  Optimal control parameters $u(\tau) = {\sigma}^k, k = 8$
$p=8$ 1 2 3 4
${\sigma}^k$ $0.019998$ $0.019997$ $0.010003$ $0.010002$
$p=8$ 5 6 7 8
${\sigma}^k$ $0.080410$ $0.080297$ 0.029837 0.02985
$p=8$ 1 2 3 4
${\sigma}^k$ $0.019998$ $0.019997$ $0.010003$ $0.010002$
$p=8$ 5 6 7 8
${\sigma}^k$ $0.080410$ $0.080297$ 0.029837 0.02985
[1]

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