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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:

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##### References:
General layout of the main channel combined with a dead-end micro-channel
Target distribution and optimal parameters
Optimal colloid density distribution at fixed time point and numerical errors
Optimal spatiotemporal evolution of colloid transport
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 ${\sigma}^k$ $0.019997$ $0.010004$ $0.080603$ $0.029775$
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
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