JIMO
Sensor deployment for pipeline leakage detection via optimal boundary control strategies
Chao Xu Yimeng Dong Zhigang Ren Huachen Jiang Xin Yu
Journal of Industrial & Management Optimization 2015, 11(1): 199-216 doi: 10.3934/jimo.2015.11.199
We consider a multi-agent control problem using PDE techniques for a novel sensing problem arising in the leakage detection and localization of offshore pipelines. A continuous protocol is proposed using parabolic PDEs and then a boundary control law is designed using the maximum principle. Both analytical and numerical solutions of the optimality conditions are studied.
keywords: Offshore pipeline network leakage detection optimal control. continuation approach multi-agent system partial differential equations Lagrangian sensor
JIMO
Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels
Tehuan Chen Chao Xu Zhigang Ren
Journal of Industrial & Management Optimization 2018, 14(3): 1251-1269 doi: 10.3934/jimo.2018052

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.

keywords: Diffusiophoresis dead-end micro-channel colloid transport control parameterization variational method pde-constrained optimization computational optimal control
DCDS
The $C$-regularized semigroup method for partial differential equations with delays
Xin Yu Guojie Zheng Chao Xu
Discrete & Continuous Dynamical Systems - A 2016, 36(9): 5163-5181 doi: 10.3934/dcds.2016024
This paper is devoted to study the abstract functional differential equation (FDE) of the following form $$\dot{u}(t)=Au(t)+\Phi u_t,$$ where $A$ generates a $C$-regularized semigroup, which is the generalization of $C_0$-semigroup and can be applied to deal with many important differential operators that the $C_0$-semigroup can not be used to. We first show that the $C$-well-posedness of a FDE is equivalent to the $\mathscr{C}$-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator $\mathscr{C}$ is related with the operator $C$ and will be defined in the following text. Then, by making use of a perturbation result of $C$-regularized semigroup, a sufficient condition is provided for the $C$-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.
keywords: perturbation functional differential equation $C$-regularized semigroup $C$-well-posedness partial differential equation.

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