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Discrete & Continuous Dynamical Systems - S

June 2020 , Volume 13 , Issue 6

Issue on advances in theory and real world applications of control and dynamic optimization

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Preface: Advances in theory and real world applications of control and dynamic optimization
Changjun Yu, Honglei Xu and Kok Lay Teo
2020, 13(6): ⅰ-ⅲ doi: 10.3934/dcdss.2020094 +[Abstract](197) +[HTML](51) +[PDF](89.39KB)
A linear optimal feedback control for producing 1, 3-propanediol via microbial fermentation
Honghan Bei, Lei Wang, Yanping Ma, Jing Sun and Liwei Zhang
2020, 13(6): 1623-1635 doi: 10.3934/dcdss.2020095 +[Abstract](688) +[HTML](336) +[PDF](355.36KB)

In this paper, we consider a multistage feedback control strategy for the production of 1, 3-propanediol(1, 3-PD) in microbial fermentation. The feedback control strategy is widely used in industry, and to the best of our knowledge, this is the first time it is applied to 1, 3-PD. The feedback control law is assumed to be linear of the concentrations of biomass and glycerol, and the coefficients in the controller are continuous. A multistage feedback control law is obtained by using the control parameterization method on the coefficient functions. Then, the optimal control problem can be transformed into an optimal parameter selection problem. The time horizon is partitioned adaptively. The corresponding gradients are derived, and finally, our numerical results indicate that the strategy is flexible and efficient.

Optimization model and solution method for dynamically correlated two-product newsvendor problems based on Copula
Songhai Deng, Zhong Wan and Yanjiu Zhou
2020, 13(6): 1637-1652 doi: 10.3934/dcdss.2020096 +[Abstract](602) +[HTML](313) +[PDF](483.99KB)

In this paper, a two-product newsvendor problem is taken into consideration, where the demands of products are correlated random variables and the buyer is risk-averse. Some important qualitative properties of the constructed model are analyzed, particularly the gradient information of the model is obtained and incorporated into solution method of the model. Based on the theory of Copulas, an efficient algorithm, called the feasible-direction based BFGS algorithm, is developed for solution of the constrained optimization model. Case study shows the efficiency of model and algorithm, and numerical results demonstrate that compared with the situation of independent demands, the total order quantity reduces against the high dynamic interrelation coefficient of two demands with the same degree of risk-aversion, and the optimal order quantity decreases as the degree of risk-aversion becomes greater.

A multi-stage method for joint pricing and inventory model with promotion constrains
Li Deng, Wenjie Bi, Haiying Liu and Kok Lay Teo
2020, 13(6): 1653-1682 doi: 10.3934/dcdss.2020097 +[Abstract](756) +[HTML](383) +[PDF](531.31KB)

In this paper, we consider a joint pricing and inventory problem with promotion constrains over a finite planning horizon for a single fast-moving consumer good under monopolistic environment. The decision on the inventory is realized through the decision on inventory replenishment, i.e., decision on the quantity to be ordered. The demand function takes into account all reference price mechanisms. The main difficulty in solving this problem is how to deal with the binary logical decision variables. It is shown that the problem is equivalent to a quadratic programming problem involving binary decision variables. This quadratic programming problem with binary decision variables can be expressed as a series of conventional quadratic programming problems, each of which can be easily solved. The global optimal solution can then be obtained readily from the global solutions of the conventional quadratic programming problems. This method works well when the planning horizon is short. For longer planning horizon, we propose a multi-stage method for finding a near-optimal solution. In numerical simulation, the accuracy and efficiency of this multi-stage method is compared with a genetic algorithm. The results obtained validate the applicability of the constructed model and the effectiveness of the approach proposed. They also provide several interesting and useful managerial insights.

Time-scaling transformation for optimal control problem with time-varying delay
Di Wu, Yanqin Bai and Fusheng Xie
2020, 13(6): 1683-1695 doi: 10.3934/dcdss.2020098 +[Abstract](687) +[HTML](323) +[PDF](327.48KB)

This paper focuses on the solution to nonlinear time-delay optimal control problems with time-varying delay subject to canonical equality and inequality constraints. Traditional control parameterization in conjunction with time-scaling transformation could optimize control parameters and switching times at the same time when the time-delays in the dynamic system are taken as constants. The purpose of this paper is to extend this method to solve the dynamic systems with time-varying delay. We introduce a hybrid time-scaling transformation that converts the given time-delay system into an equivalent system defined on a new time horizon with fixed switching times. Meanwhile, we obtain the value of time-delay state utilizing the relationship between the new time scale and the original one. After computing the gradients of the cost and constraints with respect to the control heights and its durations, we could solve the equivalent optimal control problem using gradient based optimization method.

Time-delay optimal control of a fed-batch production involving multiple feeds
Chongyang Liu and Meijia Han
2020, 13(6): 1697-1709 doi: 10.3934/dcdss.2020099 +[Abstract](651) +[HTML](297) +[PDF](466.27KB)

In this paper, we consider time-delay optimal control of 1, 3-propan-ediol (1, 3-PD) fed-batch production involving multiple feeds. First, we propose a nonlinear time-delay system involving feeds of glycerol and alkali to formulate the production process. Then, taking the feeding rates of glycerol and alkali as well as the terminal time of process as the controls, we present a time-delay optimal control model subject to control and state constraints to maximize 1, 3-PD productivity. By a time-scaling transformation, we convert the optimal control problem into an equivalent problem with fixed terminal time. Furthermore, by applying control parameterization and constraint transcription techniques, we approximate the equivalent problem by a sequence of finite-dimensional optimization problems. An improved particle swarm optimization algorithm is developed to solve the resulting optimization problems. Finally, numerical results show that 1, 3-PD productivity increases considerably using the obtained optimal control strategy.

Optimal control of hybrid manufacturing systems by log-exponential smoothing aggregation
Kobamelo Mashaba, Jianxing Li, Honglei Xu and Xinhua Jiang
2020, 13(6): 1711-1719 doi: 10.3934/dcdss.2020100 +[Abstract](571) +[HTML](283) +[PDF](280.99KB)

This paper studies new optimal control policies for solving complex decision-making problems encountered in industrial hybrid systems in a manufacturing setting where critical jobs exist in a busy structure. In such setting, different dynamical systems interlink each other and share common functions for smooth task execution. Entities arriving at shared resources compete for service. The interactions of industrial hybrid systems become more and more complex and need a suitable controller to achieve the best performance and to obtain the best possible service for each of the entities arriving at the system. To solve these challenges, we propose an optimal control policy to minimize the operational cost for the manufacturing system. Furthermore, we develop a hybrid model and a new smoothing algorithm for the cost balancing between the quality and the job tardiness by finding optimal service time of each job in the system.

Earth pressure field modeling for tunnel face stability evaluation of EPB shield machines based on optimization solution
Yi An, Zhuohan Li, Changzhi Wu, Huosheng Hu, Cheng Shao and Bo Li
2020, 13(6): 1721-1741 doi: 10.3934/dcdss.2020101 +[Abstract](720) +[HTML](328) +[PDF](2941.74KB)

Earth pressure balanced (EPB) shield machines are large and complex mechanical systems and have been widely applied to tunnel engineering. Tunnel face stability evaluation is very important for EPB shield machines to avoid ground settlement and guarantee safe construction during the tunneling process. In this paper, we propose a novel earth pressure field modeling approach to evaluate the tunnel face stability of large and complex EPB shield machines. Based on the earth pressures measured by the pressure sensors on the clapboard of the chamber, we construct a triangular mesh model for the earth pressure field in the chamber and estimate the normal vector at each measuring point by using optimization solution and projection Delaunay triangulation, which can reflect the change situation of the earth pressures in real time. Furthermore, we analyze the characteristics of the active and passive earth pressure fields in the limit equilibrium states and give a new evaluation criterion of the tunnel face stability based on Rankine's theory of earth pressure. The method validation and analysis demonstrate that the proposed method is effective for modeling the earth pressure field in the chamber and evaluating the tunnel face stability of EPB shield machines.

Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem
Yibing Lv, Tiesong Hu and Jianlin Jiang
2020, 13(6): 1743-1755 doi: 10.3934/dcdss.2020102 +[Abstract](704) +[HTML](299) +[PDF](289.54KB)

In this paper, a linear bilevel multiobjective programming problem is concerned. Based on the method of replacing the lower level problem with its optimality conditions, and taking the complementary constraints as the penalty term of the upper level objectives, we obtain the exact penalized multiobjective programming problem \begin{document}$ (P_{K}) $\end{document}. The concept of equilibrium point of problem \begin{document}$ (P_{K}) $\end{document} is introduced and its properties are analyzed. Thereafter, we propose a penalty method-based equilibrium point algorithm, which only needs to solve a series of linear programming problems, for the linear bilevel multiobjective programming problem. Numerical results showing viability of the equilibrium point approach are presented.

An alternating minimization method for matrix completion problems
Yuan Shen and Xin Liu
2020, 13(6): 1757-1772 doi: 10.3934/dcdss.2020103 +[Abstract](858) +[HTML](322) +[PDF](455.84KB)

Matrix completion problems have applications in various domains such as information theory, statistics, engineering, etc. Meanwhile, solving matrix completion problems is not a easy task since the nonconvex and nonsmooth rank operation is involved. Existing approaches can be categorized into two classes. The first ones use nuclear norm to take the place of rank operation, and any convex optimization algorithms can be used to solve the reformulated problem. The limitation of this class of approaches is singular value decomposition (SVD) is involved to tackle the nuclear norm which significantly increases the computational cost. The other ones factorize the target matrix by two slim matrices. Fast algorithms for solving the reformulated nonconvex optimization problem usually lack of global convergence, meanwhile convergence guaranteed algorithms require restricted stepsize. In this paper, we consider the matrix factorization model for matrix completion problems, and propose an alternating minimization method for solving it. The global convergence to a stationary point or local minimizer is guaranteed under mild conditions. We compare the proposed algorithm with some state-of-the-art algorithms in solving a bunch of testing problems. The numerical results illustrate the efficiency and great potential of our algorithm.

A new iterative identification method for damping control of power system in multi-interference
Miao Yu, Haoyang Lu and Weipeng Shang
2020, 13(6): 1773-1790 doi: 10.3934/dcdss.2020104 +[Abstract](663) +[HTML](299) +[PDF](612.7KB)

In this paper, we consider the closed-loop model of a power system in a multi-interference environment. For a multi-interference power system, the closed-loop identification is a difficult task. Yet, the model identification error can degrade the effect of the damping control. This could lead to instability of the power grid. Thus, for the closed-loop identification, we propose an iterative online identification algorithm based on the recursive least squares method and the v-gap distance. The convergence of the algorithm is proved by using direct method. The proposed algorithm is applied to the New England system, for which the results obtained are compared with those obtained using the prediction error method and the Runge-Kutta method. From the simulation study being carried out on the IEEE 39-bus New England system, we observe that by using the iterative identification algorithm proposed in this paper, the output response time is reduced by about half when compared with those obtained by using the prediction error method and the Runge-Kutta method. Also, the number of oscillations in the output response is less. These clearly indicate that the algorithm proposed can effectively suppress low frequency oscillation. As for the amplitudes of the output responses produced by the three methods, they are basically the same.

On evolution quasi-variational inequalities and implicit state-dependent sweeping processes
Samir Adly and Tahar Haddad
2020, 13(6): 1791-1801 doi: 10.3934/dcdss.2020105 +[Abstract](720) +[HTML](326) +[PDF](311.4KB)

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint \begin{document}$ {C(\cdot, u)} $\end{document} depends upon the unknown state \begin{document}$ u $\end{document}, which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set \begin{document}$ C $\end{document} does not depend explicitly on the state \begin{document}$ u $\end{document} (i.e. \begin{document}$ C: = C(t) $\end{document}) given in [4], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

Stabilization of a discrete-time system via nonlinear impulsive control
Shaohong Fang, Jing Huang and Jinying Ma
2020, 13(6): 1803-1811 doi: 10.3934/dcdss.2020106 +[Abstract](650) +[HTML](311) +[PDF](313.81KB)

An impulsive control is one of the important stabilizing control strategies and exhibits many strong system performances such as shorten action time, low power consumption, effective resistance to uncertainty. This paper develops a nonlinear impulsive control approach to stabilize discrete-time dynamical systems. Sufficient conditions for asymptotical stability of discrete-time impulsively controlled systems are derived. Furthermore, an Ishi chaotic neural network is effectively stabilized by a designed nonlinear impulsive control.

An optimal pid tuning method for a single-link manipulator based on the control parametrization technique
Bin Li, Xiaolong Guo, Xiaodong Zeng, Songyi Dian and Minhua Guo
2020, 13(6): 1813-1823 doi: 10.3934/dcdss.2020107 +[Abstract](736) +[HTML](311) +[PDF](401.63KB)

A control parametrization based optimal PID tuning scheme for a single-link manipulator is developed in this paper. The performance specifications of the control system are formulated as continuous state inequality constraints. Then, the PID optimal tuning problem of the single-link manipulator can be formulated as an optimal parameter selection problem subject to continuous inequality constraints. These continuous inequality constraints are handled by the constraint transcription method together with a local smoothing technique. In such a way, the transformed problem becomes an optimal parameter selection problem in a canonical form, which can be solved efficiently by control parametrization method. Since approach is using the gradient-based method, the corresponding gradient formulas for the cost function and the constraints are derived, respectively. The effectiveness of the proposed method is demonstrated by numerical simulations.

A time-scaling technique for time-delay switched systems
Linna Li, Changjun Yu, Ning Zhang, Yanqin Bai and Zhiyuan Gao
2020, 13(6): 1825-1843 doi: 10.3934/dcdss.2020108 +[Abstract](719) +[HTML](356) +[PDF](392.86KB)

In this paper, we consider a class of optimal control problems governed by nonlinear time-delay switched systems, in which the system parameters and switching times between different subsystems are decision variables to be optimized. We propose a new computational approach to deal with the computational difficulties caused by variable switching times. The original time-delay switched system is firstly transformed into an equivalent switched system defined on a new time horizon where the switching times are fixed, but each of the subsystems contain a variable time-delay that depends on the durations of each sub-system in the original system. By deriving the analytical form for the variable time-delay in the new time horizon, we can solve the new time-delay switched system. Then, gradient-based optimization algorithm can be applied to solve the equivalent problem efficiently. Numerical results show that this new approach is effective.

An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints
Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang and Ming Yu
2020, 13(6): 1845-1865 doi: 10.3934/dcdss.2020109 +[Abstract](723) +[HTML](319) +[PDF](432.91KB)

Adjoint methods applied to solve optimal control problems (OCPs) have a restriction that the number of constraints shall be less than that of optimization variables. Otherwise, they are less efficient than the forward methods. This paper proposes an efficient adjoint method to solve OCPs for index-\begin{document}$ 1 $\end{document} differential algebraic systems with continuous-time inequality constraints. The continuous-time inequality constraints are not discretized on time grid but transformed into integrals and penalized in the cost through an exact penalty function. Thus, all the constraints except for box constraints on optimization variables can be removed. Furthermore, a lifted implicit Runge-Kutta (IRK) integrator with adjoint sensitivity propagation is employed to accelerate the function and gradient evaluation procedure. Based on a sensitivity update technique, the number of Newton iterations involved in forward simulation can be reduced to one. Besides this, Lagrange interpolation is applied to approximate the states not on collocation points such that integrals in the penalty function can be evaluated on the same grid for forward simulation. Complexity analysis shows that, for the proposed algorithm, computation involved in the sensitivity propagation is comparable to that of forward one. Numerical simulations on the optimal maneuvering a Delta robot demonstrate that the computational speed of the proposed adjoint algorithm is comparable to that of our previous one, which is based on the lifted IRK integrator and forward sensitivity propagation.

Design of green bonds by double-barrier options
Shuhua Zhang, Zhuo Yang and Song Wang
2020, 13(6): 1867-1882 doi: 10.3934/dcdss.2020110 +[Abstract](721) +[HTML](340) +[PDF](492.08KB)

Green finance is an innovative model that can promote sustainable economic development. The green bonds also develop gradually as a part of green finance. The green bonds are designed to fund the projects of positive environmental impact. If the green bonds are superior to other debt securities, they will attract more investors' participation in green energy projects. Thus, the design of green bonds is crucial to the development of green bonds market. This article assumes that the floating rate of green bonds is linked to carbon price, and carbon price is described by a jump diffusion process. The carbon price fluctuation can lead to interest rate fluctuation of green bonds. We set two boundary values of carbon price, and the coupon rate is revalued when the carbon price reaches the boundary. The higher the carbon price is, the higher the coupon rate is to be paid by issuers. Thus, the boundary can impel issuers to boost energy savings and emission-reduction, and the higher interest rate will also attract more investors to invest in green bonds. The lower the carbon price is, the lower the interest rate is to be paid by issuers. Accordingly, the boundary may encourage issuers to boost emission reduction. This design can monitor and incentivize issuers to make more contribution to green finance. Furthermore, the design is characterized by the double-barrier option, such that the interest rate of green bonds can be obtained by double-barrier option pricing. Subsequently, the central difference method and the composite trapezoidal formula are employed to obtain the numerical solution. Finally, we conduct the sensitivity analysis of the model.

2019  Impact Factor: 1.233

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