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The time-scaling transformation technique used within the computational framework of control parameterization serves as an effective method for the optimization of control switching time as well as the control value for time-delay optimal control problems. However, the conventional time-scaling transformation stipulates that the switching times for all control components must be identical, which may not be feasible in real-world scenarios. To address this limitation, we introduce the sequential time-scaling transformation technique for solving time-delay optimal control problems. By sequentially applying control parameterization and time-scaling transformation to each of the control variable, the new technique enables the configuration of a varying number of segments for the control, and the switching time for each control component can be adaptively selected. Numerical results demonstrate that our approach not only achieves superior cost performance but also offers enhanced flexibility in control strategy. • We propose a sequential time-scaling transformation technique for time-delay optimal control problem. • The switching times for each of the control components can be adaptively optimized. • We give the derivation on the partial derivatives of the cost(constraint) with respect to all the decision variables. • The gradients of the cost (constraint) are derived based on the co-state method. • The proposed new method can obtain better solutions within less time than conventional time scaling transformation method. [ABSTRACT FROM AUTHOR]

The online reputation evaluation problem is an urgent issue in global business that makes use of e-commerce. However, in most online reputation evaluation systems, the lack of incentive mechanisms to attract online traders to submit feedback voluntarily has been a bottleneck. Thus, it is difficult to ensure that the feedback received is reliable. Furthermore, it may not even be achievable to receive a continuous inflow of feedbacks. This paper presents an online reputation dynamic feedback incentive (DFI) model in the form of an optimal control problem. Based on dynamic properties of reputation evaluation, three cases of the DFI model over a long-term period are considered. Our study shows that the establishment of an effective feedback incentive mechanism is a way to solve the bottleneck of online reputation evaluation systems. Finally, a case study is carried out so as to illustrate the effectiveness of the approach. [ABSTRACT FROM AUTHOR]