Implicit-Explicit Runge-Kutta schemes for numerical discretization of optimal control problems

Implicit-explicit (IMEX) Runge-Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and non-stiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX Runge-Kutta methods in the context of optimal control problems. The analysis of the schemes is based on the continuous optimality system. Using suitable transformations of the adjoint equation, order conditions up to order three are proven as well as the relation between adjoint schemes obtained through different transformations is investigated. Conditions for the IMEX Runge-Kutta methods to be symplectic are also derived. A numerical example illustrating the theoretical properties is presented

Self-Evaluation on students’ English speaking fluency at eleventh grade of MA al-Amien 1 Pragaan Sumenep

This research discuss about Self-Evaluation on Students’ English Speaking Fluency at Eleventh Grade of MA Al-Amien 1 Pragaan Sumenep which focuses on two aspects. The aspects are to know the implementation of self-evaluation in MA Al-Amien 1 Pragaan Sumenep and the students responses toward self-evaluation as formative asessment on students’ English speaking fluency. The research concerns about the ability of students MA Al-Amien 1 Pragaan in speaking English, one of the problem is eventhough the students are confidence to speak English but they still have hesitation and repetition in their speaking. Many students in eleventh grade of MA-Al-Amien 1 Pragaan who get trouble in term of speaking fluency, also the problem of the student who is structurally competent but cannot communicate appropriately. This problem should get more attention from the school, especially teachers. Self-evaluation is an important part of self-regulation. It encourages students to reflect on their own learning, assess their own strengths and weaknesses, and identify their own goals for learning. Teachers do this by asking students to reflect on their work. By applying the self-evaluation, the teacher could make the students become self- regulated learner. Knowing the use of self-evaluation at MA Al-Amien 1 Pragaan Sumenep, the researcher conducted a research at that school. By applying descriptive qualitative approach, the researcher used instrument such as observation, interview the teacher, and questionnaire to collect the data. From the result of study, it shows that the implementation of self-evaluation on students English speaking fluency at XI Excellent A and B was well impemented. The students aware about their mistakes that they found after completing self-evaluation. The students responses toward the implementation of self-evaluation were self-evaluation practice is always useful for the students learning process of speaking, the students always got motivation in learning English speaking from the self-evaluation, there are progress in speaking after self-evaluation is implemented. Finally, based on the data, the researcher concludes that self-evaluation gives useful contribution to the learning process so that this study is recommended for the English teachers

Multi-leader-follower potential games

In this paper, we discuss a particular class of Nash games, where the participants of the game (the players) are divided into two groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996 and has beneficial properties to reformulate the bilevel Nash game. We develope necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results. In this paper, we discussed analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games. The application of these theoretical results to various fields of applications are a future research topic. Moreover, they are meant to serve as a starting point for the developement of efficient numerical solution methods for multi-leader-follower games

Dynamic boundary control games with networks of strings

Consider a star-shaped network of strings. Each string is governed by the wave equation. At each boundary node of the network there is a player that performs Dirichlet boundary control action and in this way influences the system state. At the central node, the states are coupled by algebraic conditions in such a way that the energy is conserved. We consider the corresponding antagonistic game where each player minimizes a certain quadratic objective function that is given by the sum of a control cost and a tracking term for the final state. We prove that under suitable assumptions a unique Nash equilibrium exists and give an explicit representation of the equilibrium strategies