Study of The Lazy Nature of Physics Students Using The Quadratic Optimal Control Method

The lazy nature of learning is an obstacle in teaching and learning activities. The higher level of laziness will reduce the students learning achievement. This problem has motivated the author to make a mathematical model of student learning's lazy nature and analyze it. The lazy nature of learning is modeled to a discrete-time system ( 1) ( ) ( ) N k AN k BU k + = + with ( ) N k , denotes a state vector of the lazy nature, ( ) U k represents an input vector system in time, and denotes a coefficient matrix of ( ) N k and ( ) U k . The optimal quadratic control method is applied in the model to analyze the system's stability and give the interpretation of the model based on the graphics info obtained during the simulation process. The feedback controller ( ) K k is obtained as a result of optimizing the quadratic objective function. ( ) K k a feedback controller's existence makes the lazy nature of learning physics students decrease from time to time, and the system becomes stable.


INTRODUCTION
The lazy nature of learning is an obstacle in teaching and learning activities (Kessels and Heyder 2020). In the digital era, students are faced with activities that can reduce the desire to learn, such as online games and social media activities (Acosta and Denham 2018). Each of these activities harms student learning patterns. Some of the negative effects of social media are student lose the habit of communicating face to face (Siddiqui and Singh 2016), the students lose concentration while studying because they open social media frequently (Raut and Patil 2016), and student need more time to complete assignments (Flanigan and Babchuck 2015). These three examples show that social media has negative impacts and fosters a lazy nature of learning reflected in various student behaviors.
This research intends to study the lazy nature of student learning through a mathematical approach (Sari et al. 2019). The model will be constructed using a discrete-time system (Liu, Yan, and Wei 2014). The Discrete-time system has been widely used to model daily events such as a traffic model (Rachim 2017), an economic model in Hu and Tu (2015), (Canto et al. 2008), population growth model (Haberman 1998), and several other models. Based on these studies, a discrete-time system is considered appropriate to model the lazy nature of student learning.
The lazy behavior modeled in a discrete-time system is then given the optimal quadratic control to analyze the system's stability and controllability (Bontempi, Birattari, and and Bersini 2010). Furthermore, the analysis of the model will be presented based on the graphical info generated. The quadratic optimal control method is quite widely used to analyze systems such as the macroeconomic model (Engwerdah et al. 2009), network and Markov chain model (Kordonis and Papavassilopaulos 2014), model in game theory (Musthofa et al. 2015), and system descriptor model (Engwerda et al. and  Salmah 2008). The next section will discuss the research methodology and theoretical basis needed in this research.

METHODS
The research method used in this study is mathematical modeling based on the survey of related literature and accompanied by mathematical theorems. The study is organized out through three stages: (1) Study literature to find suitable models for modeling the lazy nature of learning, (2) Observes the lazy nature of student learning and the factors, and (3) Construct mathematical models and determine the value of parameters model based on factors obtained during the observation. This research aims to study students of Physics Study Program class 2019/2020 State Islamic University Sultan Maulana Hasanuddin Banten. The Mathematical models that will be built follow the flow chart defined by Pagalay (2009). The lazy nature of learning model in this paper is constructed by using a discrete time system (Ogata 1995 Furthermore, the feedback control applied in the discrete time system to analyze how the stability and controllability of the system. The feedback controller is obtained by optimizing the quadratic objective function that form defined in EQUATION (2) (Ogata 1995).
where S and Q represent positive semi definite hermitian matrices with nn  dimension, also R denotes positive hermitian matrix with rr  dimension.

RESULTS AND DISCUSSION
The modeling process starts with identifying the problem. Based on the observation the lazy nature of learning students is influenced by several factors such as the temptation to use social media, the density of activities on campus, dating with boy or girlfriend, uncomfortable classroom environment, and inadequate infrastructure. These factors cause the lazy nature of student learning can increase over time. It is assumed that each of student has a different level of laziness based on observations during the study and it represented by the diagonal matrix A with 30 30  dimension.
which interprets student productivity in learning. This component is formulated to reduce the lazy nature of student learning expressed by a discrete-time system. According to the problem identification and assumptions, the discrete-time system of lazy nature follows this diagram.  (Ogata, 1995 : 403) Based on a discrete time system of lazy nature in FIGURE 2. The mathematical modeling for the lazy nature of student learning can be modeled into a discrete time system in EQUATION (3).
(3) Furthermore, defined the quadratic objective function to minimize performance index of the system that defined in EQUATION (2) such as (4)

Matrices , , , ,
A B S Q and R represent 30x30 the diagonal matrix that entries contain parameters in  The parameters in TABLE 1 are substituted into EQUATION (3) and (4) to solve the optimal quadratic control problem and transform the basic problem into a linear programming problem with EQUATION (3) as the objective function and EQUATION (4) represent the constraint. The Lagrange multiplier method is used to solve this problem and defined the Lagrange function as in EQUATION (5). The new objective function was obtained in EQUATION 6 such that the linear programming problem with constraint has transformed into the linear programming problem without restraint. The necessary condition for a partial derivative of L has been applied to solve EQUATION (6). The results are the EQUATION (7), (8), and (9).
According to the results, the percentages of the lazy nature of learning physics students have decreased from time to time and asymptotically stable. This outcome establishes the feedback controller K(k) that applied to the system and has successfully stabilized the system. TABLE 3 presents the value of feedback controller K(k) during the simulation period. The feedback controller K(k) is 30 x 30 matrix that the diagonal entries contain the parameters in TABLE 3 for each student. During the observation, some things can increase students' motivation to counteract their lazy nature of learning, such as parents' motivation, comfortable learning environment, improvement in worship, and supportive learning colleagues (Kessels and Heyder 2020). The feedback controller K(k) can be interpreted as the percentages of those good things. We can also give the students the recommendation to apply the value of K(k) their daily lives, so their lazy nature of learning can be decreased to zero and stable as the simulation process results.

CONCLUSION
The previous section discussed the lazy nature of learning physics students that modeled into a discrete-time system and applied the optimal quadratic control method. The system is modeled thirty physics students, which are expressed in the 30 x 30 matrix. The simulation process has shown the percentages of each student's lazy nature of learning have decreased from time to time and stable. The decrease was caused by giving the feedback controller K(k), which is applied to the system. The feedback controller K(k) can be interpreted as good things that can improve students' motivation. This research is limited to modeling the lazy nature of learning from thirty physics students. Future research can increase the number of subjects and construct a larger matrix size and input more external factors into the model.