## 1 Introduction  
\"Degree of controllability\" is a scalar measure of controllability to check how controllable a given system is [1; 2; 3]. It reflects the efficiency of the control effort. The check of controllability only provides an answer to the binary question of whether or not a dynamical system is completely controllable. However, in practice we always need to know how controllable a given system is. Therefore, several authors proposed the concept of degree of controllability (DOC) as a measure of the controllability of a given system [1; 2; 3]. The degree of controllability analysis can help us choose the optimal parameters, number, and locations of actuators for a control system, which is of substantial importance in the aerospace industries to solve optimal control problems (the actuator placement in flexible spacecraft, sensor and actuator positioning for active vibration control, etc.) [4; 5; 6; 2]. In addition, the study of the importance or lack of importance of each control variable to the various outputs of the system is useful for decoupling or model reduction in systems [2; 7]. There are many definitions of the degree of controllability. The most widely-known definition is related to the minimum control input energy required to transfer any initial state \(x_{0}\) to the origin [1; 2; 3; 4; 7]. If a system requires smaller input energy than others to regulate the system, it can be considered more controllable [3]. [PERSON] et al. first discussed the method for determining the degree of controllability in the work [1], where the authors considered the weighted trace of the controllability Gramian as a measure of controllability. After that, some authors proposed a series of new criteria to determine the degree of controllability and observability [2; 7; 8; 9; 10; 11; 12]. In [2], [PERSON] calculated the degree of observability under the condition of the minimum energy problem. The author considered the maximum eigenvalue, trace, and determinant of the inverse Gramian matrix as a control measure. Instead of analyzing the minimum energy problem, [PERSON] used the recovery region to determine the degree of controllability. The authors chose a suitable terminal time \(T\) to solve this problem [7; 9; 13; 14]. For modal systems, [PERSON] demonstrated the relationship between observation vectors, control vectors, and system eigenvectors. This approach has an obvious physical meaning and is easy to realize [10]. However, all the above-mentioned methods are based on an open-loop control system without considering the optimal feedback, and the control law does not change over time \(u_{k}=const\). In this work, we have discussed the degree of controllability in the framework of the linear quadratic optimal control (LQR) system. We provide a degree of controllability in a closed-form for feedback control, where the control law is calculated by minimizing the cost function, \(u_{k}=-K_{k}x_{k}\). We considered the efficiency of the control effort in this case as the control performance of the response process (amplitude and convergence time of the initial state \(x_{0}\) ) and the cost of input energy, which can be directly reflected by the value of quadratic performance index \(P\). The cost function (or performance index \(P\)) plays a very important role in the LQR control scenario. First of all, it affects the control strategy (control law): we find the optimal solutions \(u_{k}\) by minimizing the cost function \(J_{k}\). Secondly, the cost function is also an indicator to evaluate the control performance of a given system. [PERSON] and [PERSON] studied the degree of observability directly from the filtering results of the linear quadratic optimal estimation (LQE) problem [11; 12]. Then, we could consider the degree of controllability in LQR (linear quadratic optimal regulation) problem as a dual task.