I. Introduction
In contrast to the linear quadratic regulator, the universal and efficient design framework of optimal quadratic regulation for nonlinear systems remains a control engineering challenge. The lack of proper structure for a general nonlinear system makes this design problem challenging. There have been several attempts to provide such a systematic approach, including convex optimization-based Sum-of-Squares (SoS) programming [1], [2] and differential geometric-based feedback linearization control [3], [4]. The introduction of operator theoretic methods from the ergodic theory of dynamical systems provides another opportunity for the development of systematic methods for the design of feedback controllers [5]. The operator theoretic methods offer a linear representation for a nonlinear dynamical system. This linear representation of the nonlinear system is made possible by shifting the focus from state space to space of functions using two linear and dual operators, namely, the Perron-Frobenius (P-F) and Koopman operators. The work involving the third author [6]–[8] proposed a systematic linear programming-based approach involving transfer P-F operator for the optimal control of nonlinear systems. This contribution was made possible by exploiting the linearity and the positivity properties of the P-F operator. Similarly, application of P-F operator for control of chaotic nonlinear dynamics is discussed in [9], [10]. In [11] the P-F operator-based feedback control is applied to the mobile sensor networks for dynamic target detection. A flexible modeling approach based on the P-F operator is developed for optimal control of fluid mixing in [12]. More recently, there has been increased research activity on the use of Koopman operator for the analysis and control of nonlinear systems [13]–[19]. This recent work is mainly driven by the ability to approximate the spectrum (i.e., eigenvalues and eigenfunctions) of the Koopman operator from time-series data [20]–[23]. The data-driven approach for computing the spectrum of the Koopman operator is attractive as it opens up the possibility of employing operator theoretic methods for data-driven control. In fact, research works in [16], [18], [24]–[28] are proposing to develop Koopman operator-based data-driven methods for the design of optimal control and model predictive control for nonlinear and partial differential equations as well. In [29]–[31], the benefit of Koopman operator in reduced order modeling has been used for nonlinear PDE and flow fluid control. The Koopman operator is showing the advantage in low-dimensional bilinear approximation of PDEs in [32].