Recently, control theory using machine learning, which is useful for the control of unknown systems, has attracted significant attention. This study focuses on such a topic with optimal control problems for unknown nonlinear systems. Because optimal controllers are designed based on mathematical models of the systems, it is challenging to obtain models with insufficient knowledge of the systems. Kernel functions are promising for developing data-driven models with limited knowledge. However, the complex forms of such kernel-based models make it difficult to design the optimal controllers. The design corresponds to solving Hamilton-Jacobi (HJ) equations because their solutions provide optimal controllers. Therefore, the aim of this study is to derive certain kernel-based models for which the HJ equations are solved in an exact sense, which is an extended version of the authors' former work. The HJ equations are decomposed into tractable algebraic matrix equations and nonlinear functions. Solving the matrix equations enables us to obtain the optimal controllers of the model. A numerical simulation demonstrates that kernel-based models and controllers are successfully developed.

In this paper, two new clustering algorithms based on fuzzy c-means for data with tolerance using kernel functions are proposed. Kernel functions which map the data from the original space into higher dimensional feature space are introduced into the proposed algorithms. Nonlinear boundary of clusters can be easily found by using the kernel functions. First, two clustering algorithms for data with tolerance are introduced. One is based on standard method and the other is on entropy-based one. Second, the tolerance in feature space is discussed taking account into soft margin algorithm in Support Vector Machine. Third, two objective functions in feature space are shown corresponding to two methods, respectively. Fourth, Karush-Kuhn-Tucker conditions of two objective functions are considered, respectively, and these conditions are re-expressed with kernel functions as the representation of an inner product for mapping from the original pattern space into a higher dimensional feature space. Fifth, two iterative algorithms are proposed for the objective functions, respectively. Through some numerical experiments, the proposed algorithms are discussed.

Kernel-based learning algorithms have been successfully applied in various problem domains, given appropriate kernel functions. In this paper, we discuss the problem of designing kernel functions for binary regression and show that using a bell-shaped cosine function as a kernel function is optimal in some sense. The rationale of this result is based on the Karhunen-Loeve expansion, i.e., the optimal approximation to a set of functions is given by the principal component of the correlation operator of the functions.