A G-RKHS of bounded nonlinear operators for nonlinear systems control

Abstract

A generalized reproducing kernal Hilbert space (G-RKHS) of nonlinear Lipschitz operators is constructed for systems and control engineering applications. Specifically, the uniform topology is first introduced into the totality of one-parameter families of nonlinear Lipschitz operators to form a uniformly normed linear space, and then a generalized Bochner integral is introduced to define an operator-valued inner product structure and an induced norm for the space. It is shown that any closed and separable subspace of the resultant inner product space is a G-RKHS, which is a new mathematical structure. A generalized Fock space for the specific family of bounded nonlinear Volterra operators for multi-input/multi-output (MIMO) control systems can be constructed in the same manner. An application of the approach to a feedback design problem involving optimal disturbance rejection for general nonlinear MIMI control systems formulated in a Banach space setting is indicated. >