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

A generalized reproducing kernel 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 generalized reproducing kernel Hilbert space (G-RKHS), which is a new mathematical structure that we are introducing in this paper. Moreover, a generalized Fock space for the specific family of bounded nonlinear Volterra operators is precisely constructed. Base on these results, a generalized Fock space over vector-valued bounded nonlinear Volterra operators for multi-input/multi-output (MIMO) control systems can be constructed in the same manner. Finally, an application of the new theory to a feedback design problem for the optimal disturbance rejection issue for general nonlinear MIMO control systems formulated in a Banach space setting is indicated.

© 1989 IEEE