Convergence of series expansions for some infinite dimensional nonlinear systems

Volterra series expansions have been extensively used to solve and represent the dynamics of weakly nonlinear finite dimensional systems. Such expansions can be recovered by using the regular perturbation method and choosing the input of the system as the perturbation: the state (or the output) is then described by a series expansion composed of homogeneous contributions with respect to the input, from which kernels of convolution type can be deduced. This paper provides an extension (based on this approach) to a class of semilinear infinite dimensional systems, nonlinear in state and affine in input. As a main result, computable bounds of the convergence radius of the series are established. They characterize domains on which the series defines a mild solution of the system. The convergence criterion is established for bounded signals (infinite norms on finite or infinite time intervals) as follows: first, norm estimates of the series expansion terms are derived; second, the singular inversion theorem is used to deduce an easily computable bound of the convergence radius. In the formalism proposed here, non zero initial conditions can be also considered as a perturbation so that no precomputation of nominal trajectories is required in practice. The relevance of the method is illustrated on an academic example.