Invariance of the Reachable Set under Nonlinear Perturbations

Abstract

For the abstract controlled Volterra equation \[( * )\qquad x(t) = \bar x(t) + \int_0^t {{\boldsymbol{\psi}} (t,s)[\phi (s,x(s)) + {\bf B}(s)v(s)]ds} \] on $[0,T]$, we consider the reachable set $\mathcal{K}_\phi : = \{ {x(T):( * ){\text{ for some }}v \in \mathfrak{B}} \}$. Viewing $\phi ( \cdot , \cdot )$ as a nonlinear perturbation of an otherwise linear control problem, conditions are obtained under which $\mathcal{K}_\phi = \mathcal{K}_0 $ for a suitable class $\mathfrak{F}$ of such nonlinear perturbations $\phi $ whenever the linear problem is known to have a reachable set invariant under affine perturbations: $\phi ( \cdot , \cdot ) = g \in \mathfrak{Z}$. The results generalize those obtained by Naito [7] for control of the heat equation.