  This paper proposes a tractable family of remainder-form mixed-monotone
decomposition functions that are useful for over-approximating the image set of
nonlinear mappings in reachability and estimation problems. In particular, our
approach applies to a new class of nonsmooth nonlinear systems that we call
either-sided locally Lipschitz (ELLC) systems, which we show to be a superset
of locally Lipschitz continuous systems, thus expanding the set of systems that
are formally known to be mixed-monotone. In addition, we derive lower and upper
bounds for the over-approximation error and show that the lower bound is
achievable with our proposed approach. Moreover, we develop a set inversion
algorithm that along with the proposed decomposition functions, can be used for
constrained reachability analysis and interval observer design for continuous
and discrete-time systems with bounded noise.
