### A converse Lyapunov theorem and robustness with respect to unbounded perturbations for exponential dissipativity.

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We investigate a Lotka-Volterra predator-prey model with state dependent impulsive effects, in which the control strategies by releasing natural enemies and spraying pesticide at different thresholds are considered. We present some sufficient conditions to guarantee the existence and asymptotical stability of semi-trivial periodic solutions and positive periodic solutions.

A system of ordinary differential equations modelling an electric circuit with a thermistor is considered. Qualitative properties of solution are studied, in particular, the existence and nonexistence of time-periodic solutions (the Hopf bifurcation).

We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-$\mathrm{\mathcal{K}\mathcal{L}}$ estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-$\mathrm{\mathcal{K}\mathcal{L}}$ estimate, exists if and only if the class-$\mathrm{\mathcal{K}\mathcal{L}}$ estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether...