I have worked intensely on converse results for Lyapunov functions and functionals. 

Many converse Lyapunov results (including a general result for the existence of Lyapunov functionals in abstract dynamical systems) are given in the book:

• I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems, Springer-Verlag, London (Series: Communications and Control Engineering), 2011.

Here is a list of papers containing converse Lyapunov results for various classes of systems:

• I. Karafyllis and J. Tsinias, “A Converse Lyapunov Theorem for Non-Uniform in Time Global Asymptotic Stability and Stabilization by Means of Time-Varying Feedback”, SIAM Journal Control and Optimization, 42(3), 2003, pp. 936-965.

• I. Karafyllis, “Non-Uniform in Time Robust Global Asymptotic Output Stability”, Systems & Control Letters, 54(3), 2005, pp. 181-193.

• I. Karafyllis, “Lyapunov Theorems for Systems Described by Retarded Functional Differential Equations”, Nonlinear Analysis: Theory, Methods and Applications, 64(3), 2006, pp. 590-617.

• I. Karafyllis, “Non-Uniform in Time Robust Global Asymptotic Output Stability for Discrete-Time Systems”, International Journal of Robust and Nonlinear Control, 16(4), 2006, pp. 191-214.

• I. Karafyllis, P. Pepe and Z.-P. Jiang, “Global Output Stability for Systems Described by Retarded Functional Differential Equations: Lyapunov Characterizations”, European Journal of Control, 14(6), 2008, pp. 516-536.

• I. Karafyllis, P. Pepe and Z.-P. Jiang, “Input-to-Output Stability for Systems Described by Retarded Functional Differential Equations”, European Journal of Control, 14(6), 2008, pp. 539-555.

• I. Karafyllis and P. Pepe, “A Note on Converse Lyapunov Results for Neutral Systems”, Recent Results on Nonlinear Time Delayed Systems, I. Karafyllis, M. Malisoff, F. Mazenc and P. Pepe (Eds.), Advances in Delays and Dynamics, Vol. 4, Springer, 2015.

• P. Pepe and I. Karafyllis, “Converse Lyapunov-Krasovskii Theorems for Systems Described by Neutral Functional Differential Equations in Hale’s Form”, International Journal of Control, 86(2), 2013, pp. 232-243.

• I. Karafyllis and M. Krstic, “On the Relation of Delay Equations to First-Order Hyperbolic Partial Differential Equations”, ESAIM Control, Optimisation and Calculus of Variations, 20(3), 2014, pp. 894 - 923.

• P. Pepe, I. Karafyllis and Z.-P. Jiang, “Lyapunov-Krasovskii Characterization of the Input-to-State Stability for Neutral Systems in Hale’s Form”, Systems & Control Letters, 102, 2017, pp. 48-56.

• A. Chaillet, I. Karafyllis, P. Pepe and Y. Wang, “The ISS Framework for Time-Delay systems: A Survey”, Mathematics of Control, Signals, and Systems, 35, 2023, pp. 237–306.

• E. Loko, A. Chaillet and I. Karafyllis, “Building Coercive Lyapunov-Krasovskii Functionals Based on Razumikhin and Halanay Approaches”, International Journal of Robust and Nonlinear Control, 34(10), 2024, pp. 6372-6392.

Chapter 7 in the book 

• I. Karafyllis and M. Krstic, Predictor Feedback for Delay Systems: Implementations and Approximations, Birkhäuser, Boston (Series: Mathematics, Systems & Control: Foundations & Applications), 2017.

contains a converse Lyapunov result for systems described by Integral Delay Equations (IDEs).