Are nonlocal Lagrangian systems fatally unstable?

We prove that higher-derivative and genuinely nonlocal Lagrangian systems can be Lyapunov-stable even when their Hamiltonians lack a lower bound. Explicit free and coupled Pais-Uhlenbeck oscillators, together with a genuine nonlocal model, are analysed to identify the precise conditions under which stability holds. These counterexamples point out the logical gap in the “Ostrogradsky instability” claims and provide benchmarks for constructing efficient stable higher-derivative theories.