  We observe that a wide class of higher-derivative systems admits a bounded
integral of motion that ensures the classical stability of dynamics, while the
canonical energy is unbounded. We use the concept of a Lagrange anchor to
demonstrate that the bounded integral of motion is connected with the
time-translation invariance. A procedure is suggested for switching on
interactions in free higher-derivative systems without breaking their
stability. We also demonstrate the quantization technique that keeps the
higher-derivative dynamics stable at quantum level. The general construction is
illustrated by the examples of the Pais-Uhlenbeck oscillator, higher-derivative
scalar field model, and the Podolsky electrodynamics. For all these models, the
positive integrals of motion are explicitly constructed and the interactions
are included such that keep the system stable.
