  For one-dimensional vibrating cavity systems appearing in the standard
illustration of the dynamical Casimir effect, we propose an approach to the
construction of exact closed-form solutions. As new results, we obtain
solutions that are given for arbitrary frequencies, amplitudes and time
regions. In a broad range of parameters, a vibrating cavity model exhibits the
general property of exponential instability. Marginal behavior of the system
manifests in a power-like growth of radiated energy.
