Conditions for strict dissipativity of infinite-dimensional generalized linear-quadratic problems

We derive sufficient conditions for strict dissipativity for optimal control of linear evolution equations on Hilbert spaces with a cost functional including linear and quadratic terms. We show that strict dissipativity with a particular storage function is equivalent to ellipticity of a Lyapunov-like operator. Further we prove under a spectral decomposition assumption of the underlying generator and an orthogonality condition of the resulting subspaces that this ellipticity property holds under a detectability assumption. We illustrate our result by means of an example involving a heat equation on a one-dimensional domain.