The talk deals with the eigenvalue decay of solution operators to operator Lyapunov equations, a relevant topic in the context of model reduction for parabolic control problems. We mainly focus on the Gramian operator that arises in the context of control and observation of heat processes in infinite time. By improving existing energy and observability estimates for parabolic equations, we obtain both upper and lower bounds on the convergence rate of the eigenvalues of the Gramian operator towards zero. Both bounds follow the same polynomial decay rate, up to a multiplicative constant, which ensures their optimality. This confirms the slow decay of the eigenvalues and limits the efficiency of model reduction. The theoretical findings are supported by numerical results.

We consider a weighted discrete graph over , i.e. is a countable discrete set, is a function on which induces a measure and is an edge weight. Then the corresponding Laplacian is a non-negative self-adjoint operator. We investigate whether (weak) observability estimates for the corresponding heat equation exist, i.e. we study if, for given final time and a subset of , the norm of the solution of the heat equation at time can be bounded by the portions of the solution on up to time and the norm of the initial condition. We will also comment on the impact of such estimates w.r.t. controllability. This is joint work with Peter Stollmann (Chemnitz) and Martin Tautenhahn (Leipzig).