We focus on Lyapunov-based output feedback control for a class of distributed parameter systems with spatiotemporal dynamics described by input-affine semilinear dissipative partial differential equations (DPDEs). The control problem is addressed via adaptive model order reduction. Galerkin projection is applied to discretize the DPDE and derive low-dimensional reduced order models (ROMs). The empirical basis functions needed for this discretization are updated using adaptive proper orthogonal decomposition (APOD) which needs measurements of the complete profile of the system state (called snapshots) at revision times. The main objective of this paper is to minimize the demand for snapshots from the spatially distributed sensors by the control structure while maintaining closed-loop stability and performance. A control Lyapunov function is defined and its value is monitored as the system evolves. Only when the value violates a closed-loop stability threshold, snapshots are requested for a brief period by APOD after which the ROM is updated and the controller is reconfigured. The proposed approach is applied to stabilize the Kuramoto-Sivashinsky equation.