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This work focuses on control of distributed processes modeled by linear parabolic partial differential equations (PDEs) with constrained and quantized control inputs. Using a suitable finite-dimensional model that captures the PDE's dominant dynamics, we first characterize the inherent conflict in the control design objectives when both control constraints and quantization are simultaneously present, and the implications of this conflict for the spatial placement of the control actuators. At the heart of this conflict is the fact that control constraints limit the set of initial conditions starting from where stability can be achieved, while quantization constrains the set of terminal states that the system can be steered to. Using Lyapunov-based techniques, we explicitly characterize both the stability and terminal regions in terms of the control constraints, the quantization levels and the actuator spatial locations. The analysis reveals that the actuator configuration with the largest stability region also possesses the largest terminal region. This implies that steering the closed-loop state from large initial conditions to arbitrarily small terminal sets may not be possible using a single actuator configuration. To resolve this conflict, we devise an actuator scheduling strategy that orchestrates a finite number of transitions between different actuator configurations based on where the closed-loop state is with respect to the stability and terminal regions at any given time. The theoretical results are illustrated using a diffusion-reaction process example.