Modern radars are highly flexible, using digital antennas which can dynamically change the radar beam shape and position through electronic control. Radar surveillance is performed by emitting sequentially different radar beams. Optimization of radar surveillance requires finding, among a collection of available radar beams with different shapes and positions, a minimal subset of radar beams which covers the surveillance space, ensuring detection while minimizing the required scanning time. Optimal radar surveillance can be modelled by grid covering, a specific geometric case of set covering where the universe set is laid out on a grid, representing the radar surveillance space, which must be covered using available subsets, representing the radar beams detection areas. While the set cover problem is generally difficult to solve optimally, certain geometric cases can be optimized in polynomial time. This paper studies the theoretical complexity of grid cover problems used for modelling radar surveillance, proving that unidimensional grids can be covered by strongly polynomial algorithms based on dynamic programming, whereas optimal covering of bidimensional grids is generally non-deterministic polynomially (NP) hard.