A Piecewise-linear control (PLC) law, based on LQ theory, is derived. Ths law raises an LQ gain as the controlled error converges towards the origin. It can be computed off-line and it guarantees that input bounds are never exceeded without causinginput saturation. A succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest LQ gain possible in the presence of the input bounds. The formulation gives rise to an iteration function whose solution is a fixed point (FP). The FP defines the optimal gain and the invariant set, which comprise ellipsoids. FP existence, uniqueness and convergence properties are proven. We explain the algorithms for the generation and subsequent implementation of the PLC law. The notion of a switching function is introduced. Consideration is given to the computational aspects of the FP. The PLC scheme is demonstrated for the linearized dynamics of the simple pendulum and the PUMA 560 robot. A discrete time PLC law can also be derived using the discrete time algebraic Riccati equation.