This paper presents a novel technique for control of systems with bounded nonlinearity, convex state constraints, and control constraints. The technique is particularly useful for problems whose control constraints may be written as convex sets or the union of convex sets. The problem is reduced to finding bounding solutions associated with linear systems, and it is shown that this can be done with efficient second-order cone program (SOCP) solvers. The nonlinear control may then be interpolated from the bounding solutions. Three engineering problems are solved. These are the Van der Pol oscillator with bounded control and with quantized control, a pendulum driven by a DC motor with bounded voltage control, and a lane change maneuver with bounded rotational control acceleration. For each problem, the resulting second-order cone program solves in approximately 0.1 s or less. It is concluded that the technique provides an efficient means of solving certain control problems with control constraints.