The attainable sets of the nonlinear control systems with integral constraint on the control functions are considered. It is assumed that the behavior of control system is described by a differential equation which is nonlinear with respect to phase-state vector and control vector. The admissible control functions are chosen from the closed ball centered at the origin with radius mu(0) in L-p([t(0),theta];R-m)(p is an element of (1, +infinity)). Precompactness of the solutions set is specified, and dependence of the attainable sets on the initial conditions and on the parameters of the control system is studied. Copyright (c) 2008 K.G. Guseinov and A.S. Nazlipinar.