Approximate dynamic programming formulation implemented with an Adaptive Critic (AC) based neural network (NN) structure has evolved as a powerful alternative technique that eliminates the need for excessive computations and storage requirements needed for solving the Hamilton-Jacobi-Bellman (HJB) equations. A typical AC structure consists of two interacting NNs. In this paper, a novel architecture, called the Cost Function Based Single Network Adaptive Critic (J-SNAC) is used to solve control-constrained optimal control problems. Only one network is used that captures the mapping between states and the cost function. This approach is applicable to a wide class of nonlinear systems where the optimal control (stationary) equation can be explicitly expressed in terms of the state and costate variables. A non-quadratic cost function is used that incorporates the control constraints. Necessary equations for optimal control are derived and an algorithm to solve the constrained-control problem with J-SNAC is developed. Benchmark nonlinear systems are used to illustrate the working of the proposed technique. Extensions to optimal control-constrained problems in the presence of uncertainties are also considered.