The study of optimal controller design for nonlinear systems has attracted increasing interest over the last few years. Although optimal controller design has been fully developed for dynamic linear systems, the development of its nonlinear counterpart is still an open problem for research. The problem is in our inability to find the solution of the Hamilton-Jacobi-Bellman (HJB) equation for optimal control of nonlinear systems. Most current solutions involve numerical approximation of the HJB equation. However, such algorithms usually require the solution to a two-point boundary value problem, which is not applicable for on-line implementation. This dissertation presents two new design approaches for both inverse optimal control and H_∞ inverse optimal control of a class of dynamic neural networks (DNN). The DNN structure considered here is a high-order dynamic approximation that is viewed as a generic equivalent model for a class of nonlinear systems, which can be adjusted on-line. For these DNN systems, two methods are developed for their inverse optimal control, one for DNNs with equal number of inputs and states, and another for DNNs with arbitrary number of inputs. For the DNN models with additive disturbances, two H_∞ inverse optimal control laws are derived using nonlinear damping and a variation of Sontag's formula, which achieve inverse optimality with a prescribed level of disturbance attenuation. The stability and optimality of the proposed control laws are shown using Lyapunov technique and the HJB equation. Computer simulations are carried out to support the theoretical findings. Application of the proposed methods to optimal control design for nonlinear systems via dynamic neural networks (DNN) is also considered. Adjustable DNNs with stable training rules are considered for on-line modeling of the unknown systems. The proposed methods of stabilizing inverse optimal controls and robust H_∞ inverse optimal controls are applied. The results of several numerical simulations on the application of the proposed techniques in the controller design for unknown nonlinear systems, demonstrate the effectiveness of the proposed strategies.