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This paper presents a method for designing closed-loop time-optimal controllers. The fact that the gradient of the isochrone surface is coincident with the direction of the co-state vector for any given state is exploited to arrive at a feedback time-optimal controller. A radial basis function based neural network is used to represent the isochrone surface, which then provides us with a closed form expression for the gradients of the isochrone surface as a function of the system states. Numerical simulation of the proposed technique for a second order system indicates a control profile that is characterized by chattering. A boundary layer is used to minimize the chattering of the control. IntroductionDesign of optimal controllers with control constraints lead to open-loop controllers. Errors in system model and external disturbances leads to degradation of performance of the optimal controller which can be minimized by implementing the time-optimal controllers in feedback form. The numerically intensive procedure to solve for the time-optimal controller makes its real-time implementation impossible excepts for some simple systems. An approach to determine the near-time optimal controller has been proposed by Junkins et al.[l] where a near-time optimal controller for the rigid body motion is determined and a Lyapunov based feedback tracking controller is superimposed on the rigid-body open-loop controller. Their work exploits the fact that a smooth controller will not excite the high-frequency dynamics of the system.The two-point boundary value problem for the time-optimal controller reveals that the control can be expressed as a function of the co-states. If the co-states can be estimated as a function of the states, then one can implement the timeoptimal controller in a closed-loop. Athans and Falb[2] demonstrate that the co-state is given by the gradient (provided it exists), of the isochrone surface for any state. Determination of the isochrone surface in closed form is not possible for any but the simplest system. Luh and Shafran[3] describe a technique to obtain an approximate functional expression that describes the minimal time isochrones of the system. It involved least squares fits to the system states on various isochrones. Radial basis function based neural networks have been used by Vadali et a1.[4] to estimate the optimal controller as a function of the system state and time-to-go. Encouraging results provided by that paper motivate us to use a radial basis function based neural network to approximate the isochrone surface which is then used to estimate the co-states and thus the control. This paper proposes the use of a radial basis function based neural network to approximate the isochrone surface and use the gradients of