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Accurate control of robots at high speeds requires a control system that can take into account the kinodynamic interactions of the robot with the environment. Prior works on learning inverse kinodynamic (IKD) models of robots have shown success in capturing the complex kinodynamic effects. However, the types of control problems these approaches can be applied to are limited only to that of following pre-computed kinodynamically feasible trajectories. In this paper we present Optim-FKD, a new formulation for accurate, high-speed robot control that makes use of a learned forward kinodynamic (FKD) model and non-linear least squares optimization. Optim-FKD can be used for accurate, high speed control on any control task specifiable by a non-linear least squares objective. Optim-FKD can solve for control objectives such as path following and time-optimal control in real time, without needing access to pre-computed kinodynamically feasible trajectories. We empirically demonstrate these abilities of our approach through experiments on a scale one-tenth autonomous car. Our results show that Optim-FKD can follow desired trajectories more accurately and can find better solutions to optimal control problems than baseline approaches.