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The first evolution equation is derived under the Variation Evolving Method (VEM) that seeks optimal solutions with the variation evolution principle. To improve the performance, its compact form is developed. By replacing the states and costates variation evolution with that of the controls, the dimension-reduced Evolution Partial Differential Equation (EPDE) only solves the control variables along the variation time to get the optimal solution, and its definite conditions may be arbitrary. With this equation, the scale of the resulting Initial-value Problem (IVP), transformed via the semi-discrete method, is significantly reduced. Illustrative examples are solved and it is shown that the compact form evolution equation outperforms the primary form in the precision, and the efficiency may be higher for the dense discretization. Moreover, in discussing the connections to the classic iteration methods, it is uncovered that the computation scheme of the gradient method is the discrete implementation of the third evolution equation, and the compact form of the first evolution equation is a continuous realization of the Newton type iteration mechanism., Comment: arXiv admin note: substantial text overlap with arXiv:1802.04663, arXiv:1801.10486, arXiv:1801.01383, arXiv:1802.02140, arXiv:1801.07395

The Variation Evolving Method (VEM) that originates from the continuous-time dynamics stability theory seeks the optimal solutions with variation evolution principle. After establishing the first and the second evolution equations within its frame, the third evolution equation is developed. This equation only solves the control variables along the variation time to get the optimal solution, and its definite conditions may be arbitrary since the equation can eliminate possible infeasibilities. With this equation, the dimension of the resulting Initial-value Problem (IVP), transformed via the semi-discrete method, is greatly reduced. Therefore it might relieve the computation burden in seeking solutions. Illustrative examples are solved and it is shown that the proposed equation may produce more precise numerical solutions than the second evolution equation, and its computation time may be shorter for the dense discretization., Comment: Key words: Optimal control, dynamics stability, variation evolution, evolution partial differential equation, the third evolution equation, initial-value problem. arXiv admin note: substantial text overlap with arXiv:1801.10486, arXiv:1801.01383, arXiv:1801.07395, arXiv:1712.09702 and text overlap with arXiv:1802.02140, arXiv:1709.02242

The Variation Evolving Method (VEM), which seeks the optimal solutions with the variation evolution principle, is further developed to be more flexible in solving the Optimal Control Problems (OCPs) with terminal constraint. With the first-order stable dynamics to eliminate the infeasibilities, the Modified Evolution Partial Differential Equation (MEPDE) that is valid in the infeasible solution domain is proposed, and a Lyapunov functional is constructed to theoretically ensure its validity. In particular, it is proved that even with the infinite-time convergence dynamics, the violated terminal inequality constraints, which are inactive for the optimal solution, will enter the feasible domain in finite time. Through transforming the MEPDE to the finite-dimensional Initial-value Problem (IVP) with the semi-discrete method, the OCPs may be solved with common Ordinary Differential Equation (ODE) numerical integration methods. Illustrative examples are presented to show the effectiveness of the proposed method., Comment: arXiv admin note: substantial text overlap with arXiv:1801.07395, arXiv:1801.01383; text overlap with arXiv:1712.09702, arXiv:1709.02242

Enlightened from the inverse consideration of the stable continuous-time dynamics evolution, the Variation Evolving Method (VEM) analogizes the optimal solution to the equilibrium point of an infinite-dimensional dynamic system and solves it in an asymptotically evolving way. In this paper, the compact version of the VEM is further developed for the computation of Optimal Control Problems (OCPs) with terminal constraint. The corresponding Evolution Partial Differential Equation (EPDE), which describes the variation motion towards the optimal solution, is derived, and the costate-free optimality conditions are established. The explicit analytic expressions of the costates and the Lagrange multipliers adjoining the terminal constraint, related to the states and the control variables, are presented. With the semi-discrete method in the field of PDE numerical calculation, the EPDE is discretized as finite-dimensional Initial-value Problems (IVPs) to be solved, with common Ordinary Differential Equation (ODE) numerical integration methods., Comment: arXiv admin note: substantial text overlap with arXiv:1709.02242, arXiv:1712.09702, arXiv:1711.02998, arXiv:1703.10263

The feedback linearization method is further developed for the controller design on general nonlinear systems. Through the Lyapunov stability theory, the intractable nonlinear implicit algebraic control equations are effectively solved, and the asymptotically tracking performance is guaranteed. Moreover, it is proved that the controller may be used in an inverse-free version to the set-point control. With this method, a nonlinear aircraft outer-loop trajectory controller is developed. For the concern regarding the controller's robustness, the integral control technique is combined to counteract the adverse effect from modeling errors. Simulation results verify the well performance of the proposed controller.