The Third Evolution Equation for Optimal Control Computation
This work provides an incremental improvement for researchers computing optimal control solutions, potentially reducing computational burden.
The paper introduces a third evolution equation within the Variation Evolving Method (VEM) framework for optimal control, which only solves for control variables, reducing the dimension of the resulting initial-value problem. The proposed equation yields more precise numerical solutions and shorter computation times for dense discretization compared to the second evolution equation.
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.