A symplectic indirect approach for a class of nonlinear optimal control problems of differential‐algebraic systems

Abstract

Differential-algebraic equations (DAEs) can model constrained dynamical systems and processes from practical engineering. Therefore, research on nonlinear optimal control problems of DAEs is of theoretical significance for optimal control of constrained systems, which can generate reference trajectories and control inputs for online control strategies. In terms of the numerical solution of this type of problem, research on indirect numerical methods is still insufficient and less research focuses on symplectic-preserving methods. In this article, a symplectic indirect approach is proposed for optimal control problems subject to index-1 DAEs. Necessary conditions of the optimal control problem constitute a Hamiltonian boundary value problem (HBVP) and there exists a symplectic structure in the Hamiltonian system. In the proposed approach, based on specified properties of generating functions, discrete equations can preserve the symplectic structure of the Hamiltonian system. In the iterative solution, the Jacobian matrices of the discrete equations are sparse and symmetric, which are very significant to save memory and improve efficiency in practical computation. In numerical examples, the proposed approach can provide highly accurate state variables and control inputs with fewer iterations. More accurate cost functional can be obtained. Problems from the chemistry process also can be solved effectively, it verifies the problem-solving ability of the proposed approach. K E Y W O R D Sconstrained systems, differential-algebraic equations, nonlinear optimal control, symplectic methods INTRODUCTIONOriginating from practical engineering applications such as motion planning of robots and optimal strategies of processes, optimal control problems 1-4 have been widely researched in the past decades. Generally, there are a defined cost functional and dynamical constraints in the optimal control problem. The aim is to find optimal state variables and control inputs that fulfill dynamical constraints and minimize the cost functional. The cost functional is a scalar function with respect to state variables and control inputs, which is defined according to practical requirements. For example, the task is to 2712

Robustness of Hopfield Neural Networks Described by Differential Algebraic Systems of Index-1 under the Conditions of Deviation Argument and Stochastic Disturbance

“…By interpreting the DASs as feedback interconnections between a pure differential equation and an algebraic equation, the stability analysis is analogous to a small gain-like condition [32]. A symplectic indirect method is proposed for the optimal control problem with index-1 DASs in [33].…”