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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. [ABSTRACT FROM AUTHOR]

A symplectic algorithm with nonuniform grids is proposed for solving the hypersensitive optimal control problem using the density function. The proposed method satisfies the first-order necessary conditions for the optimal control problem that can preserve the structure of the original Hamiltonian systems. Furthermore, the explicit Jacobi matrix with sparse symmetric character is derived to speed up the convergence rate of the resulting nonlinear equations. Numerical simulations highlight the features of the proposed method and show that the symplectic algorithm with nonuniform grids is more computationally efficient and accuracy compared with uniform grid implementations. Besides, the symplectic algorithm has obvious advantages on optimality and convergence accuracy compared with the direct collocation methods using the same density function for mesh refinement. [ABSTRACT FROM AUTHOR]

A new reliable structure-preserving algorithm for computingH∞norm of linear continuous-time periodic systems is proposed in this paper. In the computation of theH∞norm, no Riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic Hamiltonian system will be evaluated. First, the monodromy matrix is expressed as the product of state transition matrices of the Hamiltonian system. Second, these state transition matrices, which have been proved to be symplectic matrices, are evaluated by a structure-preserving Magnus series method. Then, in order to preserve the standard symplectic form of the monodromy matrix, the structure-preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix. At last, the effectiveness and the high accuracy of the proposed structure-preserving algorithm are demonstrated by numerical examples. [ABSTRACT FROM PUBLISHER]

A new reliable algorithm for computing -norm of linear time-varying periodic (LTP) systems in continuous-time domain is proposed in this article. The solving of -norm is firstly transformed into the solving of the periodic Lyapunov differential equation (PLDE). Then, the key point in this article is that a new method based on the interval mixed energy matrix and the interval transition matrix computed by the structure-preserving Magnus series method is proposed for solving the PLDE. With the solutions of PLDE, the -norm of LTP systems is evaluated by a simple first-order ordinary differential equation. Finally, the effectiveness and the high accuracy of the proposed algorithms are demonstrated by two numerical examples. [ABSTRACT FROM AUTHOR]

A semianalytical spectral element method (SEM) is proposed for electromagnetic simulations of 3-D layered structures. 2-D spectral elements are employed to discretize the cross section of a layered structure, and the Legendre transformation is then used to cast the semidiscretized problem from the Lagrangian system into the Hamiltonian system. A Riccati equation-based high precision integration method is utilized to perform integration along the longitudinal direction, which is the undiscretized direction, to generate the stiffness matrix of the whole layered structure. The final system of equations by the semianalytical SEM will take the form of a set of linear equations with a block tri-diagonal matrix, which can be solved efficiently by the block Thomas algorithm. Numerical examples demonstrate the high efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equations into infinite dimensional Hamiltonianian system and its concrete form are obtained. Then by combining this method with Wu's method, a new method of constructing general solution of a class of mechanical equations is got, which several examples show very effective. [ABSTRACT FROM AUTHOR]