We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss–Legendre method ha...

An important aspect of numerically approximating the solution of an infinite-horizon optimal control problem is the manner in which the horizon is treated. Generally, an infinite-horizon optimal control problem is approximated with a finite-horizon problem. In such cases, regardless of the finite duration of the approximation, the final time lies an infinite duration from the actual horizon at ...

Abstract. This paper pr esents an energ y-optimal trajectory planning method fo r spacecraft fo rmation reconfiguration in deep space environment using continuous lo w-thrust propulsion system. First, we emplo y the Legendre pseudospectral method (LPM) to transform the optimal reconfiguration problem to a parameter optimization nonlinear programming (NLP) problem. Then, to avoid the computation...

Let p be an odd prime and ζ be a primitive p-root of unity. For any integer a prime to p, let ( p ) denote the Legendre symbol, which is 1 if a is a square mod p, and is −1 otherwise. Using Euler’s Criterion that a(p−1)/2 = ( p ) mod p, it follows that the Legendre symbol gives a homomorphism from the multiplicative group of nonzero elements Fp of Fp = Z/pZ to {±1}. Gauss’s law of quadratic rec...

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of the continuous-time optimal control problem. Us...

We analyze the consistency, stability, and convergence of an hp discontinuous Galerkin spectral element method of Kopriva [J. Comput. Phys., 128 (1996), pp. 475–488] and Kopriva, Woodruff, and Hussaini [Internat. J. Numer. Methods Engrg., 53 (2002), pp. 105–122]. The analysis is carried out simultaneously for acoustic, elastic, coupled elastic-acoustic, and electromagnetic wave propagation. Our...

A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of ...

We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss–Legendre method ha...

A state-defect constraint pairing graph coarsening method is described for improving computational efficiency during the numerical factorization of large sparse Karush–Kuhn–Tucker matrices that arise from the discretization of optimal control problems via an Legendre–Gauss–Radau orthogonal collocation method. The method takes advantage of the particular sparse structure of the Karush–Kuhn–Tucke...