Lagrange polynomial

Abstract: We present a Legendre pseudospectral method for directly estimating the costates of the Bolza problem encountered in optimal control theory. The method is based on calculating the state and control variables at the Legendre‐Gauss‐Lobatto (LGL) points. An Nth degree Lagrange polynomial approximation of these variables allows a conversion of the optimal control problem into a standard nonlinear programming (NLP) problem with the state and control values at the LGL points as optimization parameters. By applying theKarush ‐Kuhn‐Tucker (KKT) theorem to the NLP problem, we show that the KKT multipliers satisfy a discrete analog of the costate dynamics including the transversality conditions. Indeed, we prove that the costates at the LGL points are equal to the KKT multipliers divided by the LGL weights. Hence, the direct solution by this method also automatically yields the costates by way of the Lagrange multipliers that can be extracted from an NLP solver. One important advantage of this technique is that it allows a very simple way to check the optimality of the direct solution. Numerical examples are included to demonstrate the method.

Abstract: With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an m-bonacci orbital angular momentum (OAM) pump, Lagrange interpolation polynomials, and reverse Huffman-Fibonacci-tree coding. To be exact, we employ entangled states prepared by m -bonacci sequences to detect eavesdropping. Meanwhile, we encode m -bonacci sequences in Lagrange interpolation polynomials to generate the shares of a secret with reverse Huffman-Fibonacci-tree coding. The advantages of the proposed scheme is that it can detect eavesdropping without joint quantum operations, and permits secret sharing for an arbitrary but no less than threshold-value number of classical participants with much lower bandwidth. Also, in comparison with existing quantum secret sharing schemes, it still works when there are dynamic changes, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants. Finally, we provide security analysis of the new hybrid quantum secret sharing scheme and discuss its useful features for modern applications.