A meshless method for optimal control problem of Volterra-Fredholm integral equations using multiquadratic radial basis functions

In this paper, a numerical method is proposed for solving optimal control problem of Volterra integral equations using radial basis functions (RBFs) for approximating unknown function. Actually, the method is based on interpolation by radial basis functions including multiquadrics (MQs), to determine the control vector and the corresponding state vector in linear dynamic system while minimizing the quadratic cost functional. In addition for greater precision, the included integrals in Volterra integral equation and the cost functional are approximated using Legendre-Gauss-Lobatto nodes and weights. These nodes are considered as collocations points. The optimal control problem is reduced to a minimization so that the  control vector and the state vector are considered as an approximation of solution vectors based on radial basis functions. Two numerical examples are presented and results are compared with the analytical solutions to demonstrate the applicability  and accuracy of the proposed method.