Solving a non-standard Optimal Control royalty payment problem using a new modified shooting method

This paper considers a non-standard Optimal Control problem that has an application in economics. The primary focus of this research is to solve the royalty problem, which has been categorized as a non-standard Optimal Control problem, where the final state value and its functional performance index value are unknown. A new continuous necessary condition is investigated for the final state value so that it will convert the final costate value into a non-zero value. The research analyzes the seven-stage royalty piecewise function, which is then approximated to continuous form with the help of the hyperbolic tangent function and solves the problem by using a new modified shooting method. This modified shooting method applies Sufahani-Ahmad-Newton-Brent-Royalty Algorithm and Sufahani-Ahmad-Powell-Brent-Royalty Algorithm. For a validation process, the results are compared with the existing methods such as Euler, Runge-Kutta, Trapezoidal, and Hermite-Simpson approximations, and the results show that the proposed method yields an accurate terminal state value.