A control parameterization method for solving combined fractional optimal parameter selection and optimal control problems

Many real-world decision problems can be naturally modeled as fractional optimal parameter selection and fractional optimal control problems. Therefore, in this paper, we consider a class of combined fractional optimal parameter selection and optimal control problems involving nonlinear fractional systems with Caputo fractional derivatives and subject to canonical equality and inequality constraints. We first approximate this problem by a set of finite-dimensional optimization problems using the control parameterization method, where both the heights of parameterized controls and system parameters are taken as decision variables. We then show that the gradients of the cost and constraint functions with respect to the decision variables can be expressed as the solutions of a series of auxiliary fractional systems, which can be solved together with the original fractional system forward in time, simultaneously. Furthermore, we present a third-order numerical scheme for solving both the original and auxiliary fractional systems. On this basis, a gradient-based optimization algorithm is developed to solve the resulting optimization problems. Finally, we demonstrate the effectiveness and applicability of the developed algorithm through five non-trivial examples, one of which involves the optimal treatment of human immunodeficiency virus.