On fractional infinite-horizon optimal control problems with a combination of conformable and Caputo-Fabrizio fractional derivatives.

Fractional calculus is a powerful and effective tool for modeling of nonlinear systems. In this paper, we first introduce a modification of conformable fractional derivative to solve fractional infinite horizon optimal control problems. We point out that the term t in definition of conformable derivative has some disadvantages and thus must be refined. Using an interesting property of relationship between the presented fractional derivative and the usual first-order derivative, fractional dynamic system in the infinite horizon optimal control problem is transformed into a non-fractional one. By a suitable change of variable, the obtained infinite horizon problem is reduced to a finite-horizon one. According to the Pontryagin minimum principle for optimal control problems and by constructing an error function, an unconstrained minimization problem is then defined. In the achieved minimization problem, trial solutions for state, co-state and control functions are utilized where these trial solutions are constructed by using two-layered perceptron neural network. Some numerical results are solved to explain our main results. Two applicable examples as stabilization and chaos control of fractional order systems are also provided.