In this paper, an intelligence method based on single layer legendre neural network is proposed to solve fractional optimal control problems where the dynamic control system depends on Caputo fractional derivatives. First, with the help of an approximation, the Caputo derivative is replaced to integer order derivative. 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 optimization problem, trial solutions are used for state, costate and control functions, where these trial solutions are constructed by using Legendre polynomial based functional link artificial neural network. In the following, error back propagation algorithm is used for updating the network parameters (weights). At the end, some illustrative examples are included to demonstrate the validity and capability of the proposed method. Three applicable examples about chaos control of Malkus waterwheel, finance fractional chaotic models and fractional-order geomagnetic field models are also considered.
In this paper, we describe a new neural network model for solving a class of non-smooth optimization problems with min–max objective function. The basic idea is to replace the min–max function by a smooth one using an entropy function. With this smoothing technique, the non-smooth problem is converted into an equivalent differentiable convex programming problem. A neural network model is then constructed based on Karush–Kuhn–Tucker optimality conditions. It is investigated that the proposed neural network is stable in the sense of Lyapunov and can converge to an exact optimal solution of the original problem. As an application in economics, we use the proposed scheme to a min–max portfolio optimization problems. The effectiveness of the method is demonstrated by several numerical simulations.
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