In this article, we study a class of nonlinear fractional differential equations with mixed-type boundary conditions. The fractional derivatives are involved in the nonlinear term and the boundary conditions. By using the properties of the Green function, the fixed point index theory and the Banach contraction mapping principle based on some available operators, we obtain the existence of positive solutions and a unique positive solution of the problem. Finally, two examples are given to demonstrate the validity of our main results.
In this paper, the measurement data loss is considered and two data-driven stochastic optimal iterative learning control (DDSOILC) methods are presented directly for nonlinear network systems. Specifically, an iterative dynamic linearization (IDL) is adopted to construct the linear incremental input output relationship of the repetitive nonlinear network system between two consecutive iterations. In the sequel, a lifted IDL is obtained by defining two super vectors of inputs and outputs over the entire finite time interval. Then, a lifted IDL-based DDSOILC scheme is proposed where the random data loss is described by a Bernoulli distribution of random variable. The results are extended by using a non-lifted IDL where the input-output relationship is described pointwisely. The learning gains of the proposed two methods are iteration-time-variant and can be iteratively estimated using real-time data. The proposed two methods do not depend on any explicit model. Moreover, the proposed non-lifted IDL-based DDSOILC can use more control information than the proposed lifted IDL-based one, and thus it can achieve a better control performance. Both theoretical analysis and simulations verify the efficiency and applicability of the two proposed methods.INDEX TERMS Data-driven control, stochastic optimal ILC, nonlinear network systems, measurement data loss.
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