Numerical Solution And L₁ Regularization Of Fractional Order Optimal Problems

Fractional calculus theory was original discussed in the field of mathematics, and then widely studied in the natural and scientific areas.As an important component of control disciplines, fractional control theory has a unique advantage. Fractional order control system can describe the internal mechanism of industrial process more accurately than integer order system. Therefore, models based on fractional calculus are more accurate than integer one. For the fractional control system, the optimal control law can achieve safe and accurate working conditions.Therefore, numerical solution for fractional optimal control problem is valuable. As an important part of fractional order optimization theory,fractional regularization theory has great value in the field of identification and image processing. Currently, plenty of studies exist in fractional order optimization and regularization problems, but there still some shortcomings in the solving methods of optimization problem, and the accuracy and complexity of the algorithm still need to be improved.On the basis of former research, in-depth researches on the solution of fractional order system optimization problem and fractional order total variation regularization problem are done in this thesis:1. Aiming at the fractional order optimal control problems, a numerical solution algorithm based on piecewise linear interpolation discretization theory is proposed. The proposed algorithm is applied to the solution of multi-term fractional order optimal control problems.Compared with the traditional discretization and solving algorithms, the piecewise linear interpolation discretization algorithm can obtain more accurate numerical for linear time-invariant and time-varying fractional order optimal control problems. According to the algorithm convergence,the efficiency of the algorithm is improved. In addition, for the multi-term fractional order optimal control problems, that is, the state equation of the control system contains both fractional and integer order differential terms. By using the proposed algorithm, higher solution precision and better performance index can be achieved.2. The problems of fractional total-variation l₁ regularization is also studied in this thesis. The solution and application of fractional total-variation model is emphasized and discussed. A fractional total-variation l₁ regularization model is established for the inverse problem of image restoration. The model is solved by the split Bregman iterative algorithm, which improves the accuracy of the existing algorithm and has fast convergence. In addition, an adaptive strategy based on fractional order and fractional mask operator is proposed. Then,the adaptive strategy for two regularization parameters is obtained. The standard image library ?SIL? is used in the simulation experiments, to verify the validity and rationality of the proposed algorithm, as well as the better convergence and robustness. Simulation experiments on the LIDC-IDRI medical image database verify the applicability of the proposed method in medical image processing.This thesis focuses on the theoretical study and analysis of optimal problems which contains fractional calculus. The proposed algorithms improve the accuracy of the existing methods. Different numerical simulation experiments are designed. In addition, this paper verifies the effectiveness and applicability of the proposed method.