In this dissertation we focus on fractional-order dynamical systems and classify these problems as optimal control of system described by fractional derivative, fractional-order nonlinear differential equations, optimal control of systems described by variable-order differential equations and delay fractional optimal control problems. These problems are solved by using the spectral method and reducing the problem to a system of algebraic equations. In fact for the optimal control problems described by fractional and variable-order equations, the variables are approximated by chosen wavelets with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem is converted to an optimization problem, which can be solved numerically. We have applied the new generalized wavelets to approximate the fractional-order nonlinear differential equations such as Riccati and Bagley-Torvik equations. Then, the solution of this kind of problem is found using the collocation method. For solving the fractional optimal control described by fractional delay system, a new set of hybrid functions have been constructed. Also, a general and exact formulation for the fractional-order integral operator of these functions has been achieved. Then we utilized it to solve delay fractional optimal control problems directly. The convergence of the present method is discussed. For all cases, some numerical examples are presented and compared with the existing results, which show the efficiency and accuracy of the present method.

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Wavelet methods possess versatile properties for statistical applications. We would like to explore the advantages of using wavelets in the analyses in two different research areas. First of all, we develop an integrated tool for online detection of network anomalies. We consider statistical change point detection algorithms, for both local changes in the variance and for jumps detection, and propose modified versions of these algorithms based on moving window techniques. We investigate performances on simulated data and on network traffic data with several superimposed attacks. All detection methods are based on wavelet packets transformations. We also propose a Bayesian model for the analysis of high-throughput data where the outcome of interest has a natural ordering. The method provides a unified approach for identifying relevant markers and predicting class memberships. This is accomplished by building a stochastic search variable selection method into an ordinal model. We apply the methodology to the analysis of proteomic studies in prostate cancer. We explore wavelet-based techniques to remove noise from the protein mass spectra. The goal is to identify protein markers associated with prostate-specific antigen (PSA) level, an ordinal diagnostic measure currently used to stratify patients into different risk groups.