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The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method. First, a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay. Then, stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system, and after that the Lyapunov function of the averaged Itô equation is taken as the optimal linear combination of the corresponding independent first integrals in involution. Finally, the stability of the system is determined by using the largest eigenvalue of the linearized system. Two examples are used to illustrate the proposed procedure and the effects of delayed time on the Lyapunov stability are discussed as well. Lyapunov stability, Lyapunov function, time delay, stochastic averaging Citation:Ling Q, Jin X L, Huang Z L. Stochastic stability of quasi-integrable Hamiltonian systems with time delay by using Lyapunov function method. Sci

Machine learning is a field of study where the computer can learn for itself without a human explicitly hardcoding the knowledge for it. These algorithms make up the backbone of machine learning. This paper aims to study the field of machine learning and its algorithms. It will examine different types of machine learning models and introduce their most popular algorithms. The methodology of this paper is a literature review, which examines the most commonly used machine learning algorithms in the current field. Such algorithms include Nave Bayes, Decision Tree, KNN, and K-Mean Cluster. Nowadays, machine learning is everywhere and almost everyone using a technology product is enjoying its convenience. Applications like spam mail classification, image recognition, personalized product recommendations, and natural language processing all use machine learning algorithms. The conclusion is that there is no single algorithm that can solve all the problems. The choice of the use of algorithms and models must depend on the specific problem.

A procedure for designing a feedback control to asymptotically stabilize, with probability one, quasi-generalized Hamiltonian systems subject to stochastically parametric excitations is proposed. First, the motion equations of controlled systems are reduced to lower-dimensional averaged Itô stochastic differential equations by using the stochastic averaging method. Second, a dynamic programming equation for the averaged system with an appropriate performance index (with undetermined parameters in cost function) is established based on the dynamic programming principle, and the optimal control law is derived from a minimization condition with respect to control. Third, the Lyapunov function method is adopted to evaluate the stability boundary of asymptotic stability with probability one for the uncontrolled/ controlled systems. Finally, the parameters in cost function are selected to guarantee the sufficient stability of the controlled systems. Numerical results for a nine-dimensional mathematical system and a three-dimensional practical system, which describes a structure including viscoelastic element, illustrate the effectiveness of the feedback control strategy, and stability domains can be obviously enlarged when imposing the feedback controls on the original systems.