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Lyapunov Functionals and Stochastic Stability Analyses for Highly Random Nonlinear Functional Epidemic Dynamical Systems with Multiple Distributed Delays
Published in Hemen Dutta, Mathematical Methods in Engineering and Applied Sciences, 2020
The most widely used stability concept in control theory is Lyapunov stability [34,37,54]. Lyapunov stability for infectious disease dynamic systems describes the behavior of the trajectories of the SDE or ODE epidemic dynamic system near the equilibria of the system. Techniques applied to analyze Lyapunov stability determine the qualitative behavior of the trajectories of the stochastic or deterministic dynamic systems near the equilibria, without knowledge of explicit solutions. The LyapunovDirect Method [34,37,54], as an example of Lyapunov stability techniques, involves constructing an "energy function, in the neighborhood of the equilibrium. This function is also called a Lyapunov function, which metaphorically measures the "energy of the trajectories of the SDE or ODE system near the steady state. If the rate of change of the energy function with respect to the solutions of the system is negative over time, and so indicating that the "energy of the system near the equilibrium is decreasing, then all trajectories that start near the steady state remain near the steady state and can asymptotically converge to the steady state, almost surely, under more special conditions for the Lyapunov function. The extension of the Lyapunov Direct Methods to analyze the stability of the equilibria of delay SDEs or ODEs are collectively called Lyapunov functionals techniques (cf. [34,37,54]). Lyapunov stability techniques have wide applications in infectious disease dynamic systems (cf. [11,12,18,29,30,32,41,47]).
Fundamentals of biology and thermodynamics
Published in Mohammad E. Khosroshahi, Applications of Biophotonics and Nanobiomaterials in Biomedical Engineering, 2017
(Lyapunov stability gives a definition of asymptotic stability for more general dynamical systems, and all exponentially stable systems are also asymptotically stable). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. Remembering that the definition of asymptotic is that it is a line that approaches a curve but never touches it. Consider a simple scalar equation y′(t) = ay (t). The solution is, of course, y (t) = y0 eat, where y0 = y (0). In particular, y (t) ≡ 0 is a solution. What happens if we start at some point other that 0? If a < 0, then every solution approaches 0 as t → ∞. We say that the zero solution is (globally) asymptotically stable. Figure 5.7 shows the graphs of a few solutions and the direction field of the equation, i.e., the arrows have the same slope as the solution that passes through the tail point.
Advanced Control Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
These conditions constitute what is known as the Lyapunov stability. Another notion found frequently in the description of the Lyapunov stability of systems is that of asymptotic stability. Briefly, the asymptotic stability refers to long-term stability as time t → ∞. An asymptotically stable system is a stable system in the Lyapunov sense, but, when perturbed from the equilibrium position, in the long term, it will converge to its original position. Generally, it requires that, in addition to Lyapunov stability, there exist a number R > 0 depending on the initial condition at t0 such that if
Optimal fuzzy controller based on non-monotonic Lyapunov function with a case study on laboratory helicopter
Published in International Journal of Systems Science, 2019
Shahrzad Behzadimanesh, Alireza Fatehi, Siavash Fakhimi Derakhshan
Stability analysis is an important issue to design control systems. The Lyapunov stability theorem is the main applicable theorem for stability analysis or stabilisation of control systems. One of the prominent properties of T-S fuzzy systems is the ability to use state feedback structure in order to analyse the stability and design stabilised controller by the Lyapunov theorem.
Reentry attitude tracking via coupling effect-triggered control subjected to bounded uncertainties
Published in International Journal of Systems Science, 2018
Zongyi Guo, Jianguo Guo, Jun Zhou, Jinlong Zhao, Bin Zhao
The Lyapunov stability theory is the fundamental method to analyse the stability of the overall system. Note that the negativity of the Lyapunov function shows the moving orientation towards the origin of the system, therefore, we can analyse the influences of couplings on the derivative of the Lyapunov function. Let us recall Definition 4.1. The coupling effect indicator represents the influence of one sub-system on the other sub-system. Specifically, the coupling effect is determined by whether coupling is helpful to steer the Lyapunov function move to the origin. Select a Lyapunov function for sub-system (22) as , and we can obtain that through designing the virtual input , where the matrix is Hurwitz, and Φ is a function of the coupling which needs to be designed. That clearly shows that the sign of is related with the function . Then, the following two cases are discussed: Case 1: . Based on the design of Φ function, it is obtain that , which leads to . In this case, the coupling on is beneficial and useful in terms of control, and thus can be injected into the controller design.Case 2: . We can choose that via the Φ function, and thus . In this case, the coupling is harmful and thus it is eliminated.
Power-law stability of Hausdorff derivative nonlinear dynamical systems
Published in International Journal of Systems Science, 2020
The Lyapunov stability of nonlinear dynamical systems usually can be determined via the Lyapunov direct method. In this section, based on the Lyapunov direct method, the Power-law stability of nonlinear dynamical systems including the Hausdorff time derivative is investigated.