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Population Models with State-Dependent Delays
Published in Ovide Arino, David E. Axelrod, Marek Kimmel, Mathematical population dynamics, 2020
The classical existence and uniqueness theorems applied to (9) require G(·) to be Lipschitzian on C. Even the method of steps imposes a condition like the Lipschitz continuity on the function G(·) to ensure uniqueness in the stepwise ordinary differential equation obtained from the initial condition (El’sgol’ts and Norkin, 1973). As pointed out by Halanay and Yorke (1971), it is unreasonable to expect such a theory to hold with equations containing state-dependent delays. For instance, in an equation such as () x′(t)=b[x(t)]−b[x(t−L[x(t)])]
Advanced Control Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
Similarly, the asymptotic stability of nonlinear systems can be defined as the long-term measure of the Lyapunov stability as time t → ∞. Basically, this requires that the states that started close to a stable equilibrium will converge to the equilibrium in the long term. Since under the Lipschitz continuity of f the solution of the nonlinear dynamics is a continuous function of the initial condition, given any t0 ≥ 0, and any finite T, the map s(.,x(t0) that takes the initial condition into the solution trajectory in Cn[t0, T] is continuous, then in the long term as t → ∞, Cn [t0, ∞] becomes a linear space of continuous n-vector valued function on [t0, ∞]. A Banach space BCn[t0, ∞] can be defined as a subset of Cn[t0, T] consisting of bounded continuous functions such that
Coordinate Gradient Descent Methods
Published in Yulei Wu, Fei Hu, Geyong Min, Albert Y. Zomaya, Big Data and Computational Intelligence in Networking, 2017
Furthermore, Assumption 1(ii) is typical for convex optimization and covers many applications, see e.g., [12, 13, 15, 16]. Coordinatewise Lipschitz continuity of the gradient of function F implies that F can be upper bounded by a quadratic function along the ith coordinate. Indeed, using the mean value theorem in the integral form, we have:
Stability analysis for nonlinear closed loop system structure
Published in Systems Science & Control Engineering, 2020
Wang Jianhong, Ricardo A. Ramirez-Mendoza
Comment: Two inequalities (4) and (5) are commonly applied in nonlinear control analysis, as the true input, noise and output are bounded to be in a tight subset, i.e. these nonlinear functions used to describe the plant and controller are all differentiable function in this tight subset. From the real analysis theory, differentiable satisfies Lipschitz continuity.