scholarly journals Stability of differential inclusions: A computational approach

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Vadim Azhmyakov
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Differential Inclusions ◽  
Auxiliary Problem ◽  
Saddle Points ◽  
Computational Approach ◽  
Saddle Point Problems ◽  
Difference Inclusions ◽  
Type Algorithm ◽  
Gradient Type

We present a technique for analysis of asymptotic stability for a class of differential inclusions. This technique is based on the Lyapunov-type theorems. The construction of the Lyapunov functions for differential inclusions is reduced to an auxiliary problem of mathematical programming, namely, to the problem of searching saddle points of a suitable function. The computational approach to the auxiliary problem contains a gradient-type algorithm for saddle-point problems. We also extend our main results to systems described by difference inclusions. The obtained numerical schemes are applied to some illustrative examples.

Recursive Lyapunov Functions

1989 ◽  
Vol 111 (4) ◽  
pp. 641-645 ◽  
Author(s):  
Andrzej Olas
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Exponential Stability ◽  
Performance Measures ◽  
Lyapunov Functions ◽  
Stability Problem ◽  
Autonomous Systems ◽  
Sequence Of Functions

The paper presents the concept of recursive Lyapunov function. The concept is applied to investigation of asymptotic stability problem of autonomous systems. The sequence of functions {Uα(i)} and corresponding performance measures λ(i) are introduced. It is proven that λ(i+1) ≤ λ(i) and in most cases the inequality is a strong one. This fact leads to a concept of a recursive Lyapunov function. For the very important applications case of exponential stability the procedure is effective under very weak conditions imposed on the function V = U(0). The procedure may be particularly applicable for the systems dependent on parameters, when the Lyapunov function determined from one set of parameters may be employed at the first step of the procedure.

scholarly journals An efficient three-term conjugate gradient-type algorithm for monotone nonlinear equations

2020 ◽  
Author(s):  
Jamilu Sabi'u ◽  
Abdullah Shah
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Numerical Results ◽  
Search Direction ◽  
Projection Technique ◽  
Type Algorithm ◽  
Gradient Type

In this article, we proposed two Conjugate Gradient (CG) parameters using the modified Dai-{L}iao condition and the descent three-term CG search direction. Both parameters are incorporated with the projection technique for solving large-scale monotone nonlinear equations. Using the Lipschitz and monotone assumptions, the global convergence of methods has been proved. Finally, numerical results are provided to illustrate the robustness of the proposed methods.

A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem

2019 ◽  
Vol 84 (2) ◽  
pp. 485-512 ◽  
Author(s):  
Cristian Daniel Alecsa ◽  
Szilárd Csaba László ◽  
Adrian Viorel
Keyword(s):  
Minimization Problem ◽  
Nonconvex Minimization ◽  
Type Algorithm ◽  
Gradient Type

scholarly journals State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1424 ◽  
Author(s):  
Angelo Alessandri ◽  
Patrizia Bagnerini ◽  
Roberto Cianci
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Estimation Error ◽  
Stability Conditions ◽  
Nonlinear Dynamical ◽  
State Observation ◽  
State Observers ◽  
The Stability

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established.

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