# Questions tagged [lyapunov-functions]

This tag is for questions relating to Lyapunov function, which is a scalar function defined on the phase space, which can be used to prove the stability of an equilibrium point. The Lyapunov function method is applied to study the stability of various differential equations and systems.

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### Necessity of the hypotheses of Lyapunov asymptotic stability theorem

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption. The statement we ...
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### The lyapunov function of gradient system

Given a dynamical system $$\frac{dx}{dt}=-\nabla f(x)$$ which $x=0$ is the only equilibrium point, i.e. $-\nabla f(x)|_{x=0}=0$. I am reading this tutorial, and it states: $f(x)$ is a lyapunov ...
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### Lyapunov function to prove globally asymptotically stable

I have the system $x'=-x^3+2y^3$ and $y'=-2xy^2$. I need to prove that the point $(0,0)$ is asymptotically globally stable. Here's what I did: if we have a Lyapunov function $v(x,y)=ax^2+bxy+cy^2$, ...
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### Finding Lyapunov function for particular system

I am trying to find Lyapunov function for $$\begin{cases}\dot{t} = y\\\dot{y} = t^2-t\end{cases}$$ I tried common examples but, maybe I was wrong in my computations, couldn't derive anything. Could ...
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### Lyapunov function for a second order system involving trigonometric functions

I am studying the stability of the following system: \begin{aligned} \dot{x}_{1} &= -x_{1}^{2} - \sin x_{2}\\ \dot{x}_{2} &= x_{1} - \frac{\cos x_{2}}{x_{1}}\\ \end{aligned} The system ...
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### Is the solution to $\theta''+0.021\,\text{sgn}(\theta')\sqrt{|\theta'|}+0.02\sin(\theta)=0,\,\theta_0=\pi/2,\,\theta'_0=0$ of finite duration?

Is the solution to $\ddot{\theta}+0.021\,\text{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|}+0.02\sin(\theta)=0,\,\,\theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \quad\text{(Eq. 1)}$ of finite duration? I ...
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### Picking a Lyapunov function that is dependent on time

I have the following system: $$\begin{cases} \dot{x_{1}}(t) = x_{2} - x_{1}^{3}\\ \dot{x_{2}}(t) = -0.5 x_{1} - x_{2} + d\sin(t) \end{cases}$$ where $d$ is some constant. I want to show $(0,0)$ is ...
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### Lyapunov function of a transformed system

I have a system, which has the form $\frac{da_i}{dt}=\nu(\eta_i)(\sigma(\eta_i)-a_i)$, where $\eta=WA+V$ and $A,V,\eta\in\mathbb{R}^N$ are vectors and $W\in\mathbb{R}^{N\times N}$ is some square ...
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### Does Asymptotic Stability Imply the Existence of a Lyapunov Function for a Nonlinear System?

For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some ...
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### LMI-Solution Invariant to the Initial Conditions

It is known from R. Bellman that the value of the functional $J = \int_{0}^{\infty}xWx \ dt, \ W>0$ along the solution of the linear time-invariant system $\dot x = Ax, \ x(0)=x_0$, with Schur ...
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