Showing the equilibrium point to be globally exponentially stable using Lyapunov indirect method. The 2019 Stack Overflow Developer Survey Results Are InShow that the real part and imaginary part of a solution of a differential equation are also real solutionsDemonstrate by the Lyapunov methodLyapunov function instead of linearizationDetermining stability of equilibrium points for a non linear systemAsymptotic stability without LyapunovConditions for which all orbits are periodic through varying constants.Center manifold of nonhyperbolic fixed pointShowing that a centre of the 2D linear system $dotmathbfx = A mathbf x$ is Lyapunov stableHow do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?Find one domain of attraction for this system
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Showing the equilibrium point to be globally exponentially stable using Lyapunov indirect method.
The 2019 Stack Overflow Developer Survey Results Are InShow that the real part and imaginary part of a solution of a differential equation are also real solutionsDemonstrate by the Lyapunov methodLyapunov function instead of linearizationDetermining stability of equilibrium points for a non linear systemAsymptotic stability without LyapunovConditions for which all orbits are periodic through varying constants.Center manifold of nonhyperbolic fixed pointShowing that a centre of the 2D linear system $dotmathbfx = A mathbf x$ is Lyapunov stableHow do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?Find one domain of attraction for this system
$begingroup$
We have the system $ddotq + dotq + g(dotq,q) + q = 0, forall t geq 0$.
$x = beginbmatrix x_1\ x_2endbmatrix = beginbmatrix q\ dotqendbmatrix$
$dotx = Ax + h(x)$ with the initial condition $x(0) = x_0 in BbbR^2$
We are given that $g(0,0)=0$; $g$ is lipschitz continuous meaning we have
$||g(x) - g(y)|| leq mu ||x-y|| forall x,y in BbbR^2$
I am thinking how to apply Lyapunov's indirect method here.
$V(x) = x^* P x, P>0$ such that $A^* P + PA =I$ using Lyapunov indirect method I am trying to determine $mu>0$ such that $0$ is globally exponentially stable.
Also hoping that we get the same result using Gronwalls inequality, getting the inequality is tricky I think due the presence of unknown function $g(dotq,q)$.
real-analysis ordinary-differential-equations dynamical-systems stability-in-odes lyapunov-functions
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
$endgroup$
add a comment |
$begingroup$
We have the system $ddotq + dotq + g(dotq,q) + q = 0, forall t geq 0$.
$x = beginbmatrix x_1\ x_2endbmatrix = beginbmatrix q\ dotqendbmatrix$
$dotx = Ax + h(x)$ with the initial condition $x(0) = x_0 in BbbR^2$
We are given that $g(0,0)=0$; $g$ is lipschitz continuous meaning we have
$||g(x) - g(y)|| leq mu ||x-y|| forall x,y in BbbR^2$
I am thinking how to apply Lyapunov's indirect method here.
$V(x) = x^* P x, P>0$ such that $A^* P + PA =I$ using Lyapunov indirect method I am trying to determine $mu>0$ such that $0$ is globally exponentially stable.
Also hoping that we get the same result using Gronwalls inequality, getting the inequality is tricky I think due the presence of unknown function $g(dotq,q)$.
real-analysis ordinary-differential-equations dynamical-systems stability-in-odes lyapunov-functions
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
$endgroup$
add a comment |
$begingroup$
We have the system $ddotq + dotq + g(dotq,q) + q = 0, forall t geq 0$.
$x = beginbmatrix x_1\ x_2endbmatrix = beginbmatrix q\ dotqendbmatrix$
$dotx = Ax + h(x)$ with the initial condition $x(0) = x_0 in BbbR^2$
We are given that $g(0,0)=0$; $g$ is lipschitz continuous meaning we have
$||g(x) - g(y)|| leq mu ||x-y|| forall x,y in BbbR^2$
I am thinking how to apply Lyapunov's indirect method here.
$V(x) = x^* P x, P>0$ such that $A^* P + PA =I$ using Lyapunov indirect method I am trying to determine $mu>0$ such that $0$ is globally exponentially stable.
Also hoping that we get the same result using Gronwalls inequality, getting the inequality is tricky I think due the presence of unknown function $g(dotq,q)$.
real-analysis ordinary-differential-equations dynamical-systems stability-in-odes lyapunov-functions
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
$endgroup$
We have the system $ddotq + dotq + g(dotq,q) + q = 0, forall t geq 0$.
$x = beginbmatrix x_1\ x_2endbmatrix = beginbmatrix q\ dotqendbmatrix$
$dotx = Ax + h(x)$ with the initial condition $x(0) = x_0 in BbbR^2$
We are given that $g(0,0)=0$; $g$ is lipschitz continuous meaning we have
$||g(x) - g(y)|| leq mu ||x-y|| forall x,y in BbbR^2$
I am thinking how to apply Lyapunov's indirect method here.
$V(x) = x^* P x, P>0$ such that $A^* P + PA =I$ using Lyapunov indirect method I am trying to determine $mu>0$ such that $0$ is globally exponentially stable.
Also hoping that we get the same result using Gronwalls inequality, getting the inequality is tricky I think due the presence of unknown function $g(dotq,q)$.
real-analysis ordinary-differential-equations dynamical-systems stability-in-odes lyapunov-functions
real-analysis ordinary-differential-equations dynamical-systems stability-in-odes lyapunov-functions
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
edited Apr 7 at 20:32
Rodrigo de Azevedo
13.2k41962
13.2k41962
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
asked Apr 6 at 23:33
BAYMAXBAYMAX
3,00921225
3,00921225
add a comment |
add a comment |
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