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Why is stability of D.E. defined on infinite interval?
Clash Royale CLAN TAG#URR8PPP
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I will cite the following definition (from an O.D.E textbook) of the stability of a solution:
Definition. Let $f:[0,infty)timesOmegatomathbbR^n$ be continuous and locally Lipschitz on $OmegasubseteqmathbbR^n$. Then a solution $phi:[0,infty)tomathbbR^n$ of the system:
$$
x'(t)=f(t,x(t)), forall tin [0,infty)
$$
is called stable if for every $ageq 0$ there exists $mu(a)>0$ such that for every $xiinOmega$ with $||xi-phi(a)||<mu(a)$, the unique saturated solution $x$, of the problem
$$
begincases x'(t)=f(t,x(t)), forall tin [0,infty) \ x(a)=xiendcases
$$
is defined on $[a,+infty)$ and for every $epsilon>0$,there exists $delta(epsilon,a)in (0,mu(a)]$ such that for each $xiinOmega$ with $||xi-phi(a)||<delta(epsilon,a)$ we have that:
$$||x(t)-phi(t)||<epsilon, forall tin [a,infty).$$
My question is: Why can't we define the stability of a solution on a bounded interval $[a,b]$?? Why we must consider only systems on $[a,+infty)$?
differential-equations stability-in-odes stability-theory
asked Jul 31 at 20:06
Bogdan
59829
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up vote
1
down vote
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I will cite the following definition (from an O.D.E textbook) of the stability of a solution:
Definition. Let $f:[0,infty)timesOmegatomathbbR^n$ be continuous and locally Lipschitz on $OmegasubseteqmathbbR^n$. Then a solution $phi:[0,infty)tomathbbR^n$ of the system:
$$
x'(t)=f(t,x(t)), forall tin [0,infty)
$$
is called stable if for every $ageq 0$ there exists $mu(a)>0$ such that for every $xiinOmega$ with $||xi-phi(a)||<mu(a)$, the unique saturated solution $x$, of the problem
$$
begincases x'(t)=f(t,x(t)), forall tin [0,infty) \ x(a)=xiendcases
$$
is defined on $[a,+infty)$ and for every $epsilon>0$,there exists $delta(epsilon,a)in (0,mu(a)]$ such that for each $xiinOmega$ with $||xi-phi(a)||<delta(epsilon,a)$ we have that:
$$||x(t)-phi(t)||<epsilon, forall tin [a,infty).$$
My question is: Why can't we define the stability of a solution on a bounded interval $[a,b]$?? Why we must consider only systems on $[a,+infty)$?
differential-equations stability-in-odes stability-theory
asked Jul 31 at 20:06
Bogdan
59829
1
The answer is twofold. There is an ideological side of it and there is a technical side. Technically all solutions are stable when you are considering finite intervals of time. This is managed by a theorem about continuous dependence of solution on initial condition. The ideological side is that primarily people are interested in long-term evolution of systems: what happens, in what state system will be when you wait for time $T$, when $T$ is long enough and ideally $T rightarrow infty$? The usual definitiion of Lyapunov stability just captures the essence of this question.
– Evgeny
Jul 31 at 22:43
@Evgeny: Why write such a good answer as a comment?
– Hans Lundmark
Aug 1 at 8:37
@HansLundmark To help somehow first and try not to forget to add useful details later :) And because my "answer" has caveat: there are plenty of papers that are focused on finite-time properties of dynamical systems. But I need to find few good references for this, and this can take quite long..
– Evgeny
Aug 1 at 9:25
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I will cite the following definition (from an O.D.E textbook) of the stability of a solution:
Definition. Let $f:[0,infty)timesOmegatomathbbR^n$ be continuous and locally Lipschitz on $OmegasubseteqmathbbR^n$. Then a solution $phi:[0,infty)tomathbbR^n$ of the system:
$$
x'(t)=f(t,x(t)), forall tin [0,infty)
$$
is called stable if for every $ageq 0$ there exists $mu(a)>0$ such that for every $xiinOmega$ with $||xi-phi(a)||<mu(a)$, the unique saturated solution $x$, of the problem
$$
begincases x'(t)=f(t,x(t)), forall tin [0,infty) \ x(a)=xiendcases
$$
is defined on $[a,+infty)$ and for every $epsilon>0$,there exists $delta(epsilon,a)in (0,mu(a)]$ such that for each $xiinOmega$ with $||xi-phi(a)||<delta(epsilon,a)$ we have that:
$$||x(t)-phi(t)||<epsilon, forall tin [a,infty).$$
My question is: Why can't we define the stability of a solution on a bounded interval $[a,b]$?? Why we must consider only systems on $[a,+infty)$?
differential-equations stability-in-odes stability-theory
asked Jul 31 at 20:06
Bogdan
59829
I will cite the following definition (from an O.D.E textbook) of the stability of a solution:
Definition. Let $f:[0,infty)timesOmegatomathbbR^n$ be continuous and locally Lipschitz on $OmegasubseteqmathbbR^n$. Then a solution $phi:[0,infty)tomathbbR^n$ of the system:
$$
x'(t)=f(t,x(t)), forall tin [0,infty)
$$
is called stable if for every $ageq 0$ there exists $mu(a)>0$ such that for every $xiinOmega$ with $||xi-phi(a)||<mu(a)$, the unique saturated solution $x$, of the problem
$$
begincases x'(t)=f(t,x(t)), forall tin [0,infty) \ x(a)=xiendcases
$$
is defined on $[a,+infty)$ and for every $epsilon>0$,there exists $delta(epsilon,a)in (0,mu(a)]$ such that for each $xiinOmega$ with $||xi-phi(a)||<delta(epsilon,a)$ we have that:
$$||x(t)-phi(t)||<epsilon, forall tin [a,infty).$$
My question is: Why can't we define the stability of a solution on a bounded interval $[a,b]$?? Why we must consider only systems on $[a,+infty)$?
differential-equations stability-in-odes stability-theory
asked Jul 31 at 20:06
Bogdan
59829
asked Jul 31 at 20:06
Bogdan
59829
asked Jul 31 at 20:06
Bogdan
59829
asked Jul 31 at 20:06
Bogdan
59829
59829
1
The answer is twofold. There is an ideological side of it and there is a technical side. Technically all solutions are stable when you are considering finite intervals of time. This is managed by a theorem about continuous dependence of solution on initial condition. The ideological side is that primarily people are interested in long-term evolution of systems: what happens, in what state system will be when you wait for time $T$, when $T$ is long enough and ideally $T rightarrow infty$? The usual definitiion of Lyapunov stability just captures the essence of this question.
– Evgeny
Jul 31 at 22:43
@Evgeny: Why write such a good answer as a comment?
– Hans Lundmark
Aug 1 at 8:37
@HansLundmark To help somehow first and try not to forget to add useful details later :) And because my "answer" has caveat: there are plenty of papers that are focused on finite-time properties of dynamical systems. But I need to find few good references for this, and this can take quite long..
– Evgeny
Aug 1 at 9:25
add a comment |Â
1
The answer is twofold. There is an ideological side of it and there is a technical side. Technically all solutions are stable when you are considering finite intervals of time. This is managed by a theorem about continuous dependence of solution on initial condition. The ideological side is that primarily people are interested in long-term evolution of systems: what happens, in what state system will be when you wait for time $T$, when $T$ is long enough and ideally $T rightarrow infty$? The usual definitiion of Lyapunov stability just captures the essence of this question.
– Evgeny
Jul 31 at 22:43
@Evgeny: Why write such a good answer as a comment?
– Hans Lundmark
Aug 1 at 8:37
@HansLundmark To help somehow first and try not to forget to add useful details later :) And because my "answer" has caveat: there are plenty of papers that are focused on finite-time properties of dynamical systems. But I need to find few good references for this, and this can take quite long..
– Evgeny
Aug 1 at 9:25
1
1
The answer is twofold. There is an ideological side of it and there is a technical side. Technically all solutions are stable when you are considering finite intervals of time. This is managed by a theorem about continuous dependence of solution on initial condition. The ideological side is that primarily people are interested in long-term evolution of systems: what happens, in what state system will be when you wait for time $T$, when $T$ is long enough and ideally $T rightarrow infty$? The usual definitiion of Lyapunov stability just captures the essence of this question.
– Evgeny
Jul 31 at 22:43
The answer is twofold. There is an ideological side of it and there is a technical side. Technically all solutions are stable when you are considering finite intervals of time. This is managed by a theorem about continuous dependence of solution on initial condition. The ideological side is that primarily people are interested in long-term evolution of systems: what happens, in what state system will be when you wait for time $T$, when $T$ is long enough and ideally $T rightarrow infty$? The usual definitiion of Lyapunov stability just captures the essence of this question.
– Evgeny
Jul 31 at 22:43
@Evgeny: Why write such a good answer as a comment?
– Hans Lundmark
Aug 1 at 8:37
@Evgeny: Why write such a good answer as a comment?
– Hans Lundmark
Aug 1 at 8:37
@HansLundmark To help somehow first and try not to forget to add useful details later :) And because my "answer" has caveat: there are plenty of papers that are focused on finite-time properties of dynamical systems. But I need to find few good references for this, and this can take quite long..
– Evgeny
Aug 1 at 9:25
@HansLundmark To help somehow first and try not to forget to add useful details later :) And because my "answer" has caveat: there are plenty of papers that are focused on finite-time properties of dynamical systems. But I need to find few good references for this, and this can take quite long..
– Evgeny
Aug 1 at 9:25
add a comment |Â
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1
The answer is twofold. There is an ideological side of it and there is a technical side. Technically all solutions are stable when you are considering finite intervals of time. This is managed by a theorem about continuous dependence of solution on initial condition. The ideological side is that primarily people are interested in long-term evolution of systems: what happens, in what state system will be when you wait for time $T$, when $T$ is long enough and ideally $T rightarrow infty$? The usual definitiion of Lyapunov stability just captures the essence of this question.
– Evgeny
Jul 31 at 22:43
@Evgeny: Why write such a good answer as a comment?
– Hans Lundmark
Aug 1 at 8:37
@HansLundmark To help somehow first and try not to forget to add useful details later :) And because my "answer" has caveat: there are plenty of papers that are focused on finite-time properties of dynamical systems. But I need to find few good references for this, and this can take quite long..
– Evgeny
Aug 1 at 9:25