Solution for first order ODE with discontinuous right hand side












1












$begingroup$


I'm now approaching for the first time to first order differential equations with discontinuous right hand side.



Let $Asubset mathbb{R}timesmathbb{R}^n$ and $f=f(t, x):Ato mathbb{R}^n$. Fix $(t_0,x_0)in A$ and consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(t_0)=x_0end{cases}.qquad[A]$$ It is well known that if $f$ is continuous then Peano's Theorem guarantee the existence of a local classical solution to $[A]$, which is also a $C^1$ function.



The notion of Carathéodory solution for an initial problem like $[A]$ arises when $f$ is a general function, not necessarily continuous.



Def. An absolutely continuous function $y:[t_0,t_0+a]to mathbb{R}^n$ ($a in mathbb{R},a>0$), such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is said to be a Carathedory solution for $[A]$ if $$y(t)=x_0+int_{t_0}^{t}f(s,y(s)dsqquad t in [t_0,t_0+a].$$ By equivalent definitions of absolute continuity for functions on closed and bounded intervals it follow immediately that if $y:[t_0,t_0+a]to mathbb{R}^n$, such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is an absolutely continuous function, then $$text{y is a Carathédory solution of}, [A] iff ,text y(t_0)=x_0,text{and}, y'(t)=f(t,y(t)), text {for a.e.}, t in [t_0,t_0+a].$$



By Carathédory existence theorem we know that, under suitable conditions on $f$, there is at least a solution of $[A]$, in the weak sense as above.



Now let's suppose we are given a function $f:[0,+infty)times mathbb{R}^nto mathbb{R}^n$ and a point $x_0 in mathbb{R}^n$. Consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(0)=x_0end{cases}.qquad[B]$$ In many books I read that under the hypothesis:



(i)$f$ is bounded



(ii) the map $tto f(x, t)$ is measurable for each $x$



(iii) Exists $L>0$ such that $$|f(t,x_1)-f(t,x_2)|leq L|x_1-x_2| qquad forall (t,x_1)(t,x_2) in [0,+infty)times mathbb{R}^n $$
then there exist a unique global solution $y_{x_0}:[0,+infty)to mathbb{R}^n$ to $[B]$.



The problem is that I'm missing what "solution" in this case means (the interval of definition of $y_{x_0}$ is not closed and it is also unbounded). I think that it could be understood like a function $y:[0,+infty)to mathbb{R}^n$ such that $y$ is absolutely continuous on every compact subinterval $[a,b]subset [0,+infty)$ and $$y(t)=x_0+int_{t_0}^{t}f(s,y(s))ds qquad t>0,$$ but I'm not sure about it. For instance, this problem arises in studying optimal control theory.



Can anyone give me some help or some good references?



Thanks a lot.






real-analysis ordinary-differential-equations control-theory optimal-control







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I'm now approaching for the first time to first order differential equations with discontinuous right hand side.



    Let $Asubset mathbb{R}timesmathbb{R}^n$ and $f=f(t, x):Ato mathbb{R}^n$. Fix $(t_0,x_0)in A$ and consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(t_0)=x_0end{cases}.qquad[A]$$ It is well known that if $f$ is continuous then Peano's Theorem guarantee the existence of a local classical solution to $[A]$, which is also a $C^1$ function.



    The notion of Carathéodory solution for an initial problem like $[A]$ arises when $f$ is a general function, not necessarily continuous.



    Def. An absolutely continuous function $y:[t_0,t_0+a]to mathbb{R}^n$ ($a in mathbb{R},a>0$), such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is said to be a Carathedory solution for $[A]$ if $$y(t)=x_0+int_{t_0}^{t}f(s,y(s)dsqquad t in [t_0,t_0+a].$$ By equivalent definitions of absolute continuity for functions on closed and bounded intervals it follow immediately that if $y:[t_0,t_0+a]to mathbb{R}^n$, such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is an absolutely continuous function, then $$text{y is a Carathédory solution of}, [A] iff ,text y(t_0)=x_0,text{and}, y'(t)=f(t,y(t)), text {for a.e.}, t in [t_0,t_0+a].$$



    By Carathédory existence theorem we know that, under suitable conditions on $f$, there is at least a solution of $[A]$, in the weak sense as above.



    Now let's suppose we are given a function $f:[0,+infty)times mathbb{R}^nto mathbb{R}^n$ and a point $x_0 in mathbb{R}^n$. Consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(0)=x_0end{cases}.qquad[B]$$ In many books I read that under the hypothesis:



    (i)$f$ is bounded



    (ii) the map $tto f(x, t)$ is measurable for each $x$



    (iii) Exists $L>0$ such that $$|f(t,x_1)-f(t,x_2)|leq L|x_1-x_2| qquad forall (t,x_1)(t,x_2) in [0,+infty)times mathbb{R}^n $$
    then there exist a unique global solution $y_{x_0}:[0,+infty)to mathbb{R}^n$ to $[B]$.



    The problem is that I'm missing what "solution" in this case means (the interval of definition of $y_{x_0}$ is not closed and it is also unbounded). I think that it could be understood like a function $y:[0,+infty)to mathbb{R}^n$ such that $y$ is absolutely continuous on every compact subinterval $[a,b]subset [0,+infty)$ and $$y(t)=x_0+int_{t_0}^{t}f(s,y(s))ds qquad t>0,$$ but I'm not sure about it. For instance, this problem arises in studying optimal control theory.



    Can anyone give me some help or some good references?



    Thanks a lot.






    real-analysis ordinary-differential-equations control-theory optimal-control







    share|cite|improve this question









    $endgroup$















      1












      1








      1


      2



      $begingroup$


      I'm now approaching for the first time to first order differential equations with discontinuous right hand side.



      Let $Asubset mathbb{R}timesmathbb{R}^n$ and $f=f(t, x):Ato mathbb{R}^n$. Fix $(t_0,x_0)in A$ and consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(t_0)=x_0end{cases}.qquad[A]$$ It is well known that if $f$ is continuous then Peano's Theorem guarantee the existence of a local classical solution to $[A]$, which is also a $C^1$ function.



      The notion of Carathéodory solution for an initial problem like $[A]$ arises when $f$ is a general function, not necessarily continuous.



      Def. An absolutely continuous function $y:[t_0,t_0+a]to mathbb{R}^n$ ($a in mathbb{R},a>0$), such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is said to be a Carathedory solution for $[A]$ if $$y(t)=x_0+int_{t_0}^{t}f(s,y(s)dsqquad t in [t_0,t_0+a].$$ By equivalent definitions of absolute continuity for functions on closed and bounded intervals it follow immediately that if $y:[t_0,t_0+a]to mathbb{R}^n$, such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is an absolutely continuous function, then $$text{y is a Carathédory solution of}, [A] iff ,text y(t_0)=x_0,text{and}, y'(t)=f(t,y(t)), text {for a.e.}, t in [t_0,t_0+a].$$



      By Carathédory existence theorem we know that, under suitable conditions on $f$, there is at least a solution of $[A]$, in the weak sense as above.



      Now let's suppose we are given a function $f:[0,+infty)times mathbb{R}^nto mathbb{R}^n$ and a point $x_0 in mathbb{R}^n$. Consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(0)=x_0end{cases}.qquad[B]$$ In many books I read that under the hypothesis:



      (i)$f$ is bounded



      (ii) the map $tto f(x, t)$ is measurable for each $x$



      (iii) Exists $L>0$ such that $$|f(t,x_1)-f(t,x_2)|leq L|x_1-x_2| qquad forall (t,x_1)(t,x_2) in [0,+infty)times mathbb{R}^n $$
      then there exist a unique global solution $y_{x_0}:[0,+infty)to mathbb{R}^n$ to $[B]$.



      The problem is that I'm missing what "solution" in this case means (the interval of definition of $y_{x_0}$ is not closed and it is also unbounded). I think that it could be understood like a function $y:[0,+infty)to mathbb{R}^n$ such that $y$ is absolutely continuous on every compact subinterval $[a,b]subset [0,+infty)$ and $$y(t)=x_0+int_{t_0}^{t}f(s,y(s))ds qquad t>0,$$ but I'm not sure about it. For instance, this problem arises in studying optimal control theory.



      Can anyone give me some help or some good references?



      Thanks a lot.






      real-analysis ordinary-differential-equations control-theory optimal-control







      share|cite|improve this question









      $endgroup$




      I'm now approaching for the first time to first order differential equations with discontinuous right hand side.



      Let $Asubset mathbb{R}timesmathbb{R}^n$ and $f=f(t, x):Ato mathbb{R}^n$. Fix $(t_0,x_0)in A$ and consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(t_0)=x_0end{cases}.qquad[A]$$ It is well known that if $f$ is continuous then Peano's Theorem guarantee the existence of a local classical solution to $[A]$, which is also a $C^1$ function.



      The notion of Carathéodory solution for an initial problem like $[A]$ arises when $f$ is a general function, not necessarily continuous.



      Def. An absolutely continuous function $y:[t_0,t_0+a]to mathbb{R}^n$ ($a in mathbb{R},a>0$), such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is said to be a Carathedory solution for $[A]$ if $$y(t)=x_0+int_{t_0}^{t}f(s,y(s)dsqquad t in [t_0,t_0+a].$$ By equivalent definitions of absolute continuity for functions on closed and bounded intervals it follow immediately that if $y:[t_0,t_0+a]to mathbb{R}^n$, such that $(t,y(t))in A ,forall t in [t_0,t_0+a]$, is an absolutely continuous function, then $$text{y is a Carathédory solution of}, [A] iff ,text y(t_0)=x_0,text{and}, y'(t)=f(t,y(t)), text {for a.e.}, t in [t_0,t_0+a].$$



      By Carathédory existence theorem we know that, under suitable conditions on $f$, there is at least a solution of $[A]$, in the weak sense as above.



      Now let's suppose we are given a function $f:[0,+infty)times mathbb{R}^nto mathbb{R}^n$ and a point $x_0 in mathbb{R}^n$. Consider the initial value problem $$begin{cases} y'(t)=f(t,y(t))\y(0)=x_0end{cases}.qquad[B]$$ In many books I read that under the hypothesis:



      (i)$f$ is bounded



      (ii) the map $tto f(x, t)$ is measurable for each $x$



      (iii) Exists $L>0$ such that $$|f(t,x_1)-f(t,x_2)|leq L|x_1-x_2| qquad forall (t,x_1)(t,x_2) in [0,+infty)times mathbb{R}^n $$
      then there exist a unique global solution $y_{x_0}:[0,+infty)to mathbb{R}^n$ to $[B]$.



      The problem is that I'm missing what "solution" in this case means (the interval of definition of $y_{x_0}$ is not closed and it is also unbounded). I think that it could be understood like a function $y:[0,+infty)to mathbb{R}^n$ such that $y$ is absolutely continuous on every compact subinterval $[a,b]subset [0,+infty)$ and $$y(t)=x_0+int_{t_0}^{t}f(s,y(s))ds qquad t>0,$$ but I'm not sure about it. For instance, this problem arises in studying optimal control theory.



      Can anyone give me some help or some good references?



      Thanks a lot.






      real-analysis ordinary-differential-equations control-theory optimal-control




      real-analysis ordinary-differential-equations control-theory optimal-control






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 21 at 16:41









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