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''multi-valued differential equation, differential equation with multi-valued right-hand side''
 
''multi-valued differential equation, differential equation with multi-valued right-hand side''
  
 
A relation
 
A relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\frac{dx}{dt}\in F(t,x),\tag{1}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322002.png" /> is an unknown vector function on some interval and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322003.png" /> is a set in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322004.png" />-dimensional space which depends on the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322005.png" /> and on the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322006.png" />. The solution of a differential inclusion (1) is usually understood to mean an absolutely-continuous vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322007.png" /> which satisfies the relation
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where $x=x(t)$ is an unknown vector function on some interval and $F(t,x)$ is a set in an $n$-dimensional space which depends on the number $t$ and on the vector $(x_1,\ldots,x_n)$. The solution of a differential inclusion \ref{1} is usually understood to mean an absolutely-continuous vector function $x(t)$ which satisfies the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322008.png" /></td> </tr></table>
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$$\frac{dx(t)}{dt}\in F(t,x(t))$$
  
almost-everywhere on the interval of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d0322009.png" /> under consideration. In particular, if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220010.png" /> consists of a single point, a differential inclusion becomes an ordinary differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220011.png" />. Equations of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220012.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220013.png" /> is a [[Contingent|contingent]], [[#References|[1]]], are equivalent to differential inclusions in a large number of cases.
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almost-everywhere on the interval of variation of $t$ under consideration. In particular, if the set $F(t,x)$ consists of a single point, a differential inclusion becomes an ordinary differential equation $dx/dt=F(t,x)$. Equations of the type $Dx(t)\in F(t,x(t))$ where $Dx(t)$ is a [[Contingent|contingent]], [[#References|[1]]], are equivalent to differential inclusions in a large number of cases.
  
 
Differential inclusions are generated, for example, by the problem concerning functions which satisfy a differential equation to within required accuracy
 
Differential inclusions are generated, for example, by the problem concerning functions which satisfy a differential equation to within required accuracy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220014.png" /></td> </tr></table>
+
$$\left|\frac{dx(t)}{dt}-f(t,x(t))\right|\leq\epsilon;$$
  
 
by differential inequalities
 
by differential inequalities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220015.png" /></td> </tr></table>
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$$f\left(t,x,\frac{dx}{dt}\right)\geq0;$$
  
 
by differential equations with discontinuous right-hand side [[#References|[1]]], Chapt. 2; and by problems in the theory of optimal control [[#References|[3]]], [[#References|[2]]]. The equation which is most often considered in control problems is
 
by differential equations with discontinuous right-hand side [[#References|[1]]], Chapt. 2; and by problems in the theory of optimal control [[#References|[3]]], [[#References|[2]]]. The equation which is most often considered in control problems is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\frac{dx}{dt}=f(t,x,u),\tag{2}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220017.png" /> is the vector function sought, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220018.png" /> is the control, i.e. a vector function which may be arbitrarily chosen out of all permissible controls (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220019.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220021.png" /> is a given set which may depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220022.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220023.png" />). The set of solutions of equation (2) for all permissible controls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220024.png" /> satisfies the differential inclusion (1), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220025.png" /> is the set of all values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220026.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220027.png" /> runs through the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220028.png" />.
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where $x=x(t)$ is the vector function sought, while $u=u(t)$ is the control, i.e. a vector function which may be arbitrarily chosen out of all permissible controls (i.e. such that $u(t)\in U$ for each $t$, where $U$ is a given set which may depend on $t$ and on $x=x(t)$). The set of solutions of equation \ref{2} for all permissible controls $u=u(t)$ satisfies the differential inclusion \ref{1}, where $F(t,x)$ is the set of all values of the function $f(t,x,u)$ when $u$ runs through the set $U$.
  
In the theory of differential inclusions it is usually assumed that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220029.png" /> from the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220030.png" /> under consideration the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220031.png" /> is a non-empty closed bounded set in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220032.png" />-dimensional space. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220033.png" /> is everywhere convex, and, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220034.png" />, it is an upper semi-continuous function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220035.png" /> (i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220036.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220037.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220038.png" /> is contained in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220039.png" />-neighbourhood of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220040.png" /> for all sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220041.png" />), while for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220042.png" /> it is a measurable function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220043.png" /> (i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220044.png" /> and any sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220045.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220046.png" />-dimensional space, the set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220047.png" /> for which the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220048.png" /> is non-empty is Lebesgue measurable), and if also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220049.png" /> is always contained in a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220050.png" /> where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220051.png" /> is Lebesgue integrable, then, for any initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220053.png" />, a solution of the differential inclusion exists [[#References|[4]]] and the [[Integral funnel|integral funnel]] consisting of such solutions displays the usual properties [[#References|[4]]]. The requirement that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220054.png" /> be convex may be dropped if it depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032200/d03220055.png" />. The existence of a solution is preserved [[#References|[5]]], but the properties of the integral funnels are not.
+
In the theory of differential inclusions it is usually assumed that for any $t,x$ from the domain $G$ under consideration the set $F(t,x)$ is a non-empty closed bounded set in an $n$-dimensional space. If the set $F(t,x)$ is everywhere convex, and, for any $t$, it is an upper semi-continuous function in $t$ (i.e. for any $t,x$ and any $\epsilon>0$ the set $F(t,x')$ is contained in the $\epsilon$-neighbourhood of the set $F(t,x)$ for all sufficiently small $|x'-x|$), while for any $x$ it is a measurable function of $t$ (i.e. for any $x$ and any sphere $B$ in the $n$-dimensional space, the set of values of $t$ for which the set $F(t,x)\cap B$ is non-empty is Lebesgue measurable), and if also $F(t,x)$ is always contained in a sphere $|x|\leq m(t)$ where the function $m(t)$ is Lebesgue integrable, then, for any initial conditions $x(t_0)=x_0$, $(t_0,x_0)\in G$, a solution of the differential inclusion exists [[#References|[4]]] and the [[Integral funnel|integral funnel]] consisting of such solutions displays the usual properties [[#References|[4]]]. The requirement that the set $F(t,x)$ be convex may be dropped if it depends continuously on $x$. The existence of a solution is preserved [[#References|[5]]], but the properties of the integral funnels are not.
  
 
For a review of the publications on differential inclusions and on the connection of such inclusions with control problems see [[#References|[6]]], [[#References|[7]]]. For the concept of stability of differential inclusions see [[#References|[8]]], [[#References|[1]]]; for the existence of bounded and periodic solutions, and for other properties, see [[#References|[1]]], [[#References|[6]]], [[#References|[7]]].
 
For a review of the publications on differential inclusions and on the connection of such inclusions with control problems see [[#References|[6]]], [[#References|[7]]]. For the concept of stability of differential inclusions see [[#References|[8]]], [[#References|[1]]]; for the existence of bounded and periodic solutions, and for other properties, see [[#References|[1]]], [[#References|[6]]], [[#References|[7]]].

Revision as of 16:56, 12 August 2014

multi-valued differential equation, differential equation with multi-valued right-hand side

A relation

$$\frac{dx}{dt}\in F(t,x),\tag{1}$$

where $x=x(t)$ is an unknown vector function on some interval and $F(t,x)$ is a set in an $n$-dimensional space which depends on the number $t$ and on the vector $(x_1,\ldots,x_n)$. The solution of a differential inclusion \ref{1} is usually understood to mean an absolutely-continuous vector function $x(t)$ which satisfies the relation

$$\frac{dx(t)}{dt}\in F(t,x(t))$$

almost-everywhere on the interval of variation of $t$ under consideration. In particular, if the set $F(t,x)$ consists of a single point, a differential inclusion becomes an ordinary differential equation $dx/dt=F(t,x)$. Equations of the type $Dx(t)\in F(t,x(t))$ where $Dx(t)$ is a contingent, [1], are equivalent to differential inclusions in a large number of cases.

Differential inclusions are generated, for example, by the problem concerning functions which satisfy a differential equation to within required accuracy

$$\left|\frac{dx(t)}{dt}-f(t,x(t))\right|\leq\epsilon;$$

by differential inequalities

$$f\left(t,x,\frac{dx}{dt}\right)\geq0;$$

by differential equations with discontinuous right-hand side [1], Chapt. 2; and by problems in the theory of optimal control [3], [2]. The equation which is most often considered in control problems is

$$\frac{dx}{dt}=f(t,x,u),\tag{2}$$

where $x=x(t)$ is the vector function sought, while $u=u(t)$ is the control, i.e. a vector function which may be arbitrarily chosen out of all permissible controls (i.e. such that $u(t)\in U$ for each $t$, where $U$ is a given set which may depend on $t$ and on $x=x(t)$). The set of solutions of equation \ref{2} for all permissible controls $u=u(t)$ satisfies the differential inclusion \ref{1}, where $F(t,x)$ is the set of all values of the function $f(t,x,u)$ when $u$ runs through the set $U$.

In the theory of differential inclusions it is usually assumed that for any $t,x$ from the domain $G$ under consideration the set $F(t,x)$ is a non-empty closed bounded set in an $n$-dimensional space. If the set $F(t,x)$ is everywhere convex, and, for any $t$, it is an upper semi-continuous function in $t$ (i.e. for any $t,x$ and any $\epsilon>0$ the set $F(t,x')$ is contained in the $\epsilon$-neighbourhood of the set $F(t,x)$ for all sufficiently small $|x'-x|$), while for any $x$ it is a measurable function of $t$ (i.e. for any $x$ and any sphere $B$ in the $n$-dimensional space, the set of values of $t$ for which the set $F(t,x)\cap B$ is non-empty is Lebesgue measurable), and if also $F(t,x)$ is always contained in a sphere $|x|\leq m(t)$ where the function $m(t)$ is Lebesgue integrable, then, for any initial conditions $x(t_0)=x_0$, $(t_0,x_0)\in G$, a solution of the differential inclusion exists [4] and the integral funnel consisting of such solutions displays the usual properties [4]. The requirement that the set $F(t,x)$ be convex may be dropped if it depends continuously on $x$. The existence of a solution is preserved [5], but the properties of the integral funnels are not.

For a review of the publications on differential inclusions and on the connection of such inclusions with control problems see [6], [7]. For the concept of stability of differential inclusions see [8], [1]; for the existence of bounded and periodic solutions, and for other properties, see [1], [6], [7].

References

[1] A.F. Filippov, "Differential equations with discontinuous righthand sides" , Reidel (1988) (Translated from Russian)
[2] A. Wazewski, "Systèmes de commande et équations au contingent" Bull. Acad. Polon. Sci. Ser. Math. , 9 : 3 (1961) pp. 151–155
[3] A.F. Filoppov, "On certain questions in the theory of optimal control" SIAM J. Control Ser. A , 1 : 1 (1962) pp. 76–84 Vestnik Moskov. Univ. Ser. Mat. Mekh. Astr. , 2 (1959) pp. 25–32
[4] J.L. Davy, "Properties of the solution set of a generalized differential equation" Bull. Austr. Math. Soc. , 6 : 3 (1972) pp. 379–398
[5] C. Olech, "Existence of solutions of non-convex orientor fields" Boll. Un. Mat. Ital. , 11 : 3 (1975) pp. 189–197
[6] V.I. Blagodatskikh, A.F. Filippov, "Differential inclusions and optimal control" Proc. Steklov Inst. Math. , 169 (To appear) Trudy Mat. Inst. Steklov. , 169 (To appear)
[7] J.-P. Aubin, A. Cellina, "Differential inclusions" , Univ. Paris IX (1983)
[8] E. Roxin, "Stability in general control systems" J. Diff. Equations , 1 : 2 (1965) pp. 115–150
How to Cite This Entry:
Differential inclusion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_inclusion&oldid=32873
This article was adapted from an original article by A.F. Filippov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article