View source for Differential inclusion

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''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|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

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

by differential inequalities

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

by differential equations with discontinuous right-hand side [[#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

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

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 [[#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]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.F. Filippov,   "Differential equations with discontinuous righthand sides" , Reidel  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Wazewski,   "Systèmes de commande et équations au contingent"  ''Bull. Acad. Polon. Sci. Ser. Math.'' , '''9''' :  3  (1961)  pp. 151–155</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Davy,   "Properties of the solution set of a generalized differential equation"  ''Bull. Austr. Math. Soc.'' , '''6''' :  3  (1972)  pp. 379–398</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Olech,   "Existence of solutions of non-convex orientor fields"  ''Boll. Un. Mat. Ital.'' , '''11''' :  3  (1975)  pp. 189–197</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.-P. Aubin,   A. Cellina,   "Differential inclusions" , Univ. Paris IX  (1983)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Roxin,   "Stability in general control systems"  ''J. Diff. Equations'' , '''1''' :  2  (1965)  pp. 115–150</TD></TR></table>