Differential inclusion

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

A relation

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

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 \eqref{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

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

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 \eqref{2} for all permissible controls $u=u(t)$ satisfies the differential inclusion \eqref{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 $x$ (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