Differential inclusion

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

A relation

(1)

where is an unknown vector function on some interval and is a set in an -dimensional space which depends on the number and on the vector . The solution of a differential inclusion (1) is usually understood to mean an absolutely-continuous vector function which satisfies the relation

almost-everywhere on the interval of variation of under consideration. In particular, if the set consists of a single point, a differential inclusion becomes an ordinary differential equation . Equations of the type where 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

by differential inequalities

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

(2)

where is the vector function sought, while is the control, i.e. a vector function which may be arbitrarily chosen out of all permissible controls (i.e. such that for each , where is a given set which may depend on and on ). The set of solutions of equation (2) for all permissible controls satisfies the differential inclusion (1), where is the set of all values of the function when runs through the set .

In the theory of differential inclusions it is usually assumed that for any from the domain under consideration the set is a non-empty closed bounded set in an -dimensional space. If the set is everywhere convex, and, for any , it is an upper semi-continuous function in (i.e. for any and any the set is contained in the -neighbourhood of the set for all sufficiently small ), while for any it is a measurable function of (i.e. for any and any sphere in the -dimensional space, the set of values of for which the set is non-empty is Lebesgue measurable), and if also is always contained in a sphere where the function is Lebesgue integrable, then, for any initial conditions , , 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 be convex may be dropped if it depends continuously on . 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