2. 2. 4 Differential inclusions Let X be a subset of R", and let n: X-2 be a function which maps every point of X to a subset of r". Such a function defines a differential inclusion i(t)∈m(x(1) (29) By a solution of (2. 1) on a convex subset T of R we mean a function a: TH X suck )=/ u(tyt,t2∈T for some integrable function u: THR satisfying the inclusion u(t)E n(a(t) for all t E T. It turns out that differential inclusions are a convenient, though not always adequate, way of re-defining discontinuous ode to guarantee existence of solutions It turns out that differential inclusion(2. 9 )subject to fixed initial condition r(to)=.o has a solution on a sufficiently small interval T=to, ti whenever the set-valued function n is compact conver set-valued and semicontinuous with respect to its argument(plus, as usually, o must be an interior point of X) Theorem 2.4 Assume that for some r>0 (a) the set B(x0)={x∈R":|x-ro≤r} is a subset of X (b) for every i E Br (ro) the set n(i) is conver (c) for every sequence of Tk E B,(ro) converging to a limiti E B,(ro) and for every sequence uk∈m(k) there erists a subsequence k=k(q)→∞asq→ oo such that the subsequence ik(g has a limit in n(a) Then the supremum M=sup{|叫:n∈n(),∈Dn(xo,to)} finite, and tf=minto+r/M, to +r] there erists a solution a: [to, t H R of (2.9)satisfying a(to)=o. Moreover, any such solution also satisfies lx(t)sol s r for all tE [to, tf] The discontinuous differential equation i(t)=-sgn(a(t))+c,7 2.2.4 Differential inclusions Let X be a subset of Rn, and let � : X � 2Rn be a function which maps every point of X to a subset of Rn. Such a function defines a differential inclusion x˙ (t) ⊂ �(x(t)). (2.9) By a solution of (2.1) on a convex subset T of R we mean a function x : T ∈� X such that � t2 x(t2) − x(t1) = u(t)dt � t1, t2 ⊂ T t1 for some integrable function u : T ∈� Rn satisfying the inclusion u(t) ⊂ �(x(t)) for all t ⊂ T. It turns out that differential inclusions are a convenient, though not always adequate, way of re-defining discontinuous ODE to guarantee existence of solutions. It turns out that differential inclusion (2.9) subject to fixed initial condition x(t0) = x0 has a solution on a sufficiently small interval T = [t0, t1] whenever the set-valued function � is compact convex set-valued and semicontinuous with respect to its argument (plus, as usually, x0 must be an interior point of X). Theorem 2.4 Assume that for some r > 0 (a) the set Br(x0) = {x ⊂ Rn : |x − x0| ∀ r} is a subset of X; (b) for every x¯ ⊂ Br(x0) the set �(¯x) is convex; (c) for every sequence of x¯k ⊂ Br(x0) converging to a limit x¯ ⊂ Br(x0) and for every sequence u¯k ⊂ �(¯xk) there exists a subsequence k = k(q) � → as q � → such that the subsequence u¯k x). (q) has a limit in �(¯ Then the supremum M = sup{|u¯ ¯ | : u ⊂ �(¯ ¯ x), x ⊂ Dr(x0, t0)} is finite, and, for tf = min{t0 + r/M, t0 + r}, there exists a solution x : [t0, tf ] ∈� Rn of (2.9) satisfying x(t0) = x0. Moreover, any such solution also satisfies |x(t) − x0| ∀ r for all t ⊂ [t0, tf ]. The discontinuous differential equation x˙ (t) = −sgn(x(t)) + c