for all 0<hs ho. Now from the dynamic programming principle(6)we have (so, To) L(E(s), u(s)ds +V(so+h, 5(so+h)) which with(57) implies o(S0+h,、(s0+h)-叫(s0,xo)_1 h L(x(s),u(s)ds≤ Send h→0 to obtain 0 o(so, co)-Vo(so, ro)f(o, u)-L(, u)<0 Now maximize over u to obtain at p(so, To)+sup -Vo(so, ro)f(ao, u)-L(ro, u)<0 This proves that v is a viscosity subsolution Supersolution property. Let E C((to, ti)x R") and suppose that V-o attains a local minimum at(so, o); so there exists r>0 such that V(t,x)-o(t,x)≥V(s0,xo)-0(s0,x0)|x-xo|<r,|t-so|<r.(62) gain by OdE estimates, there exists ho >0 such that E(so+)-tol <r for all< hs ho and all u( ) E Uso, t1, where $()denotes the corresponding state trajectory with E(so)=To Assume the supersolution property is false, i. e. there exists a>0 such that P(so, co)+sup-Vo(so, c o)f(ao, u)-L(o, u))<-3ah<0, (64) where 0< h< ho. Now(64) implies 0 ,5(s)-Vo(s,5(s)f((s),u(s))-L(5(s),(s)≤-2ah<0.(65) for all s E[so, So+h] and all u()EUso, ti, for h >0 sufficiently small By the dynamic programming formula( 6), there exists uo()EUso,t1 such that soth V(50,xo)2/L(5(s),0()ds+V(50+h,56(0+1)-ah(60for all 0 < h ≤ h0. Now from the dynamic programming principle (6) we have V (s0, x0) ≤ Z s0+h s0 L(ξ(s), u(s)) ds + V (s0 + h, ξ(s0 + h)) (58) which with (57) implies −( φ(s0 + h, ξ(s0 + h)) − φ(s0, x0)) h ) − 1 h Z s0+h s0 L(x(s), u(s)) ds ≤ 0. (59) Send h → 0 to obtain − ∂ ∂tφ(s0, x0) − ∇φ(s0, x0)f(x0, u) − L(x0, u) ≤ 0. (60) Now maximize over u to obtain − ∂ ∂tφ(s0, x0) + sup u∈U {−∇φ(s0, x0)f(x0, u) − L(x0, u)} ≤ 0. (61) This proves that V is a viscosity subsolution. Supersolution property. Let φ ∈ C 1 ((t0, t1) × Rn ) and suppose that V − φ attains a local minimum at (s0, x0); so there exists r > 0 such that V (t, x) − φ(t, x) ≥ V (s0, x0) − φ(s0, x0) ∀ |x − x0| < r, |t − s0| < r. (62) Again by ODE estimates, there exists h0 > 0 such that |ξ(s0 + h) − x0| < r (63) for all 0 ≤ h ≤ h0 and all u(·) ∈ Us0,t1 , where ξ(·) denotes the corresponding state trajectory with ξ(s0) = x0. Assume the supersolution property is false, i.e. there exists α > 0 such that − ∂ ∂tφ(s0, x0) + sup u∈U {−∇φ(s0, x0)f(x0, u) − L(x0, u)} ≤ −3αh < 0, (64) where 0 < h < h0. Now (64) implies − ∂ ∂tφ(s, ξ(s)) − ∇φ(s, ξ(s))f(ξ(s), u(s)) − L(ξ(s), u(s)) ≤ −2αh < 0, (65) for all s ∈ [s0, s0 + h] and all u(·) ∈ Us0,t1 , for h > 0 sufficiently small. By the dynamic programming formula (6), there exists u0(·) ∈ Us0,t1 such that V (s0, x0) ≥ Z s0+h s0 L(ξ0(s), u0(s)) ds + V (s0 + h, ξ0(s0 + h)) − αh (66) 16