2.2.2 Optimal Control Value Function is a Viscosity Solution Dynamic programming. The dymamic programming principle states that for every t, ti v(t, r)=inf/L((s), u(s)ds+v(r, a(r) We now prove this(by a standard technique in optimal control FixrE[ t, ti], and u( Ut tr. Let a() denote the corresponding trajectory with initial state r(t)=T, and consider (r, r(r). Let e>0 and choose u1(E Ur,t1, with trajectory T1()on r, ti] with i(r)=a(r) be such that V(r,(r)2J(,r(r);u1()-E Define u2(s) u(s)t≤s<r u1(s) ≤t1 (48) with trajectory T2(), 2(t)=a. Now T2(s=r(s),sEt, rl, and T2(s)=T1(s),sE[r, til V(r,x)≤J(t, f L((s), u(s))ds+v((ti) C L((s),u(s))ds+L(1(s),u1(s))ds +v/(a(ti)) CL(a(s), u(s))ds+V(r,(r))+e using(47). Therefore V(t,x)≤inf L(r(s),u(s)ds+v(r, a(r))+e Since e>0 was arbitrary, we have V(t,x)≤inf L(a(s), u(s))ds+v(r, z(r)) u( Elt This proves one half of (6) For the second half of (6), let u(EUt, tr, and let a( be the corresponding trajectory with a(t)=.. Then (t,x;u()=hL(x(),(s)ds+v(r(1) ft L((s),u(s)ds+ L(r(s), u(s)ds +v(r(ti)) >L(e(s),u(s)ds+v(r, a(r) N J(t, r;u()) inf I L((s), u(s)ds+v(r, i(r)))2.2.2 The Optimal Control Value Function is a Viscosity Solution Dynamic programming. The dynamic programming principle states that for every r ∈ [t, t1], V (t, x) = inf u(·)∈Ut,r Z r t L(x(s), u(s)) ds + V (r, x(r)) . (6) We now prove this (by a standard technique in optimal control). Fix r ∈ [t, t1], and u(·) ∈ Ut,t1 . Let x(·) denote the corresponding trajectory with initial state x(t) = x, and consider (r, x(r)). Let ε > 0 and choose u1(·) ∈ Ur,t1 , with trajectory x1(·) on [r, t1] with x1(r) = x(r) be such that V (r, x(r)) ≥ J(r, x(r); u1(·)) − ε. (47) Define u2(s) =  u(s) t ≤ s < r u1(s) r ≤ s ≤ t1 (48) with trajectory x2(·), x2(t) = x. Now x2(s) = x(s), s ∈ [t, r], and x2(s) = x1(s), s ∈ [r, t1]. Next, V (r, x) ≤ J(t, x; u2(·)) = R t1 t L(x(s), u(s)) ds + ψ(x(t1)) = R t r L(x(s), u(s)) ds + R r t L(x1(s), u1(s)) ds + ψ(x(t1)) = R t r L(x(s), u(s)) ds + V (r, x(r)) + ε (49) using (47). Therefore V (t, x) ≤ inf u(·)∈Ut,r Z r t L(x(s), u(s)) ds + V (r, x(r)) + ε. (50) Since ε > 0 was arbitrary, we have V (t, x) ≤ inf u(·)∈Ut,r Z r t L(x(s), u(s)) ds + V (r, x(r)) . (51) This proves one half of (6). For the second half of (6), let u(·) ∈ Ut,t1 , and let x(·) be the corresponding trajectory with x(t) = x. Then J(t, x; u(·)) = R t1 t L(x(s), u(s)) ds + ψ(x(t1)) = R r t L(x(s), u(s)) ds + R t1 r L(x(s), u(s)) ds + ψ(x(t1)) ≥ R r t L(x(s), u(s)) ds + V (r, x(r)). (52) Now minimizing, we obtain V (t, x) = infu(·)∈Ut,t1 J(t, x; u(·)) ≥ infu(·)∈Ut,t1 { R r t L(x(s), u(s)) ds + V (r, x(r))} (53) 14