Figure 2: Distance function V and another Lipschitz solution Vi Fixr∈ Q and r>0, and let|x-x<r. Choose y(2)∈丌(2), so that v(2 12-y*(a).Then V(x)≤|x-y*(2) ≤|x-2|+|z-y*(x) Since this holds for all laI-al <r we have (z)} To see that equality holds, simply take z=a. Thus establishes(22). Note that there are many minimizers z*for the RHS of(22), viz. segments of the lines joining z to points in 2.1. 4 Viscosity Solutions We turn now to the concept of viscosity solution for the HJ equation(1). The terminology comes from the vanishing viscosity method, which finds a solution V of (1)as a limit Vε→ V of solutions te AV(a)+ F(a, V(a), Vv(r))=0 The Laplacian term△v=是∑m1 azve can be used to model fluid viscosity.The definition below is quite independent of this limiting construction, and is closely related to dynamic programming; however, the definition applies also to equations that do not necessarily correspond to optimal controlV V_1 -1 1 Figure 2: Distance function V and another Lipschitz solution V1. Fix x ∈ Ω and r > 0, and let |x − z| < r. Choose y ∗ (z) ∈ π(z), so that V (z) = |z − y ∗ (z)|. Then V (x) ≤ |x − y ∗ (z)| ≤ |x − z| + |z − y ∗ (z)| = |x − z| + V (z). Since this holds for all |x − z| < r we have V (x) ≤ inf |x−z|<r {|x − z| + V (z)}. To see that equality holds, simply take z = x. Thus establishes (22). Note that there are many minimizers z ∗ for the RHS of (22), viz. segments of the lines joining x to points in π(x). 2.1.4 Viscosity Solutions We turn now to the concept of viscosity solution for the HJ equation (1). The terminology comes from the vanishing viscosity method, which finds a solution V of (1) as a limit V ε → V of solutions to − ε 2 ∆V ε (x) + F(x, V ε (x), ∇V ε (x)) = 0 (23) The Laplacian term ε 2∆V ε = ε 2 Pn i=1 ∂ 2 ∂x2 i V ε can be used to model fluid viscosity. The definition below is quite independent of this limiting construction, and is closely related to dynamic programming; however, the definition applies also to equations that do not necessarily correspond to optimal control. 10