where So( denotes the corresponding trajectory with $(so)=To, and combining this with(62) we have d(s0+,50(s0+b)-(s0,xo)1/0+ h L(50(s),to(s)ds≥-a.(67) However, integration of(65)implies 0(s0+h,50(so+b)-(0,20)、_1/+ L(50(s),uo(s)ds≤-2a which contradicts(67) since a>0. This proves the supersolution propert 2.3 Comparison and Uniqueness The most important features of the theory of viscosity solutions are the powerful com- parison and uniqueness theorems. Comparison theorems assert that inequalities holdin on the boundary and /or terminal time also hold in the entire domain. Uniqueness follows from this. Such results are important, since they guarantee unique characterization of viscosity solutions, and ensure that convergent approximations converge to the correc limit. In the context of optimal control problems, value functions are the unique viscosity solutions In this section we give a detailed proof of the comparison and uniqueness results for a class of Dirichlet problems, and apply this to equation(20) for the distance function. We Iso present without proof results for Cauchy problems of the type(7),(8) 2.3.1 Dirichlet Problem Here we follow [ 3, Chapter II] and consider the HJ equation (a)+H(, vv(a))=0 in 3, a special case of (1) To help get a feel for the ideas, suppose Vi, V2 E C(@)nc(@2)(i.e. are smooth) satisfy Vi()+H(a, Vvi())<0(subsolution) V2(a)+H(a, vv2(a))20(supersolution Vi<V2 on aQ(boundary condition) Let Io E Q be a maximum point of Vi-V2. Now if o E& (interior, not on boundary then VVi(o)=VV2(ao) and subtracting the first second line of(70) from the first giveswhere ξ0(·) denotes the corresponding trajectory with ξ(s0) = x0, and combining this with (62) we have −( φ(s0 + h, ξ0(s0 + h)) − φ(s0, x0)) h ) − 1 h Z s0+h s0 L(ξ0(s), u0(s)) ds ≥ −α. (67) However, integration of (65) implies −( φ(s0 + h, ξ0(s0 + h)) − φ(s0, x0)) h ) − 1 h Z s0+h s0 L(ξ0(s), u0(s)) ds ≤ −2α. (68) which contradicts (67) since α > 0. This proves the supersolution property. 2.3 Comparison and Uniqueness The most important features of the theory of viscosity solutions are the powerful com￾parison and uniqueness theorems. Comparison theorems assert that inequalities holding on the boundary and/or terminal time also hold in the entire domain. Uniqueness follows from this. Such results are important, since they guarantee unique characterization of viscosity solutions, and ensure that convergent approximations converge to the correct limit. In the context of optimal control problems, value functions are the unique viscosity solutions. In this section we give a detailed proof of the comparison and uniqueness results for a class of Dirichlet problems, and apply this to equation (20) for the distance function. We also present without proof results for Cauchy problems of the type (7), (8). 2.3.1 Dirichlet Problem Here we follow [3, Chapter II] and consider the HJ equation V (x) + H(x, ∇V (x)) = 0 in Ω, (69) a special case of (1). To help get a feel for the ideas, suppose V1, V2 ∈ C(Ω)∩C 1 (Ω) (i.e. are smooth) satisfy V1(x) + H(x, ∇V1(x)) ≤ 0 (subsolution) V2(x) + H(x, ∇V2(x)) ≥ 0 (supersolution) (70) in Ω and V1 ≤ V2 on ∂Ω (boundary condition). (71) Let x0 ∈ Ω be a maximum point of V1 − V2. Now if x0 ∈ Ω (interior, not on boundary) then ∇V1(x0) = ∇V2(x0) and subtracting the first second line of (70) from the first gives V1(x0) − V2(x0) ≤ 0 17