Semicontinuous solutions for Hamilton-Jacobi equations and the \(L^ \infty\)-control problem
The authors address to an \(L^\infty\) optimal control problem with end-point constraints (a fixed final point). In this case the value function \(V\) includes an end-point cost \(g(\xi(T))\) which is an extended real-valued lower semicontinuous function. This is a non-trivial extension of the previously considered cases because it requires an extension of the theory of lsc viscosity solutions to the corresponding Bellman equations. The problems arising in this direction are essentially two, the discontinuity of the Hamiltonian and some bad behaviour of the modulus of semicontinuity of the lower and upper envelopes of the Hamiltonian. This last is the most serious one because it makes impossible an extension of the proof of the uniqueness of the solution. The authors solve both of these problems, they give detailed proofs of all the required steps, explaining how the theory of viscosity solution together with the tools of nonsmooth analysis can be used to solve first-order problems. The paper is well presented and gives a clear and complete description of the state of the literature in the field.
scientific article; zbMATH DE number 938679
scientific article