On initial value and terminal value problems for Hamilton–Jacobi equation

First order partial differential equations (PDE) are often the main tool to model problems in optimal control, differential games, image processing, physics, etc. Dependent upon the particular application, the boundary conditions are specified either at the initial time instant, leading to an initial value problem (IVP), or at the terminal time instant, leading to a terminal value problem (TVP). The IVP and TVP have in general different solutions. Thus introducing a new model in terms of a first order PDE one has to consider both possibilities of IVP and TVP, unless there is a direct physical indication. In this paper we also particularly answer the following question: how should the initial value at the initial surface and the terminal value at the terminal surface be coordinated in order to generate the same solution? One may expect that for a given initial value the consistent terminal value is the value of the IVP solution at the terminal surface. The second (time-varying) example in this paper shows that, generally, this is not true for non-smooth initial conditions. We discuss also the difference between the IVP and TVP formulations, the connection between the Hamiltonians arising in IVP and TVP.