A Viability Approach to Hamilton-Jacobi Equations: Application to Concave Highway Traffic Flux Functions

This paper presents a new approach which links the solution to a particular Hamilton-Jacobi partial differential equation to the solution of an optimal control problem provided by viability theory. It constructs the solution to this partial differential equation through its hypograph, which is defined as the capture basin of a target under an auxiliary dynamics that we define. The target itself represents the hypograph of a desired function. It is applied to concave Hamiltonian functions and has implications for the control of conservation laws with concave flux functions. It is a building block towards controlling conservation laws with concave flux functions, though at this stage, the link with boundary control of hyperbolic conservation laws cannot be made explicitly.