Lax–Hopf formula and Max-Plus properties of solutions to Hamilton–Jacobi equations

We state and prove a “Lax–Hopf formula” characterizing viable capture basins of targets investigated in viability theory and derive a “Max-Plus” morphism of capture basins with respect to the target. Capture basins are used to define “viability solutions” to Hamilton–Jacobi equations satisfying “trajectory conditions” (initial, boundary or Lagrangian conditions). The Max-Plus morphism property of Lax–Hopf formula implies the fact that the solution associated with inf-convolution of trajectory conditions is the inf-convolution of the solutions for each trajectory condition. For instance, Lipschitz regularization or decreasing envelopes of trajectory condition imply the Lipschitz regulation or decreasing envelopes of the solutions.