Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data

Abstract

We consider a class of optimal control problems in which the cost to minimize comprises both a final cost and an integral term, and the data can be discontinuous with respect to the time variable in the following sense: they are continuous w.r.t. t on a set of full measure and have everywhere left and right limits. For this class of Bolza problems, employing techniques coming from viability theory, we give characterizations of the value function in the class of lower semicontinuous functions as the unique generalized solution to the corresponding Hamilton-Jacobi equation in the class of lower semicontinuous functions: if the final cost term is extended valued, the generalized solution to the Hamilton-Jacobi equation involve the concepts of lower Dini derivative and the proximal normal vectors; if the final cost term is a locally bounded lower semicontinuous function, then we can show that this has an equivalent characterization in a viscosity sense.

Reduction of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations

“…In papers [3,7,9,11,18,19,24,25] one can find similar results to Theorem 1.3. However, these results usually require that the solution is continuous and bounded and that the Hamiltonian satisfies some kind of uniform continuity conditions with constant functions c(•) and k R (•).…”

Reduction of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations

“…In papers [3,7,9,11,18,19,24,25] one can find similar results to Theorem 1.3. However, these results usually require that the solution is continuous and bounded and that the Hamiltonian satisfies some kind of uniform continuity conditions with constant functions c(•) and k R (•).…”

Path-dependent Hamilton-Jacobi equations with super-quadratic growth in the gradient and the vanishing viscosity method

“…In the context of contingent or Dini solutions for first-order standard PDEs related to Bolza problems, [13] and also [35] are very close to our approach. More recent works in this direction are [33], [5] and [6]. Regarding the possible use of viscosity solution techniques, we refer the reader to the remarks on p. 1202 in [7].…”