Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost

Abstract

We study viscosity solutions of Hamilton-Jacobi equations that arise in optimal control problems with unbounded controls and discontinuous Lagrangian. In our assumptions, the comparison principle will not hold, in general. We prove optimality principles that extend the scope of the results of [23] under very general assumptions, allowing unbounded controls. In particular, our results apply to calculus of variations problems under Tonelli type coercivity conditions. Optimality principles can be applied to obtain necessary and sufficient conditions for uniqueness in boundary value problems, and to characterize minimal and maximal solutions when uniqueness fails. We give examples of applications of our results in this direction

Degenerate Eikonal equations with discontinuous refraction index

“…While existence of a discontinuous, possibly extended real valued solution satisfying a weaker Dirichlet-type boundary condition is not a problem, being provided by the optimal control approach, problem (0.1) is not expected to have a unique solution in general, at least without a suitable definition of solution and appropriate conditions on f . In our previous paper [22], see also [11], we studied this problem and found explicit formulas for the minimal and maximal nonnegative viscosity solution, proving also a uniqueness result when (a ij ) = I N ×N and f is piecewise constant. The case of nondegenerate matrix (a ij ) was also studied, with different ideas, by Camilli-Siconolfi [7] who adopt a more stringent notion of solution and allow f ∈ L ∞ .…”

A Bellman Approach for Regional Optimal Control Problems in $\mathbb{R}^N$

“…We start by recalling the pioneering work by Dupuis [20] who uses similar methods to construct a numerical method for a calculus of variation problem with discontinuous integrand. Problems with a discontinuous running cost were addressed by either Garavello and Soravia [25,26], or Camilli and Siconolfi [15] (even in an L ∞ -framework) and Soravia [36]. To the best of our knowledge, all the uniqueness results use a special structure of the discontinuities as in [18,19,27] or an hyperbolic approach as in [3,17].…”

Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients

“…However we can still use Corollary 2.8 to state that the problem has no continuous solution. For a more detailed discussion about existence of discontinuous solutions to boundary value problems, we refer the reader to the papers by Garavello and the author [13], [33], showing for instance that u above is the unique lower semicontinuous solution.…”