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

  • Mauro Garavello
  • Pierpaolo Soravia
Original Paper


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.

2000 Mathematics Subject Classification:

35A05 35B37 49L25 35C05. 

Keywords and phrases:

Optimal control viscosity solutions discontinuous coefficients calculus of variations principles of optimality. 


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Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Dipartimento di Matematica Pura e ApplicataUniversitá di PadovaPadovaItaly

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