A remark on IVP and TVP non-smooth viscosity solutions to hamilton-jacobi equations

Arik Melikyan, Andrei Akhmetzhanov, Naira Hovakimyan

Research output: Contribution to journalConference articlepeer-review

Abstract

First-order PDEs arise in many applications in physics, control, image processing and other. Some boundary conditions for the PDE need to be formulated to close the problem. In time dependent problems very often the boundary conditions are specified at the initial time instant, and the problem is called the Initial Value Problem (IVP). Respectively, such conditions may be needed at the terminal time instant, in which case one has the Terminal Value Problem (TVP). For time invariant problems these two types of problems still exist, though the difference between them is not that explicit. For computational convenience one can transform from TVP to IVP or vice versa. In control problem a TVP for HJBI equation naturally arises, while in mechanics (physics) a IVP originally arises. Which type of the problems should be considered depends also on the fact that the sought for function (value function or action) in these problems is introduced as the function of the left or right endpoint of the cost function in integral form. As to first-order PDEs arising in image processing there is no straightforward indication what kind of problem must be considered. This paper discusses the difference between IVP and TVP solutions, the connection between Hamiltonians arising in formulation of IVP and TVP, several illustrative examples are demonstrated, one of them showing a smoothening phenomenon in the consequent solutions to IVP, TVP.

Original languageEnglish (US)
Pages (from-to)864-869
Number of pages6
JournalProceedings of the American Control Conference
Volume2
StatePublished - 2005
Externally publishedYes
Event2005 American Control Conference, ACC - Portland, OR, United States
Duration: Jun 8 2005Jun 10 2005

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

Fingerprint

Dive into the research topics of 'A remark on IVP and TVP non-smooth viscosity solutions to hamilton-jacobi equations'. Together they form a unique fingerprint.

Cite this