OBSERVING THE HEAT EQUATION ON A TORUS ALONG A DENSE GEODESIC

Several authors have considered observability problems for the heat equation and relatedpartial differential equations.A basic problem is to determine what kinds of sampling providesufficient information to uniquely determine the initial heat distribntion.We address the case wherethe temperature is measured while travelling along a curve.We consider the special case where the space is a flat torus(of arbitrary dimension)and thecurve is a geodesic.It is shown that,in this case,the observed temperature is sufficient informationto uniquely determine the initial heat distribution if and only if the geodesic is dense in the torus.In the case of a torus,Fourier analysis techniques can be used to write down the solution of theheat equation.This allows us to derive an explicit representation of the observed temperature interms of the initial distribution.We use this representation and some ideas from the theory ofalmost periodic functions to show that the Fourier coefficients of the initial distribution can berecovered from the observation.

OBSERVING THE HEAT EQUATION ON A TORUS ALONG A DENSE GEODESIC

Several authors have considered observability problems for the heat equation and related partial differential equations.A basic problem is to determine what kinds of sampling provide sufficient information to uniquely determine the initial heat distribntion.We address the case where the temperature is measured while travelling along a curve. We consider the special case where the space is a flat torus(of arbitrary dimension)and the curve is a geodesic.It is shown that,in this case,the observed temperature is sufficient information to uniquely determine the initial heat distribution if and only if the geodesic is dense in the torus. In the case of a torus,Fourier analysis techniques can be used to write down the solution of the heat equation.This allows us to derive an explicit representation of the observed temperature in terms of the initial distribution.We use this representation and some ideas from the theory of almost periodic functions to show that the Fourier coefficients of the initial distribution can be recovered from the observation.