Abstract
Let (x,t)→y (x,t),x∈[0, 1],t∈[0,T], be the solution of the diffusion equation in one spatial variable corresponding to zero initial conditions and boundary controlu∈L 2(0,T). Givenf∈L 2(0, 1), it is not possible, in general, to find a controlu such thaty(·,T)=f. We extend the space of controls in such a manner thatL 2(0,T) can be considered to be a subset of a new spaceS of control elements; this space contains elements which do provide a solution to the problem of moments associated with the problem of makingy(·,T)=f inL 2(0, 1). We show then that the action of the elements ofS can be approximated by that of control functions inL 2(0,T) in a suitable manner.
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References
Rubio, J. E., andWilson, D. A.,On a Problem of Moments of Control Theory, Journal of the Franklin Institute, Vol. 299, No. 4, 1975.
Treves, F.,Topological Vector Spaces, Distributions, and Kernels, Academic Press, New York, New York, 1967.
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Communicated by L. Cesari
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Rubio, J.E., Wilson, D.A. On the strong controllability of the diffusion equation. J Optim Theory Appl 23, 607–616 (1977). https://doi.org/10.1007/BF00933300
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DOI: https://doi.org/10.1007/BF00933300