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A class of filter problems, I: Existence theory

Abstract

A class of filter problems in which the autocorrelation function of the noise is assumed to be expressible as a convolution is related to a variational problem. The variational problem involves the minimization of the square of theL 1[0, ∞] norm of a function y plus the square of theL 2[0, ∞] norm of a functionG. The functions y andG are related by a renewal equation. The problems formulated here include as special cases previous problems solved by the authors and their students. In the present paper, the existence and uniqueness of the solution of the variational problems are established.

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References

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Additional information

Dedicated to R. Bellman

The research of the first author was supported by NSF Grants Nos. MCS-78-01106 and MCS-75-67947.

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Berkovitz, L.D., Pollard, H. A class of filter problems, I: Existence theory. J Optim Theory Appl 32, 521–536 (1980). https://doi.org/10.1007/BF00934038

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Key Words

  • Optimal filters
  • variational problems
  • renewal equations
  • existence and uniqueness