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Reachable states in boundary control of the heat equation are independent of time†

Published online by Cambridge University Press:  14 November 2011

H. O. Fattorini
H. O. Fattorini
Affiliation:
Departments of Mathematics and System Science, University of California, Los Angeles
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Synopsis

It is known that the class of all reachable states in boundary control of systems described by parabolic equations in one space dimension is independent of the time during which control is applied. This result is generalized here to systems governed by the heat equation in an arbitrary number of space variables.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 1-2 , 1978 , pp. 71 - 77
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Courant, R. and Hilbert, D.Methods of Mathematical Physics, I (New York: Interscience, 1953).Google Scholar
2Fattorini, H. O.Control in finite time of differential equations in Banach space. Comm. Pure Appl. Math. 19 (1966), 17–34.CrossRefGoogle Scholar
3. Fattorini, H. O.Boundary control of temperature distributions in a parallelepipedon. SIAM J. Control 13 (1975), 1–13.CrossRefGoogle Scholar
4Fattorini, H. O. The time-optimal control problem for boundary control of the heat equation. Proc. Sympos. Calculus of Variations and Control Theory, Univ. Wisconsin, 1975 (New York: Academic Press, 1976).Google Scholar
5Fattorini, H. O. and Russell, D. L.Exact controllability theorems for linear parabolic equations in one space variable. Arch. Rational Mech. Anal. 43 (1971), 272–292.CrossRefGoogle Scholar
6Fattorini, H. O. and Russell, D. L. Uniform bounds on biorthogonal functions for real exponen-tials with an application to control theory of parabolic equations. Quart. Appl. Math. (1974), 45–69.CrossRefGoogle Scholar
7Lions, J. L. and Magenes, E.Non-homogeneous Boundary Value Problems and Applications (New York: Springer, 1972).Google Scholar
8Lions, J. L.Optimal Control of Systems Governed by Partial Differential Equations (New York: Springer, 1971).CrossRefGoogle Scholar
9Russell, D. L.A unified boundary controllability theory for hyperbolic and parabolic differential equations. Studies in Appl. Math. 11 (1973), 189–211.CrossRefGoogle Scholar
10Russell, D. L. Exact boundary value controllability theorems for wave and heat processes in star-complemented regions. In Differential Games and Control Theory. Eds. Liu, Roxin and Steinberg, (New York: Dekker, 1974).Google Scholar
11Schwartz, L.Étude des sommes d'exponentielles, 2nd edn (Paris: Hermann, 1959).Google Scholar
12Seidman, T.Observation and prediction for the heat equation. IV: Patch Observability and Controllability. SIAM J. Control 15 (1977), 412–427.CrossRefGoogle Scholar
13Seidman, T. Time invariance of the reachable set for linear control problems. Proc. 2nd Symp. IFAC 1977, to appear.CrossRefGoogle Scholar
14Washburn, D. A semigroup theoretical treatment of boundary control problems (Univ. Calif. Los Angeles, Doctoral dissertation, 1974).Google Scholar