A singular control approach to highly damped second-order abstract equations and applications

A singular control approach to highly damped second-order abstract equations and applications In this paper we restudy, by a radically different approach, the optimal quadratic cost problem for an abstract dynamics, which models a special class of second-order partial differential equations subject to high internal damping and acted upon by boundary control. A theory for this problem was recently derived in (LLP) and (T1) (see also (T2)) by a change of variable method and by a direct approach, respectively. Unlike (LLP) and (T1), the approach of the present paper is based on singular control theory, combined with regularity theory of the optimal pair from (T1). This way, not only do we rederive the basic control-theoretic results of (LLP) and (T1)—the (first) synthesis of the optimal pair, and the (first) nonstandard algebraic Riccati equation for the (unique) Riccati operator which enters into the gain operator of the synthesis—but in addition, this method also yields new results—a second form of the synthesis of the optimal pair, and a second (still nonstandard) algebraic Riccati equation for the (still unique) Riccati operator of the synthesis. These results, which show new pathologies in the solution of the problem, are new even in the finite-dimensional case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A singular control approach to highly damped second-order abstract equations and applications

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Publisher
Springer-Verlag
Copyright
Copyright © 1997 by Springer-Verlag New York Inc.
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/BF02683338
Publisher site
See Article on Publisher Site

Abstract

In this paper we restudy, by a radically different approach, the optimal quadratic cost problem for an abstract dynamics, which models a special class of second-order partial differential equations subject to high internal damping and acted upon by boundary control. A theory for this problem was recently derived in (LLP) and (T1) (see also (T2)) by a change of variable method and by a direct approach, respectively. Unlike (LLP) and (T1), the approach of the present paper is based on singular control theory, combined with regularity theory of the optimal pair from (T1). This way, not only do we rederive the basic control-theoretic results of (LLP) and (T1)—the (first) synthesis of the optimal pair, and the (first) nonstandard algebraic Riccati equation for the (unique) Riccati operator which enters into the gain operator of the synthesis—but in addition, this method also yields new results—a second form of the synthesis of the optimal pair, and a second (still nonstandard) algebraic Riccati equation for the (still unique) Riccati operator of the synthesis. These results, which show new pathologies in the solution of the problem, are new even in the finite-dimensional case.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jul 1, 1997

References

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