Positive semigroups and algebraic Riccati equations in Banach spaces

Positive semigroups and algebraic Riccati equations in Banach spaces We generalize Wonham’s theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to $$A^*P+PA-PBB^*P+C^*C=0$$ A ∗ P + P A - P B B ∗ P + C ∗ C = 0 when (A, B) is exponentially stabilizable and (C, A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton’s iteration for concave functions is then used to approximate the solution, the approximations converge in the strong operator topology, and the convergence is monotone. The linearized equations are the well-known Lyapunov equations of the form $$A^*P+PA=-Q$$ A ∗ P + P A = - Q , and semigroup stability criterion in terms of them is also generalized. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Positive semigroups and algebraic Riccati equations in Banach spaces

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Publisher
Springer International Publishing
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0371-3
Publisher site
See Article on Publisher Site

Abstract

We generalize Wonham’s theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to $$A^*P+PA-PBB^*P+C^*C=0$$ A ∗ P + P A - P B B ∗ P + C ∗ C = 0 when (A, B) is exponentially stabilizable and (C, A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton’s iteration for concave functions is then used to approximate the solution, the approximations converge in the strong operator topology, and the convergence is monotone. The linearized equations are the well-known Lyapunov equations of the form $$A^*P+PA=-Q$$ A ∗ P + P A = - Q , and semigroup stability criterion in terms of them is also generalized.

Journal

PositivitySpringer Journals

Published: Oct 5, 2015

References

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