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Pseudospectral components and the distance to uncontrollability. (English) Zbl 1078.93008
The authors consider the control system defined by $$x' = Ax + Bu$$ for $$A \in \mathbb{R}^{p \times p}$$, $$B \in \mathbb{R}^{p \times q}$$, $$x(t) \in \mathbb{R}^{p}$$, $$u(t) \in \mathbb{R}^{q}$$. This system is called controllable if, given initial and final states $$x(0)$$ and $$x(T)$$, there exists a control function $$u(t)$$ giving a trajectory $$x(t)$$ with the given endpoints. The main results are the following:
Theorem: Given any real $$\varepsilon$$ and matrices $$P, Q \in \mathbb{C}^{m \times n}$$, $$m \leq n$$, suppose there exists no complex $$z$$ for which the singular value $$\sigma_{\min}(P + zQ)$$ is both multiple and equals $$\varepsilon$$. Then the set $$\Lambda = \{ z \in \mathbb{C} : \sigma_{\min}(P + zQ) \leq \varepsilon \}$$ has no more than $$2m(4m-1)$$ components.
Theorem: For any matrices $$P, Q \in \mathbb{C}^{m \times n}$$, $$m \leq n$$, and any real $$\varepsilon$$, the set $$\{ z \in \mathbb{C} : \sigma_{\min}(P + zQ) \leq \varepsilon \}$$ has no more than $$2m(4m-1)$$ components.
In sections 5 and 6 the authors study algorithms for computing the distance to uncontrollability.

##### MSC:
 93B05 Controllability 15A18 Eigenvalues, singular values, and eigenvectors 65F15 Numerical computation of eigenvalues and eigenvectors of matrices
##### Software:
ARPACK; eigs; Eigtool
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