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A global minimum search algorithm for estimating the distance to uncontrollability. (English) Zbl 0778.65049
For the differential system $$\dot x=Ax+Bu$$, the measure $$\mu(A,B)$$ of the nearness of a controllable pair to an uncontrollable one is given by $$(A,B)=\min\sigma_{\min}((A-\lambda I,B))$$ with respect to $$\lambda$$, where $$\sigma_{\min}$$ holds for the smallest singular value of a matrix. By using simple properties due to R. Byers [SIAM J. Sci. Stat. Comput. 9, No. 5, 875-881 (1988; Zbl 0658.65044)], the problem is converted into a minimization problem on a bounded region in the plane.
The algorithm is described, and then a bisection algorithm is given for estimating the distance to uncontrollability within an error $$\varepsilon$$. The error analysis of the approach is carefully expanded and a numerical example is provided. These algorithms are based upon partitions of regions.

##### MSC:
 65K10 Numerical optimization and variational techniques 93B05 Controllability
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