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Multiple time-scale decomposition in cheap control problems - singular control. (English) Zbl 0563.93015
The paper presents a solution to the well-known problem of asymptotic characterization of the cheap control, i.e. the function that minimizes $$J=\int^{T}_{0}(y'y+\mu^ 2u'Ru)$$ subject to $$\dot x=Ax+Bu,$$ $$y=Cx$$ where $$\mu$$ is a small scalar parameter. Applying own earlier results [Int. J. Control 37, 1259-1286 (1983; Zbl 0525.93012)] the authors transform the problem into a multiparameter singular perturbation problem: minimize $$J=\int^{T}_{0}(x'\!_ fC'\!_ fC_ fx_ f+u'Ru)$$ subject to $$\dot x_ 0=A_ 0x_ 0+A_{0f}x_ f,$$ $$\epsilon$$ $$\dot x_ f=A_ fx_ f+A_{f0}x_ 0+B_ fu, (break?) =C_ fx$$ where the matrices $$C_ f$$, $$A_ f$$ etc. are determined in an appropriate way. Then, using singular perturbation techniques an asymptotic solution of the corresponding Riccati equation is derived. On that basis, the behaviour of the optimal feedback poles, state and control trajectory, and the optimal transfer function as $$\mu$$ $$\to 0$$ are investigated. Besides its mathematical value, the obtained result has an important practical meaning: it gives a method for decomposition of the cheap regulator problem into several low dimensional regulator problems.
Reviewer: A.Dontchev

##### MSC:
 93B17 Transformations 34E15 Singular perturbations for ordinary differential equations 49M27 Decomposition methods 93C05 Linear systems in control theory
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