×

How violent are fast controls? II. (English) Zbl 0906.93007

[For Part I by the first author see ibid. 1, No. 1, 89-95 (1988; Zbl 0663.49018).] The authors consider a linear time-dependent control system \[ \dot x=Ax+Bu \] and the problem of minimizing the \(L^p\)-norm \(\| u\|_p\) subject to \[ x(0)=0, \qquad x(T)=\xi \] when \(T\rightarrow 0\). The main result is that the asymptotic part of the minimizing norm is \[ (1/T)^{k+1-1/p},\quad p\in [1,+\infty] \] where \(k=\min\{j: \xi\in \text{range} (B+AB+\cdots +A^jB\}\). Finally a representation formula is given for the minimizing control.

MSC:

93B03 Attainable sets, reachability
49N05 Linear optimal control problems
93C05 Linear systems in control theory

Citations:

Zbl 0663.49018
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. H. Clarke,Optimization and Nonsmooth Analysis, SIAM, Philadelphia, PA, 1990. · Zbl 0696.49002
[2] T. I. Seidman, How violent are fast controls?,Math. Control Signals Systems,1 (1988), 89–95. · Zbl 0663.49018 · doi:10.1007/BF02551238
[3] W. H. Wonham,Linear Multivariable Control: A Geometric Approach, Springer-Verlag, Berlin, 1985. · Zbl 0609.93001
[4] K. Yosida,Functional Analysis, Springer-Verlag, Berlin, 1980. · Zbl 0435.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.