By B. Jakubczyk
The quantity comprises papers in keeping with lectures brought through the college "Per spectives up to the mark conception" held in Sielpia, Poland on September 19-24, 1988. the purpose of the college was once to provide the state of the art presentation of contemporary achievements as weH as views in such fields of keep watch over thought as optimum regulate and optimization, linear platforms, and nonlinear platforms. therefore, the quantity comprises survey papers including displays of a few fresh effects. The designated emphasis is wear: - nonlinear platforms (algebraic and geometric methods), - optimum keep watch over and optimization (general difficulties, disbursed parameter systems), - linear structures (linear-quadratic challenge, strong stabilization). an enormous function of the college (and for this reason of the quantity) was once its rather "international" personality because it introduced jointly top regulate theoriests from West and East. All jointly the varsity used to be attended by way of 108 contributors from 18 international locations. throughout the university 21 one-hour invited lectures have been brought. additionally, 5 half-an-hour talks got and 30 contributions have been offered in frames of poster periods. the college was once equipped and supported by way of: Institute of arithmetic of the Polish Academy of Sciences, Committee of computerized keep watch over and Robotics of the Polish Academy of Sciences, - Institute of automated regulate, Warsaw collage of expertise (as Co ordinator of the fundamental examine software R.P.I.02 "Theory of keep an eye on of constant Dynamic structures and Discrete procedures" ).
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Extra resources for Perspectives in Control Theory: Proceedings of the Sielpia Conference, Sielpia, Poland, September 19–24, 1988
Satisfying 0 ~ (or global) rank of T(s). F (P) ~ and 0 rank F(P) = rank T i f f t is output stabilizable. e. 4) min iii) If t is output stabilizable, r+min has a smallest element P . Furthermore, J' (x o) = x'P-x o 0 iv) If t is (state) stabilizable, r+min has a largest element P+. • + Furthermore, Jo(x o) = x~P xo' If imaginary axis, the infimum in J: 1: has no zeros on the is attained for some (x o ) input u that results in astate trajectory that converges to zero as t .. Solutions of 0). the rank F(P) = rank T, satisfying DI, the additional eondition are ealled rank-minimizing solutions of the DI.
0 0 The statements iii) and iv), which constitute the solution of the LQ-problem, are obvious consequences of ii). of based ii) is on completion of squares. The simplest proof The monotonicity statement in i) is an immediate consequence of iv). For the free-endpoint infinite-horizon problem, a needed for the existence of an optimal control. 7 THEOREM The following statements are equivalent: i) ii) For every Xo For every IRn, E there exists u such that S;lly~2dt < Oll. there exists u such that y(t) ..
4). the remarks in Sections 6 and 7. suitan bly generalized. apply in full: if f has "sufficiently many varia- tions" (a condition of controllability type) or if the target set Y is "sufficiently large". 5) un + fot fand obtain a theorem of Kuhn-Tucker type. f}(u). o 10. Conc1usions. The approach of Clarke and Ekeland to optimal control problems has proved very suitable for the treatment of infinite dimensional problems, It has produced very general vers ions of the maximum principle. as weIl as a number of related new results.
Perspectives in Control Theory: Proceedings of the Sielpia Conference, Sielpia, Poland, September 19–24, 1988 by B. Jakubczyk