Read e-book online A Bound on the Real Stability Radius of Continuous-Time PDF

By Bobylev N. A., Bulatov V.

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Additional info for A Bound on the Real Stability Radius of Continuous-Time Linear Infinite-Dimensional Systems

Sample text

Are possible. ). 4. x0 and A. = 0 otherwise. Then J. = I, ... , all cx 1 So, only cx l ~ cxo or = cxl, then A(x o) £ A(x i ) If cx o ~ cxl, then xl € We generate an adjacent basic solution to xi, say x i + l v. 3. would be applicable here. 5. We must check whether A space has been completely decomposed by A(x i ) generated up to this point. Direct application of this criterion would lead us to nontrivial computational difficulties. 9. ) whenever we cannot find an unexplored adjacent basis to any x i 's generated up to this point, such that A(xi ) runtA ~ ~, then the decomposition is completed.

To explore all k € J by solving the problem (2-4-1). Now we have Notice that simultaneously we determine whether the constraint 'corresponding to k € J is effectively binding A(xi ). Thus we make only the transformations leading to nondominated solutions. Since the set Nex is connected, we can always make the tra- versa!. 4. We always choose k ~ J which would lead to an adjacent basis. 3. 5. , if the storage in (4) is empty, we stop. 6. Obvious. ~o- 2. Discussion of difficulties connected with the decomposition method.

O~ i=l l. e. , II (x ) n Ilk '" cj>. D. 2. = {jl,j2, ... ,jm} be the index set of basic columns and the index set of nonbasic columns with respect to J o be denoted by Jo = {jm+l,jm+2" ··,jn}· {AlA Let the k chosen. th Then i L y. O~ , J i=l]. J Ei. E > 0; J }. E o (2-3-7) column represent the pivot column, and Ypk is the pivot element The following simplified simplex tableau describes the situation: Basic Cols. th J .. kth ,,, , , -- ----1-------~Pj--------G --, ,, ,, ,, bO 0 -- ----0---- ----Ylj---------Y lk ---- Yl ,, ,0 Yp , , ,0 -- ----0---- ----Ymj---------Ymk---- Ym , ------0----: ---z.