By Rafal Ablamowicz, P. Lounesto

This quantity is devoted to the reminiscence of Albert Crumeyrolle, who died on June 17, 1992. In organizing the quantity we gave precedence to: articles summarizing Crumeyrolle's personal paintings in differential geometry, basic relativity and spinors, articles which provide the reader an idea of the intensity and breadth of Crumeyrolle's learn pursuits and effect within the box, articles of excessive medical caliber which might be of normal curiosity. In all of the components to which Crumeyrolle made major contribution - Clifford and external algebras, Weyl and natural spinors, spin buildings on manifolds, precept of triality, conformal geometry - there was monstrous development. Our desire is that the amount conveys the originality of Crumeyrolle's personal paintings, the ongoing power of the sphere he motivated, and the iconic recognize for, and tribute to, him and his accomplishments within the mathematical neighborhood. It isour excitement to thank Peter Morgan, Artibano Micali, Joseph Grifone, Marie Crumeyrolle and Kluwer educational Publishers for his or her assist in preparingthis quantity

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This quantity is devoted to the reminiscence of Albert Crumeyrolle, who died on June 17, 1992. In organizing the quantity we gave precedence to: articles summarizing Crumeyrolle's personal paintings in differential geometry, common relativity and spinors, articles which offer the reader an concept of the intensity and breadth of Crumeyrolle's learn pursuits and effect within the box, articles of excessive clinical caliber which might be of normal curiosity.

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**Extra info for Clifford algebras and spinor structures : a special volume dedicated to the memory of Albert Crumeyrolle (1919-1992)**

**Example text**

2]. The following two theorems concern optimizability of (LQ) 00 and optimizability with closed-loop stability. They are the main results of the paper. Theorem 3 Consider ~ = [A, B, C, 0] and let P(O; N, 0) be the solution at 0 for PN = 0 of the (RRE). Then the following are equivalent: (i) ~ is optimizable. (ii) The (ARE) has a symmetric PSD solution. (iii) ~ is optimizable by state feedback control i. e. Rn there holds V(x 0 , oo, uk = Kxk) < oo . (iv) ~ is output-stabilizable. Moreover under one of these conditions PI" := lim P(O; N, 0) exists, N--too (9) suchthat P~" is the minimal PSD solution of the (ARE) and P~" solves(LQr.

Hence, taking into account that (i) and (iii) are equivalent, we are reduced to show that E 2 is optimizable using state feedback if and only if (A 2 , B 2 ) is stabilizable. Now sufficiency is a standard result, so we must show necessity. e. Uk = K2x2k , K2 E mmxnz such that X2k = (A2 + B2K2)kx2o ) Yk = C2x2k and 00 L(IIYkll 2 k=O + JJukJJ 2) < CXJ • This implies for the unstable subspace of A 2 + B2 K 2 that and using Lemma 3 we get NO ([~:J, A2 + B2K2) c NO(C2, A2) = {0}. C~(A 2 + B 2K 2) = {0} and (A 2, B 2) is stabilizable.

ND(C,A). mnxn given by {6) , (15) (16) Proof: Let KIJ. be given by (7) for P = PIJ. (i) By Theorem 3, PIJ. =A+BKIJ. xkW). k=O Thus (17) 54 Hence, by Lemma 3, Ker (Pil) C NO(C, A) . The converse holds because if x 0 E NO(C, A) then with zero control, 0 = V(x 0 , oo, 0) = V 0 (x 0 , oo) = x 0Pilx 0 . (ii) Using Lemma 1 with uk = Kpxk , xk = A~x 0 ( where Ap = A + BKp ), we get 00 ,L(JJCxkl/ 2 + 1/KPxkl/ 2 ) ~ x~Pxa < oo. k==O This and Lemma 3 give (18) suchthat on L~(Ap), A~ = Ak for k 2: 0, with Ap unstable.