1996 IEEE International Conference on Systems, Man and by IEEE

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References 1. 14 , pp. 515-543, 1993. 2. : Augmented Lagmngian Techniques for Elliptic State Constrained Optimal Control Problems, to appear in SIAM Journal on Control and Optimization. 3. 'Informatique, Dunod, Paris, France, 1982. 4. : Augmented Lagrangian Methods for Nonsmooth Convex Optimization in HilbeTt Spaces, preprint, 1994. M. fr K. Kunisch Fachbereich Mathematik Technische Universitiit Berlin Str. de 49 A Priori Estimates for the Approximation of a Parabolic Boundary Control Problem ARIELA BRIANI AND MAURIZIO FALCONE Dipartimento di Matematica Universita di Pisa Dipartimento di Matematica Universita di Roma ABSTRACT.

If it does not exist, SYMMLQ generates an iterate using information obtained from the Lanczos tridiagonalization. See [14]. 1) show that Xj E Kj(K, TO) if and only if Xj = pj-l(K) TO, where Pj-l is a polynomial of degree less or equal to j - 1. 2) Here A(K) denotes the spectrum of K and II} denotes the set of all polynomials p of degree less or equal to j which satisfy p(O) = 1. g. [16]. / ~ is the condition number of K with ~=minAEA(K) 1'\1, >. =maxAEA(K) 1'\1, and U/2 J is the largest integer less or equal to j /2.

J IUnj - Unj- 1r 1 2 J=I + IU~-I - ulf- -Iu~ - ulf-) . 46 Augmented Lagrangian Algorithms for State Constrained Optimal Control Problems We finally get for kn 2: 1 ~ . ,,(ly~ - Y n + a U~ - ul r ) + )=0 kn + 1 kn + 1 -2--lqn+I - g12n + -2--lrn+1 - f 12r PI P2 1 2 PI ~ . 2 '-P2~· + 2c1An+I - plL + b - 2) LJ IAy~ - fin + -2- LJ Iy~ )=0 )=0 ·2' 2 + 21un - ul r u~lr k n + 11 qn - gin 2 + -2k n + 11 rn - f 12r :s; -2PI P2 1 1An , - J1-12+,-P21 + 2c L -2- Yn - Yn-I 12+'1 n 2 Un-I - U-1 r2 . Since P2 :s; , we deduce that if kn 2: 1 2 2 kn +l yin + alun - ul r + -2--lqn+I IYn - PI - 2 kn +l P2 2 gin + -2-Irn+I - fir P2 1 PI)IA Yn - fl 2n + -2,I/ An+ 1 - J1-1 2 '1 Un - U-1 2r + (, - 2 Yn - Un 12r + 2c n+ 2 2 2 k +1 1 k + 11 1 1 1An-Pn 12 :S;-2-lqn-gn+-2--rn-fr+-2 ~ ~ c n '-P2 +2 - 1Yn-I n 2+ 2 '1 Un-I - U-1 r2 .

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