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By Jan Bontsema, Ruth F. Curtain (auth.), Diederich Hinrichsen, Bengt Mårtensson (eds.)

Invoking the equipment of keep watch over concept to regulate a real-world plant capacity, at the foundation of obtainable a priori wisdom of the plant, layout­ ing/selecting a possible controller reaching the keep watch over target. A priori wisdom in regards to the plant is in general represented as a mathemat­ ical version, given through actual legislation or dimension info. one of these version is unavoidably doubtful, as a result of dimension mistakes, simplified types of average legislation, ignored dynamics, and the will to get a version tractable for extra computations. keep watch over of doubtful platforms is for this reason an im­ portant factor, either virtually and philosophically. essentially, assorted ways exist: Adaptive regulate and powerful keep watch over. On 4 appealing summer season days in June 1989, researchers from 12 coun­ attempts amassed jointly in Bremen, West Germany, for a workshop dedicated to those matters. forty nine talks, invited and contributed, have been awarded. the current paintings collects 18 of those talks. (Program and unique abstracts from the workshop can be found as record 209, Institute for Dynamical System.) The papers signify a huge scope of present developments in strong and adaptive keep an eye on. themes lined contain: New instructions in adaptive keep watch over, balance research of doubtful platforms, strong and adaptive stabi­ lization, numerical equipment, and topological facets of sturdy keep watch over. The papers have been conscientiously refereed and we might prefer to thank the referees for his or her efforts, that have ended in mammoth improvements.

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1 COLONIUS AND KLIEMANN A strategy motivated by deterministic considerations Figure 6 shows the level curves of the Lyapunov exponents K,p(b) as described in Example 9. Again, from the explicit formulae in [9], Section 6, /( =112 1(=0 Figure 6. e. there is a unique damping value bp such that (timevarying) uncertainties of size Up = [-p, p] are exponentially damped in an optimal way for bp • Figure 7 shows the location of these values. Note that increased damping with b does not necessarily lead to faster damping of disturbances (overdamping) for fixed p.

Our first result in this section shows that for real perturbations the concepts of stability radius based on Bohl and on Lyapunov exponents coincide. 1). 3) and fixed U E Up define the (upper) Bohl exponent as (cp. 3) with ~u(s, s) = id. t. Up be defined by rB(A) = inf{p ~ OJ sup kB(U) ~ O}. uEUp Theorem 5. 3)p satisfies (H). e. rB(A) = rL(A). PROOF. The inequality rB(A)::; rL(A) follows from kB(U) ~ >"(xo,u) for all Xo 'I- 0, u E U. 4 in [8]. First observe that for all t ~ s and all u E U STABILITY RADII AND LYAPUNOV EXPONENTS where u(· + 8) is the shifted control function u(· Hence for all U 29 + 8)(T) = U(T + 8), T E IR.

ARNOLD, W. KLIEMANN, Large deviations of linear stochastic dif- ferential equations, in Stochastic Differential Systems, ed. H. Engelbert, W. Schmidt, Lecture Notes in Control and Information Sciences Vol. 96, Springer-Verlag, 1987, 117-151. [3] L. ARNOLD, W. KLIEMANN, E. OELJEKLAUS, Lyapunovexponents of linear stochastic systems, in [6], 85-125. [4] L. ARNOLD, E. OELJEKLAUS, E. PARDOUX, Almost sure and moment stability for linear Ito equations, in [6], 129-159. [5] L. ARNOLD, G. PAPANICOLAOU, V.

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