By Irena Lasiecka, Roberto Triggiani
Volume II makes a speciality of the optimum regulate challenge over a finite time period for hyperbolic dynamical platforms. The chapters examine a few summary versions, every one prompted through a specific canonical hyperbolic dynamics, and current quite a few new effects.
Read Online or Download Control Theory for Partial Differential Equations: Volume 2, Abstract Hyperbolic-like Systems over a Finite Time Horizon: Continuous and Approximation Theories (Encyclopedia of Mathematics and Its Applications Series, Book 75) PDF
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Additional info for Control Theory for Partial Differential Equations: Volume 2, Abstract Hyperbolic-like Systems over a Finite Time Horizon: Continuous and Approximation Theories (Encyclopedia of Mathematics and Its Applications Series, Book 75)
Example text
Although the study of this fully pathological case is deferred to the technical treatment of Chapter 16 (in Volume 3), in the present Chapter 8 we consider two different subsettings on two parallel tracks. 2), respectively. Our primary emphasis is on the first case where an observation operator R is present, bounded from a state space Y to an output space Z, such that the composition R Lo is better behaved than each of its components viewed separately. 1, the operator RLo is, in fact, continuous between the input space L2(0, T; U) and the output space C([O, T]; Z).
V); (Bu, y)y = (u, B*y)u. 1) with feedback control u = -EB*x, where E > 0: x + Ax + EBB*i = 0; A. 8) V(B*) : [A! Yl - EA-! )}. 2 Main Regularity Results The usual dissipativity arguments leading to the Lumer-Phillips theorem [Pazy, 1983, p. 14] or its corollary [Pazy, 1983, p. 15] yield a generation result. 2). c. contraction semigroups eA,t and e A:t on Y, t ~ O. Proof. We limit ourselves to dissipativity. 2) V Ai . 4). 4) where, moreover, we now assume the following hypothesis. 3). With E > 0 fixed, eA,t is exponentially stable on Y: There exist constants M ~ 1 and 8 > 0 (depending on E), such that Il eA,tll £(Y) = Ile(A-fBB*)tll £(Y) -< Me- 8t , t ~ O.
2. 1.