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.