Control Theory for Partial Differential Equations: Volume 1, by Irena Lasiecka, Roberto Triggiani

Control Theory for Partial Differential Equations: Volume 1, by Irena Lasiecka, Roberto Triggiani

By Irena Lasiecka, Roberto Triggiani

This can be the 1st quantity of a accomplished and up to date therapy of quadratic optimum keep an eye on conception for partial differential equations over a finite or limitless time horizon, and similar differential (integral) and algebraic Riccati equations. The authors describe either non-stop concept and numerical approximation. They use an summary house, operator theoretic technique, in line with semigroups equipment and unifying throughout a couple of simple sessions of evolution. a few of the summary frameworks are encouraged by means of, and finally directed to, partial differential equations with boundary/point keep watch over.

Volume I comprises the summary parabolic thought (continuous idea and numerical approximation conception) for the finite and countless circumstances and corresponding PDE illustrations, and provides various new effects. those volumes will entice graduate scholars and researchers in natural and utilized arithmetic and theoretical engineering with an curiosity in optimum keep watch over difficulties.

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Extra resources for Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories (Encyclopedia of Mathematics and Its Applications Series, Book 75)

Example text

24). 26), are given explicitly in terms of the data of the problem. -s)x) ' s . 1 Use of the Liusternik's Lagrange multiplier theorem is potentially applicable to more general situations where the cost is not quadratic. 3. 2 L2"Estimatesfor {uO,yO} and ZrEstimatefor GyO(T; . ; x). 1 With reference to the optimal pair {uo( . , s; x), yO( . 3) with C T a generic constant independent of s, °:s s < T. 4), lIu D( • ,s; X)IIL,(s,T;U):S Ilu D( . 1) is proved. 1) established in (i). 4). 2 t T. 5), we have (i) lim Ilu D( stT •, s; x)lIv(s T"U) = 0, ' .

2; and Chapter 0, Eqn. 6), or Eqn. 9)). (vii) (Classical DRE) Let t be fixed and satisfy 0::: t < T. 12). 10) admits a unique continuous extension that is bounded on Y x Y. 1). 2). 3) The operator pet), given constructively by Eqn. 3). Second Case We now assume a stronger smoothing property on G in the sense that (-A*)YG*G E £(Y). 1) since 2y - 1 < y. 1). 16), additional regularity results hold true. 16). 3) of Corollary ( i) Ilu (ii) for any O:S D (. 18) ( iii) B* P( . 15)). 5). 7). 16) is satisfied.

1, Eqns. 9)). 1(v), (vi), Eqns. 1)for G below. I. 2, respectively): lim IIP(t)x - G*Gxlly = 0, lim IIG 2y - 1. 25) is a standard result [Kato, 1966, Theorems 5,28 and 5,29, p, 168].

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