# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Dissipativity for Lur’e distributed parameter control systems. (English) Zbl 1242.93038
Summary: Dissipativity of Lur’e distributed parameter control systems has been addressed. Delay-dependent sufficient conditions for the dissipativity with respect to the infinite-dimensional version of energy supply rate $\left({Q}_{1},{S}_{1},{R}_{1}\right)$ characterized exclusively by an unbounded operator ${Q}_{1}$ are established in terms of Linear Operator Inequalities (LOIs). Finally, the heat equation illustrates our result.
##### MSC:
 93B35 Sensitivity (robustness) of control systems 93C25 Control systems in abstract spaces 93C20 Control systems governed by PDE 93C05 Linear control systems 93B28 Operator-theoretic methods in systems theory
##### References:
 [1] Willems, J. C.: Dissipative dynamical system-part 1: general theory, Archive for rational mechanics and analysis 45, 321-351 (1972) · Zbl 0252.93002 · doi:10.1007/BF00276493 [2] Zhang, H.: Reliable dissipative control for stochastic impulsive systems, Automatica 44, 1004-1010 (2008) [3] Zhao, J.; Hill, D. J.: Dissipativity theory for switched systems, IEEE transactions on automatic control 53, 941-953 (2008) [4] Fridman, E.; Orlov, Y.: Exponential stability of linear distributed parameter systems with time-varying delays, Automatica 45, 194-201 (2009) · Zbl 1154.93404 · doi:10.1016/j.automatica.2008.06.006 [5] Hill, D. J.; Moylan, P. J.: Dissipative dynamical systems: basic input–output and state properties, Journal of the franklin institute 309, 327-357 (1980) · Zbl 0451.93007 · doi:10.1016/0016-0032(80)90026-5 [6] Fridman, E.: New Lyapunov–Krasovskiĭ functionals for stability of linear retarded and neutral type systems, Systems and control letters 43, 309-319 (2001) · Zbl 0974.93028 · doi:10.1016/S0167-6911(01)00114-1 [7] Khalil, H. K.: Nonlinear systems, (1996)