# 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)
Global exact controllability of a class of quasilinear hyperbolic systems. (English) Zbl 0915.93007
This paper treats a distributed parameter boundary control system described by a reducible quasilinear hyperbolic system $${\partial r\over\partial t}+\lambda(r,s) {\partial r\over\partial x}=0$$ $${\partial s \over\partial t}+ \mu(r,s){\partial s\over \partial x}=0, \quad x\in [0,1],$$ (where $\lambda(r,s) <0<\mu(r,s))$ with the nonlinear boundary conditions $$s=g (r,t)+h_1(t)\text{ at }x=0$$ $$r=f(s,t)+h_2(t)\text{ at }x=1.$$ It is shown that the system is globally exactly boundary controllable in the linear degenerate case.

##### MSC:
 93B05 Controllability 93C20 Control systems governed by PDE
Full Text:
##### References:
 [1] Chen, G.: Energy decay estimates and exact boundary controllability of the wave equations in a bounded domain. J. math. Pures appl. 58, 249-274 (1979) · Zbl 0414.35044 [2] Chewning, W. C.: Controllability of the nonlinear wave equation in several space variables. SIAM J. Control optim. 14, 19-25 (1976) · Zbl 0322.93009 [3] Cirinà, M.: Nonlinear hyperbolic problems with solutions on preassigned sets. Michigan math. J. 17, 193-209 (1970) · Zbl 0201.42702 [4] Cirinà, M.: Boundary controllability of nonlinear hyperbolic systems. SIAM J. Control 7, 198-212 (1969) · Zbl 0182.20203 [5] Fattorini, H. O.: Local controllability of nonlinear wave equation. Math. systems theory 9, 363-366 (1975) · Zbl 0319.93009 [6] Ho, L. F.: Observabilité frontière de l’équation des ondes. C. R. Acad. sci. Paris ser. I math. 302, 443-446 (1986) [7] Komornik, V.: Controlabilité exacte en un temps minimal. C. R. Acad. sci. Paris ser. I. math. 304, 223-225 (1987) [8] Lagnese, J.: Decay of solutions of wave equations in a bounded region with boundary dissipation. J. differential equations 50, 163-182 (1983) · Zbl 0536.35043 [9] Lasiecka, I.; Triggiani, R.: Uniform exponential energy decay of the wave equation in a bounded region withl2l2. J. differential equations 66, 340-390 (1987) · Zbl 0629.93047 [10] Lasiecka, I.; Triggiani, R.: Exact controllability for the wave equation with Neumann boundary control. Appl. math. Optim. 19, 243-290 (1989) · Zbl 0666.49012 [11] Lasiecka, I.; Triggiani, R.: Exact controllability of semilinear abstract systems with applications to wave and plates boundary control problems. Appl. math. Optim. 23, 109-154 (1991) · Zbl 0729.93023 [12] Lax, P. D.: Hyperbolic systems of conservation laws, II. Comm. pure appl. Math. 10, 537-556 (1957) · Zbl 0081.08803 [13] Li, T.; Yu, W.: Boundary value problems for quasilinear hyperbolic systems. Duke university mathematics series 5 (1985) · Zbl 0627.35001 [14] Li, T.: Global classical solutions for quasilinear hyperbolic systems. Research in applied mathematics 32 (1994) · Zbl 0841.35064 [15] Lions, J. L.: Exact controllability, stabilization and perturbations for distributed systems. SIAM rev. 30, 1-68 (1988) · Zbl 0644.49028 [16] Rozdestvenskii, B. L.; Janenko, N. N.: Systems of quasilinear equations and their applications to gas dynamics. Translations of mathematical monographs 55 (1983) [17] D. L. Russell, On boundary-value controllability of linear symmetric hyperbolic systems, Mathematical Theory of Control, Academic Press, New York, 312, 321 · Zbl 0214.39607 [18] Russell, D. L.: Boundary value control of the higher dimensional wave equation. SIAM J. Control 9, 29-42 (1971) · Zbl 0216.55501 [19] Russell, D. L.: Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory. J. math. Anal. appl. 40, 336-368 (1972) · Zbl 0244.93025 [20] Russell, D. L.: Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM rev. 20, 639-739 (1978) · Zbl 0397.93001 [21] Russell, D. L.; Zhang, B. -Y.: Exact controllability and stabilizability of the Korteweg--de Vries equation. Trans. amer. Math. soc. 348, 3643-3672 (1996) · Zbl 0862.93035 [22] Zuazua, E.: Stability and decay for a class of nonlinear hyperbolic problems. Asymptotic anal. 1, 161-185 (1988) · Zbl 0677.35069 [23] Zuazua, E.: Exact controllability for the semilinear wave equation. J. math. Pures appl. 69, 1-32 (1990) · Zbl 0638.49017 [24] Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed dampings. Comm. partial differential equations 15, 205-235 (1990) · Zbl 0716.35010