A cosine operator approach to modelingL 2(0,T; L 2 (?))?Boundary input hyperbolic equations

Lasiecka, Irena; Triggiani, Roberto

doi:10.1007/bf01442108

Cited by 127 publications

(70 citation statements)

References 22 publications

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“…We shall use the cosine operator theory, as presented in [4,14] and used in [2,7]. We shall use the same notations as in [2].…”

Section: Cosine Operators and The Solutions Of The Gurtin-pipkin Equamentioning

confidence: 99%

Controllability of the Gurtin-Pipkin equation

Pandolfi¹

2006

Proceedings of Control Systems: Theory, Numerics and Applications — PoS(CSTNA2005)

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The Gurtin-Pipkin equations models the evolution of thermal phenomena when the matherial keeps memory of the past. It is well known that the Gurtin-Pipkin equation has a hyperbolic type behavior so that it is natural to expect a property of exact observability for its solutions. In this talk we consider the Gurtin-Pipkin equation with control in the Diriclet boundary condition and we prove exact controllability as a consequence of the known exact controllability of the wave equation.

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“…We shall use the cosine operator theory, as presented in [4,14] and used in [2,7]. We shall use the same notations as in [2].…”

Section: Cosine Operators and The Solutions Of The Gurtin-pipkin Equamentioning

confidence: 99%

Controllability of the Gurtin-Pipkin equation

Pandolfi¹

2006

Proceedings of Control Systems: Theory, Numerics and Applications — PoS(CSTNA2005)

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“…The following regularity results are well known (for example, see [13,14]), and thus their proofs are omitted.…”

Section: ) Let V(t) = U(t) − W(t) Where W(t) = Xh(t)mentioning

confidence: 99%

Global existence and nonexistence for nonlinear wave equations with damping and source terms

Rammaha

Strei

2002

Trans. Amer. Math. Soc.

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Abstract. We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term |u| m−1 ut and a source term of the form |u| p−1 u, with m, p > 1. We show that whenever m ≥ p, then local weak solutions are global. On the other hand, we prove that whenever p > m and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.

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“…Following well-known ideas (see for instance Lasiecka and Triggiani [18], [19]) the system (4.1) can be written in the abstract form (2.2), provided we use the notation …”

Section: Applications To the Equation Of A Vibrating Stringmentioning

confidence: 99%

“…We denote by T the semigroup generated by A. Well-known computations, using the above expressions for A * and B * (see again [18], [19]) give that…”

Section: Applications To the Equation Of A Vibrating Stringmentioning

confidence: 99%

Simultaneous Exact Controllability and Some Applications

Tucsnak¹

2000

SIAM J. Control Optim.

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Abstract. We study the exact controllability of two systems by means of a common finitedimensional input function, a property called simultaneous exact controllability. Most of the time we consider one system to be infinite-dimensional and the other finite-dimensional. In this case we show that if both systems are exactly controllable in time T 0 and the generators have no common eigenvalues, then they are simultaneously exactly controllable in any time T > T 0 . Moreover, we show that similar results hold for approximate controllability. For exactly controllable systems we characterize the reachable subspaces corresponding to input functions of class H 1 and H 2 . We apply our results to prove the exact controllability of a coupled system composed of a string with a mass at one end. Finally, we consider an example of two infinite-dimensional systems: we characterize the simultaneously reachable subspace for two strings controlled from a common end. The result is obtained using a recent generalization of a classical inequality of Ingham.

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A cosine operator approach to modelingL 2(0,T; L 2 (?))?Boundary input hyperbolic equations

Cited by 127 publications

References 22 publications

Controllability of the Gurtin-Pipkin equation

Controllability of the Gurtin-Pipkin equation

Global existence and nonexistence for nonlinear wave equations with damping and source terms

Simultaneous Exact Controllability and Some Applications

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