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Well-posedness and regularity for non-uniform Schrödinger and Euler-Bernoulli equations with boundary control and observation. (English) Zbl 1235.35191

Summary: The open-loop systems of a Schrödinger equation and an Euler-Bernoulli equation with variable coefficients and boundary controls and collocated observations are considered. It is shown, with the help of a multiplier method on a Riemannian manifold, that both systems are well-posed in the sense of D. Salamon [Trans. Am. Math. Soc. 300, 383–431 (1987; Zbl 0623.93040); Math. Syst. Theory 21, No. 3, 147–164 (1989; Zbl 0668.93018)] and regular in the sense of G. Weiss [Isr. J. Math. 65, No. 1, 17–43 (1989; Zbl 0696.47040); SIAM J. Control Optimization 27, No. 3, 527–545 (1989; Zbl 0685.93043)]. The feed-through operators are found to be zero. The results imply particularly that the exact controllability of each open-loop system is equivalent to the exponential stability of the associated closed-loop system under the output proportional feedback.

MSC:

35L35 Initial-boundary value problems for higher-order hyperbolic equations
35Q93 PDEs in connection with control and optimization
35Q40 PDEs in connection with quantum mechanics
35Q31 Euler equations
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
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