Exact Controllability for Multidimensional Semilinear Hyperbolic Equations

Abstract

In this paper, we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients. For this purpose, we establish an observability estimate for the linear hyperbolic equation with an unbounded potential, in which the crucial observability constant is estimated explicitly by a function of the norm of the potential. Such an estimate is obtained by a combination of a pointwise estimate and a global Carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem.

A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

“…These references are all based upon the use of local or global Carleman estimates. Related to this, there are also general pointwise Carleman estimates that are also useful in similar inverse problems [16,17,28].…”

A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

“…It is easy to check that if b ij = γδ ij , (δ denotes the Kronecker symbol), setting d(x) = |x − x 0 | 2 for any x 0 ∈ R N \Ω, then eventually modifying d as in [11], we see that (1.11) is satisfied; in this case, 13) which is the usual portion of the boundary that arises in the framework of the multiplier method [17,21,34]. We also note that the constraints on the coefficients b ij are almost necessary in order to establish the Carleman estimates needed in the development of our proof method; without these constraints, establishing those estimates would in most cases be impossible as shown in [24].…”

“…The proof of Theorem 1.2 is based on the following Carleman estimate due to Duyckaerts, Zhang and Zuazua [10], Theorem 2.4 (see also [11], Th. 7.1, for the special case V ≡ 0):…”

“…Remark 2.2. The constructive method that we have developed for the proof of Theorem 1.2 critically relies on the Carleman estimate in [10,11] (see Lem. 2.1 above).…”

Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations

“…We also refer to Zuazua [40] and to [13] for exponential stabilization of linearly locally damped semilinear wave equations and to Coron [10] for results on nonlinear systems (and the references therein).…”