1,492 research outputs found
Null controllability for the parabolic equation with a complex principal part
This paper is addressed to a study of the null controllability for the
semilinear parabolic equation with a complex principal part. For this purpose,
we establish a key weighted identity for partial differential operators
(\a+i\b)\pa_t+\sum\limits_{j,k=1}^n\pa_k(a^{jk}\pa_j) (with real functions
\a and \b), by which we develop a universal approach, based on global
Carleman estimate, to deduce not only the desired explicit observability
estimate for the linearized complex Ginzburg-Landau equation, but also all the
known controllability/observability results for the parabolic, hyperbolic,
Schr\"odinger and plate equations that are derived via Carleman estimates
An Internal Observability Estimate for Stochastic Hyperbolic Equations
This paper is addressed to establishing an internal observability estimate
for some linear stochastic hyperbolic equations. The key is to establish a new
global Carleman estimate for forward stochastic hyperbolic equations in the
-space. Different from the deterministic case, a delicate analysis of the
adaptedness for some stochastic processes is required in the stochastic
setting
Exact controllability for multidimensional semilinear hyperbolic equations
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
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