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Quantum optimal control using the adjoint method

Alfio Borzì (2012)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

Control of quantum systems is central in a variety of present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures, including the emerging fields of quantum computation and quantum communication. In this paper, a review of recent developments in the field of optimal control of quantum systems is given with a focus on adjoint methods and their numerical implementation. In addition, the issues of exact controllability and optimal control are...

Quenching for semidiscretizations of a semilinear heat equation with Dirichlet and Neumann boundary conditions

Diabate Nabongo, Théodore K. Boni (2008)

Commentationes Mathematicae Universitatis Carolinae

This paper concerns the study of the numerical approximation for the following boundary value problem: u t ( x , t ) - u x x ( x , t ) = - u - p ( x , t ) , 0 < x < 1 , t > 0 , u x ( 0 , t ) = 0 , u ( 1 , t ) = 1 , t > 0 , u ( x , 0 ) = u 0 ( x ) > 0 , 0 x 1 , where p > 0 . We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.

Quenching time of some nonlinear wave equations

Firmin K. N’gohisse, Théodore K. Boni (2009)

Archivum Mathematicum

In this paper, we consider the following initial-boundary value problem u t t ( x , t ) = ε L u ( x , t ) + f ( u ( x , t ) ) in Ω × ( 0 , T ) , u ( x , t ) = 0 on Ω × ( 0 , T ) , u ( x , 0 ) = 0 in Ω , u t ( x , 0 ) = 0 in Ω , where Ω is a bounded domain in N with smooth boundary Ω , L is an elliptic operator, ε is a positive parameter, f ( s ) is a positive, increasing, convex function for s ( - , b ) , lim s b f ( s ) = and 0 b d s f ( s ) < with b = const > 0 . Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation α ' ' ( t ) = f ( α ( t ) ) , t > 0 , α ( 0 ) = 0 , α ' ( 0 ) = 0 , as ε goes to zero. We also show that the above result remains...

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