CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS

By Enrique Zuazua. 2006

121 citations recorded by the source corpus.

In these notes we analyze some problems related to the control- lability and observability of partial dierential equations and its space semi- discretizations. First we present the problems under consideration in the clas- sical examples of the wave and heat equations and recall some well known results. Then we analyze the 1 d wave equation with rapidly oscillating coef- ficients, a classical problem in the theory of homogenization. Then we discuss in detail the null and approximate controllability of the constant coecient heat equation using Carleman inequalities. We also show how a fixed point technique may be employed to obtain approximate controllability results for heat equations with globally Lipschitz nonlinearities. Finally we analyze the controllability of the space semi-discretizations of some classical PDE models: the Navier-Stokes equations and the 1 d wave and heat equations. We also present some open problems.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/c863c74ed46f1c9f8231b6eef4e12b80f0833736

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In Computational Mechanics

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Citations in the corpus

A weighted identity for partial differential operators of second order and its applications (2006, background): “The study of controllability/observability problem for partial differential equations (PDEs for short) began in the 1960s, for which various techniques have been developed in the last decades [9,10].”
Approximate Controllability of Semilinear Deterministic and Stochastic Evolution Equations in Abstract Spaces (2003, methodology): “On the other hand, for the case of the nonlinear term including ∇y, the approximate controllability was established only recently by Fernandez and Zuazua [16] and Zuazua [36].”
The Wave Equation: Control and Numerics (2012, background): “Of course, the same questions arise in the context of semilin ear wave equations, even with globally Lipschitz nonlinearities, a case that ha s been handled for instance in [113].”
An Introduction to the Controllability of Partial Differential Equations (2004, background): “The interested reader may learn more on this topic through the survey articles [70] and [72].”
Approximate Controllability of a Semilinear Heat Equation (2013, result): “has been studied by several authors, particularly by [8–10]; and in a general fashion in [11].”
Control theory: history, mathematical achievements and perspectives (2003, background): “For more details, the reader is referred to the books [21] and [27] and the survey papers [13], [54] and [55].”
Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods (2005, background): “Approximate controllability is equivalent to a uniqueness or unique-continuation property for the adjoint system (see [67, 113, 117]): If the solution u of (5.”
Controllability of partial differential equations (2006, methodology): “Zuazua and collaborators in a series of articles [5, 9, 16, 21, 22, 24, 30, 36, 37, 38, 39], for different discretization processes, on different examples.”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, background): “The interested reader may complement these notes with the survey articles [205], [211] for the controllability of PDE, [27] for the controllability and homogenization.”