ContentsIntroduction § 1. Statement of the problem. Main results1.1. Exact boundary controllability of the Navier-Stokes equations1.2. Local exact distributed controllability of the Navier-Stokes equations1.3. Exact controllability of the Boussinesq system1.4. Exact local controllability and approximate controllability of the Boussinesq system1.5. Some applications § 2. Carleman estimates2.1. Preliminaries2.2. Carleman estimates for solutions of system (2.9), (2.10)2.3. Definitive estimates § 3. Solubility of the exact controllability problem for the linearized Boussinesq system3.1. Statement of the problem3.2. An auxiliary extremal problem3.3. Proof of the main result § 4. Local exact controllability of the Boussinesq system § 5. Approximate controllability of the Boussinesq system: reduction to a linear system of a special form5.1. The idea of the proof5.2. Approximate controllability of the Boussinesq system § 6. Exact controllability of a linear system6.1. Proof of Lemma 5.16.2. Proof of Theorem 5.1 Bibliography

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Exact controllability of the Navier-Stokes and Boussinesq equations

This paper is concerned with entire solutions of the Fisher-KPP equation with nonlocal dispersal, i.e., u(t) =J * u - u + f(u), which is a one-dimensional nonlocal version of the Fisher-KPP equation describing the spatial spread of a mutant in a given population and the dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modeled by a convolution operator. Here the entire solutions are defined in the whole space and for all time t is an element of R. A comparison principle is employed to establish the existence of entire solutions by combining two traveling wave solutions with different speeds and coming from both ends of the real axis and some spatially independent solutions. The main difficulty is that a lack of regularizing effect occurs. This is probably the first time the existence of entire solutions of reaction equations with nonlocal dispersal has been studied. (C) 2009 Elsevier Ltd. All rights reserved.

Entire solutions in the Fisher-KPP equation with nonlocal dispersal

We prove Carleman inequalities for a second order parabolic equation when the coefficients are not bounded and norms of right hand sides are taken in the Sobolev space L(0, T ;W− 2 (Ω)), ∈ [0, 1]. Our Carleman inequality yields the unique continuation for L-solutions. We further apply these inequalities to the global exact zero controllability of a semilinear parabolic equation whose semilinear term also contains derivatives of first order of solutions and is of sub-linear growth at the infinity. Communicated by H. Okamoto. Received July 2, 2001. Revised April 3, 2002. 2000 Mathematics Subject Classification(s): 35K10, 35K55, 35K60, 93B05, 93C10, 93C20.

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=39&iss=2&rank=2

Carleman Inequalities for Parabolic Equations in Sobolev Spaces of Negative Order and Exact Controllability for Semilinear Parabolic Equations

The design of an extended Luenberger observer is considered to solve the state observation problem for semilinear distributed-parameter systems. For this, a backstepping-based technique is proposed for the design of the output injection weights by making use of the (extended) linearization of the semilinear observer error system with respect to the observer state. Stability of both the linearized and the semilinear observer error dynamics is analyzed theoretically. Moreover, an efficient sample-and-hold implementation is considered to improve the computational efficiency of the observer design. Simulation examples are provided for a bistable semilinear partial differential equation and the simplified model of a bioreactor with Monod kinetics.

On the Extended Luenberger-Type Observer for Semilinear Distributed-Parameter Systems

Finite dimensional null controllability for the semilinear heat equation

We consider null boundary controllability for one-dimensional semilinear heat equations. We obtain null boundary controllability results for semilinear equations when the initial data is bounded continuous and sufficiently small. In this work we also prove a version of the nonlinear Cauchy-Kowalevski theorem.

Null boundary controllability for semilinear heat equations

We study in this paper the positive solution of the Cauchy problem for a fast diffusion equation with source. We derive a secondary critical exponent of the behavior of the initial value at infinity for the existence of global (in time) and nonglobal solutions of the Cauchy problem. Furthermore, the large time behaviors of those global solutions are also studied.

/pdf/on-a-fast-diffusion-equation-with-source-yb4oblznci.pdf

On a fast diffusion equation with source

Entire solutions for a discrete diffusive equation

We study the null boundary controllability for a one-dimensional fourth order parabolic equation. We show that if the initial data is continuous then the fourth order parabolic equation is controllable.

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Yung Jen Lin Guo

Papers

Null boundary controllability for semilinear heat equations

On a fast diffusion equation with source

Entire solutions for a discrete diffusive equation

Null boundary controllability for a fourth order parabolic equation

HJB equation for optimal control system with random impulses