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2010 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Controllability of Partial Differential Equations Governed by Multiplicative Controls

verfasst von: Alexander Y. Khapalov

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
In the mathematical models associated with controlled distributed parameter systems evolving in bounded domains two types of controls – boundary and internal locally distributed – are typically used. These controls enter the model as additive terms (having in mind that the boundary controls can be modeled by making use of suitable additive Dirac’s functions) and have localized support. The latter is either a part of the boundary or a set within the system’s space domain. Such control can, for example, be a source in a heat/mass-transfer process or a piezoceramic actuator placed on a beam. Publications in this area are so numerous that it is simply impossible to mention them all here – allow us just to refer the reader to our very limited bibliography below, associated mostly with the immediate content of this monograph and to the references therein.
Alexander Y. Khapalov

Multiplicative Controllability of Parabolic Equations

Frontmatter
Chapter 2. Global Nonnegative Controllability of the 1-D Semilinear Parabolic Equation
Abstract
In this chapter we study the global approximate controllability of the one dimensional semilinear convection-diffusion-reaction equation governed in a bounded domain via a coefficient in the reaction term. Even in the linear case, due to the maximum principle, such system is not globally or locally controllable in any reasonable linear space. It is also well known that for the superlinear terms admitting a power growth at infinity the global approximate controllability by traditional additive controls of localized support is out of question. However, we will show that a system like that can be steered in L2(0,1) from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static (x-dependent only) multiplicative controls, applied subsequently in time, while only one such control is needed in the linear case.
Alexander Y. Khapalov
Chapter 3. Multiplicative Controllability of the Semilinear Parabolic Equation: A Qualitative Approach
Abstract
In this chapter we establish the global non-negative approximate controllability property for a rather general semilinear heat equation with superlinear term, governed in a bounded domain Ω ⊂ Rn by a multiplicative control in the reaction term like vu(x, t), where v is the control. We show that any non-negative target state in L2(Ω) can approximately be reached from any non-negative, nonzero initial state by applying at most three static bilinear L∞(Ω)-controls subsequently in time. This result is further applied to discuss the controllability properties of the nonhomogeneous version of this problem with bilinear term like v(u(x, t).θ (x)), where θ is given. Our approach is based on an asymptotic technique allowing us to distinguish and make use of the pure diffusion and/or pure reaction parts of the dynamics of the system at hand, while suppressing the effect of a nonlinear term.
Alexander Y. Khapalov
Chapter 4. The Case of the Reaction-Diffusion Term Satisfying Newton’s Law
Abstract
We discuss both the approximate and exact null-controllability of the diffusion-reaction equation governed via a coefficient in the reaction term, modeled according to Newton’s Law. Both linear and semilinear versions of this term are considered.
Alexander Y. Khapalov
Chapter 5. Classical Controllability for the Semilinear Parabolic Equations with Superlinear Terms
Abstract
In this chapter we establish the global approximate controllability of the semilinear heat equation with superlinear term, governed in a bounded domain by a pair of controls: (I) the traditional internal either locally distributed or lumped control and (II) the lumped control entering the system as a time-dependent coefficient. The motivation for the latter is due to the well known lack of global controllability properties for this class of pde’s when they are steered solely by the former controls. Our approach involves an asymptotic technique allowing us to separate and combine the impacts generated by the above-mentioned two types of controls. In particular, the addition of multiplicative control allows us to reduce the use of the additive one to the local controllability technique only.
Alexander Y. Khapalov

Multiplicative Controllability of Hyperbolic Equations

Frontmatter
Chapter 6. Controllability Properties of a Vibrating String with Variable Axial Load and Damping Gain
Abstract
We show, in a constructive way, that the set of equilibrium states like (yd,0) of a vibrating string that can approximately be reached in H1 0 (0,1)× L2(0,1) by varying its axial load and the gain of damping is dense in the subspace H1 0 (0,1)×{0} of this space.
Alexander Y. Khapalov
Chapter 7. Controllability Properties of a Vibrating String with Variable Axial Load Only
Abstract
We show that the set of equilibrium-like states (yd,0) of a vibrating string which can approximately be reached in the energy space H1 0 (0,1)× L2(0,1) from almost any non-zero initial datum, namely, (y0,y1) ε (H2(0,1)∩H1 0(0,1))× H1(0,1), (y0,y1) ≠ (0,0) by varying its axial load only is dense in the subspace H1 0 (0,1)× {0} of this space. This result is based on a constructive argument and makes use of piecewise constant-in-time control functions (loads) only, which enter the model equation as coefficients.
Alexander Y. Khapalov
Chapter 8. Reachability of Nonnegative Equilibrium States for the Semilinear Vibrating String
Abstract
We show that the set of nonnegative equilibrium-like states, namely, like (yd,0) of the semilinear vibrating string that can be reached from any nonzero initial state (y0,y1) εH1 0 (0,1)×L2(0,1), by varying its axial load and the gain of damping, is dense in the “gnonnegative” part of the subspace L2(0,1)×{0} of L2(0,1)× H-1(0,1). Our main results deal with nonlinear terms which admit at most the linear growth at infinity in y and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.
Alexander Y. Khapalov
Chapter 9. The 1-D Wave and Rod Equations Governed by Controls That Are Time-Dependent Only
Abstract
We discuss the early results from [8] on global approximate reachability of the 1-D wave equation with Dirichlet boundary conditions and the rod equation with hinged ends in the case when the multiplicative control is time-dependent only. The methods of [8] make use of the inverse function theorem and involve dealing with the associated Riesz bases of exponential time-dependent functions under the additional assumption that all the modes in the initial datum are present.
Alexander Y. Khapalov

Controllability for Swimming Phenomenon

Frontmatter
Chapter 10. Introduction
Abstract
The swimming phenomenon has been a source of great interest and inspiration for many researchers for a long time, with formal publications traced as far back as to the works of G. Borelli in 1680–1681. It is impossible, to give a complete account of all findings in this area. However, we will attempt to classify some of them as it seems relevant to the motivation of the content of this chapter.
Alexander Y. Khapalov
Chapter 11. A “Basic” 2-D Swimming Model
Abstract
We introduce a “basic” mathematical model of a swimmer in a fluid, governed within a bounded 2-D domain by the nonstationary Stokes equation. Its body consists of finitely many subsequently connected small sets each of which can act upon any adjacent set in a rotation fashion with the purpose to generate its fishlike or rowingmotion. The shape of the object is maintained by respective elastic forces.
Alexander Y. Khapalov
Chapter 12. The Well-Posedness of a 2-D Swimming Model
Abstract
We introduce a “more complex” version of the swimming model from Chapter 11, namely, for which both the linear and angular momenta are conserved. Then we discuss the issues of existence, uniqueness and continuity for its solutions.
Alexander Y. Khapalov
Chapter 13. Geometric Aspects of Controllability for a Swimming Phenomenon
Abstract
We investigate how the geometric shape of the swimming object affects the forces acting upon it in a fluid, particularly, in the case when the object at hand consists of either small rectangles or discs.
Alexander Y. Khapalov
Chapter 14. Local Controllability for a Swimming Model
Abstract
We study the local controllability of the “basic” mathematical model (11.1)–(11.3) from Chapter 11 under an extra assumption on the geometric regularity of the sets Sr(zi)’s forming the swimming object at hand.
Alexander Y. Khapalov
Chapter 15. Global Controllability for a “Rowing” Swimming Model
Abstract
We consider a swimmer whose body consists of three small narrow rectangles connected by flexible links. Our goal is to study its swimming capabilities when it applies a rowing motion in a fluid governed in a bounded domain by the nonstationary Stokes equation. Our approach explores an idea that a body in a fluid will move in the direction of least resistance determined by its geometric shape. Respectively, we assume that the means by which we can affect the motion of swimmer are the change of the spatial orientation of the aforementioned rectangles and the direction and strength of rowing motion. The main results are derived in the framework of mathematical controllability theory for pde’s and are based on a constructive technique allowing one to calculate an incrementalmotion of swimmer.
Alexander Y. Khapalov

Multiplicative Controllability Properties of the Schrodinger Equation

Chapter 16. Multiplicative Controllability for the Schrödinger Equation
Abstract
In this chapter we discuss some recent results obtained for multiplicative controllability of the Schrödinger equation. In recent years a substantial progress has been made in investigating the controllability properties of the Schrödinger equation governed by multiplicative control. In this chapter we will discuss some of these results due to Beauchard [11, 12, 14], Beauchard and Coron [16], Beauchard and Mirrahimi [18], Chambrion, Mason, Sigalotti and Boscain [26], Nersesyan [124] and others. It should be noted that the Schrödinger equation that we study below has certain property which sets it apart from the other partial differential equations considered in this monograph. Namely, the L2-norms of its solutions are conserved,regardless of the value of real-valued multiplicative control applied. Therefore, all the results below deal with controllability properties on the unit L2-sphere S,
$$ S=\{ \varphi | \varphi \in L^{2}(\Omega, C), \int_{\Omega} | \varphi(x) |^{2} dx =1 \} \in L^{2}(\Omega, C), $$
where Ω is the system’s space domain. The Schrödinger equation with real-valued control is also complex-conjugate time-reversible in the sense that if control u(t), t ε [0,T] steers it from u0 to u1 at time t = T, then control u•(t) = u(T –t) steers this equation from ū1 to ū0.
Alexander Y. Khapalov
Backmatter
Metadaten
Titel
Controllability of Partial Differential Equations Governed by Multiplicative Controls
verfasst von
Alexander Y. Khapalov
Copyright-Jahr
2010
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-12413-6
Print ISBN
978-3-642-12412-9
DOI
https://doi.org/10.1007/978-3-642-12413-6

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