Controllability of Partial Differential Equations Governed by Multiplicative Controls

1. Front Matter

   Pages i-xv

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2. Introduction

   • Alexander Y. Khapalov

   Pages 1-12

3. Multiplicative Controllability of Parabolic Equations

   1. Front Matter

      Pages 14-14

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   2. Global Nonnegative Controllability of the 1-D Semilinear Parabolic Equation

      • Alexander Y. Khapalov

      Pages 15-31

   3. Multiplicative Controllability of the Semilinear Parabolic Equation: A Qualitative Approach

      • Alexander Y. Khapalov

      Pages 33-48

   4. The Case of the Reaction-Diffusion Term Satisfying Newton’s Law

      • Alexander Y. Khapalov

      Pages 49-65

   5. Classical Controllability for the Semilinear Parabolic Equations with Superlinear Terms

      • Alexander Y. Khapalov

      Pages 67-80

4. Multiplicative Controllability of Hyperbolic Equations

   1. Front Matter

      Pages 82-82

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   2. Controllability Properties of a Vibrating String with Variable Axial Load and Damping Gain

      • Alexander Y. Khapalov

      Pages 83-104

   3. Controllability Properties of a Vibrating String with Variable Axial Load Only

      • Alexander Y. Khapalov

      Pages 105-119

   4. Reachability of Nonnegative Equilibrium States for the Semilinear Vibrating String

      • Alexander Y. Khapalov

      Pages 121-145

   5. The 1-D Wave and Rod Equations Governed by Controls That Are Time-Dependent Only

      • Alexander Y. Khapalov

      Pages 147-156

5. Controllability for Swimming Phenomenon

   1. Front Matter

      Pages 158-158

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   2. Introduction

      • Alexander Y. Khapalov

      Pages 159-164

   3. A “Basic” 2-D Swimming Model

      • Alexander Y. Khapalov

      Pages 165-170

   4. The Well-Posedness of a 2-D Swimming Model

      • Alexander Y. Khapalov

      Pages 171-193

   5. Geometric Aspects of Controllability for a Swimming Phenomenon

      • Alexander Y. Khapalov

      Pages 195-217

   6. Local Controllability for a Swimming Model

      • Alexander Y. Khapalov

      Pages 219-236

   7. Global Controllability for a “Rowing” Swimming Model

      • Alexander Y. Khapalov

      Pages 237-262

6. Multiplicative Controllability Properties of the Schrödinger Equation

   1. Front Matter

      Pages 264-264

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The goal of this monograph is to address the issue of the global controllability of partial differential equations in the context of multiplicative (or bilinear) controls, which enter the model equations as coefficients. The mathematical models we examine include the linear and nonlinear parabolic and hyperbolic PDE's, the Schrödinger equation, and coupled hybrid nonlinear distributed parameter systems modeling the swimming phenomenon. The book offers a new, high-quality and intrinsically nonlinear methodology to approach the aforementioned highly nonlinear controllability problems.

• Mathematica
• differential equation
• hyperbolic equation
• modeling
• partial differential equation
• partial differential equations

From the reviews:

“This book offers a detailed study of controllability of infinite-dimensional control systems described by partial differential equations (PDEs), for which the control input appears in a multiplicative way in the differential equations. The book has features that are not often encountered simultaneously in the rest of the literature … . the book is interesting and will have an impact on the topic of the controllability of infinite-dimensional systems. Rigorous proofs are provided for all the results contained in the book.” (Iasson Karafyllis, Mathematical Reviews, Issue 2011 h)

“In this book, the control of evolution processes governed by partial differential equations is studied. … The book is well-written and a welcome addition to the bookshelf for mathematicians with interest in control theory and also researchers in control engineering.” (Martin Gugat, Zentralblatt MATH, Vol. 1210, 2011)

• , Department of Mathematics, Washington State University, Pullman, USA

  Alexander Y. Khapalov

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