Controllability of partial differential equations governed by multiplicative controls.

*(English)*Zbl 1210.93005
Lecture Notes in Mathematics 1995. Berlin: Springer (ISBN 978-3-642-12412-9/pbk; 978-3-642-12413-6/ebook). xv, 284 p. (2010).

Controllability is a key concept in the control of dynamical systems. In this book, the control of evolution processes governed by partial differential equations is studied. The goal of the control is to steer the system from the given initial state to the desirable target state by selecting a suitable control. If this is indeed possible, from the set of successful controls that achieve this goal a control can be chosen that optimizes a certain criterion, solving what is called an optimal control problem. In this monograph, multiplicative controls are studied that enter the system equation as coefficients. Therefore, the control to state map of the process is highly nonlinear. The controllability is studied in the context of linear and semilinear parabolic and hyperbolic equations. Particular attention is given to the Schrödinger equation and nonlinear swimming models. The book has the following structure:

Part I: Multiplicative controllability of parabolic equations

Part II: Multiplicative controllability of hyperbolic equations

Part III: Controllability for swimming phenomenon

Part IV: Multiplicative controllability properties of the Schrödinger equation

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.

Part I: Multiplicative controllability of parabolic equations

Part II: Multiplicative controllability of hyperbolic equations

Part III: Controllability for swimming phenomenon

Part IV: Multiplicative controllability properties of the Schrödinger equation

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.

Reviewer: Martin Gugat (Erlangen)

##### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93B05 | Controllability |

93C20 | Control/observation systems governed by partial differential equations |