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**Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1: Contrôlabilité exacte. (Exact controllability, perturbations and stabilization of distributed systems. Vol. 1: Exact controllability).**
*(French)*
Zbl 0653.93002

Recherches en Mathématiques Appliquées, 8. Paris etc.: Masson. x, 538 p. (1988).

This book represents the first volume of an advanced monograph series on the subject of exact controllability, its dependence on perturbations and its applications to the exponential stabilization of distributed parameter systems. This first volume addresses the question of exact controllability for a large class of linear partial differential equations with various boundary conditions and control action. Exact controllability (in time T) is the property of being able to find an admissible control which will steer between two arbitrary points in the state-space (in time T). For distributed parameter systems this is a strong property which is not generically satisfied; e.g. parabolic equations or damped beam equations will usually not be exactly controllable. Moreover, there is typically a threshold time, T, below which exact controllability is not possible. In fact the property considered here is the weaker concept of exact null controllability or the property of being able to steer to the origin. However, for the examples considered in the book the two properties do coincide and so this is not such an issue. The issue at hand is that it is difficult to establish exact controllability for a given system and the property depends crucially on the type of control action (distributed or on the boundary) the boundary conditions (Neumann, Dirichlet) and on the topologies one imposes on the state and the input space. Many papers have been devoted to studying such questions for a particular example of distributed systems with specific boundary conditions and specific control action. The great contribution of this volume is that it presents a unified approach to establishing exact null controllability for a very general class of partial differential equations with a general class of boundary conditions and input control. The method is applicable to wave equations, beam equations and various plate equations with Dirichlet, Neumann or mixed boundary conditions and both distributed and boundary control.

This new Hilbert Uniqueness Method is motivated in the introduction via an optimal control problem with a penalty term. Particular solutions of this problem are provided by controls steering the initial state to the origin if the system is exactly null controllable. Associated with the control problem is a well-posed dual system which is inhomogeneous and reversed in time and which has a unique solution. This uniqueness suggests a choice for an inner product and the uniqueness ensures that it induces a norm and a Hilbert space. This Hilbert space is then designated as the new state-space and the exact null controllability problem is solved with respect to this topology. The generality and strength of the method is demonstrated by an extremely detailed analysis of several specific examples. It is important to note that infinitely many Hilbert spaces may be used as the new state-space. In addition there is a wealth of information on previous approaches to this problem and many still open problems are posed.

The second volume addresses the question of how exact controllability depends upon various perturbations (engineers would call this the robustness issue) and, in particular:

(i) Does the threshold time for exact controllability depend on perturbations?

(ii) Does the optimal control for the penalized control problem depend continuously on the perturbations?

Of course answers to these questions depend crucially on the nature of the perturbations and both stochastic perturbations and deterministic uncertain perturbations to the boundary, the control action and to the partial differential equation are considered. In addition a modification of the Hilbert Uniqueness Method is developed which is appropriate for parabolic systems. Again there is a wealth of references to previous work, on going research and ideas for open problems.

This new Hilbert Uniqueness Method is motivated in the introduction via an optimal control problem with a penalty term. Particular solutions of this problem are provided by controls steering the initial state to the origin if the system is exactly null controllable. Associated with the control problem is a well-posed dual system which is inhomogeneous and reversed in time and which has a unique solution. This uniqueness suggests a choice for an inner product and the uniqueness ensures that it induces a norm and a Hilbert space. This Hilbert space is then designated as the new state-space and the exact null controllability problem is solved with respect to this topology. The generality and strength of the method is demonstrated by an extremely detailed analysis of several specific examples. It is important to note that infinitely many Hilbert spaces may be used as the new state-space. In addition there is a wealth of information on previous approaches to this problem and many still open problems are posed.

The second volume addresses the question of how exact controllability depends upon various perturbations (engineers would call this the robustness issue) and, in particular:

(i) Does the threshold time for exact controllability depend on perturbations?

(ii) Does the optimal control for the penalized control problem depend continuously on the perturbations?

Of course answers to these questions depend crucially on the nature of the perturbations and both stochastic perturbations and deterministic uncertain perturbations to the boundary, the control action and to the partial differential equation are considered. In addition a modification of the Hilbert Uniqueness Method is developed which is appropriate for parabolic systems. Again there is a wealth of references to previous work, on going research and ideas for open problems.

Reviewer: R.Curtain

### 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 |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49K20 | Optimality conditions for problems involving partial differential equations |

49K40 | Sensitivity, stability, well-posedness |

93B03 | Attainable sets, reachability |

93B35 | Sensitivity (robustness) |

93D15 | Stabilization of systems by feedback |

35B37 | PDE in connection with control problems (MSC2000) |

### Keywords:

exact controllability; exponential stabilization; distributed parameter systems; threshold time; exact null controllability; unified approach; distributed and boundary control; Hilbert Uniqueness Method; perturbations; robustness
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\textit{J. L. Lions}, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1: Contrôlabilité exacte. (Exact controllability, perturbations and stabilization of distributed systems. Vol. 1: Exact controllability). Paris etc.: Masson (1988; Zbl 0653.93002)