## Control Theory for Partial Differential Equations: Volume 2, Abstract Hyperbolic-like Systems Over a Finite Time Horizon: Continuous and Approximation TheoriesOriginally published in 2000, this is the second volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which unifies across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 2 is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems. A few abstract models are considered, each motivated by a particular canonical hyperbolic dynamics. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems. |

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### Contents

Preface page | xv |

3A Interpolation Intermediate Sobolev Spaces and Their | xvi |

Illustrations of the Numerical Theory of Chapter 4 | xix |

Some Auxiliary Results on Abstract Equations | 645 |

Glossary of Symbols for Chapter l | 671 |

Optimal Quadratic Cost Problem Over a Preassigned Finite Time | 673 |

l9 Motivation Statement of Main Results 608 | 679 |

2A Bounded Inversion of + SV 5 V 0 | 761 |

Notes on Chapter 3 | 913 |

l0 Differential Riccati Equations under Slightly Smoothing | 919 |

Statement of the Main Results | 926 |

Proof of Theorem l0 2 3 | 936 |

SecondOrder Hyperbolic Equations with Dirichlet | 942 |

Kirchoff Equation with One Boundary Control | 989 |

EulerBernoulli Equation with One Boundary | 1019 |

Notes on Chapter l0 l059 | 1059 |

Optimal Quadratic Cost Problem over a Preassigned Finite Time | 764 |

3B Damped Elastic Operators 285 | 884 |

Simplified Hinged BC 204 | 885 |

### Common terms and phrases

adjoint applies Assume H.I assumptions H.I Banach space boundary control boundary regularity bounded operator Chapter 9 closed graph theorem compute continuous L2(0 Corollary corresponding defined desired differential Riccati equation Dirichlet boundary conditions dual duality equivalent estimate explicitly fortiori global hence Hilbert space holds true hypotheses identity integral Riccati equation interior regularity invoking Kirchhoff equation Lasiecka and Triggiani last step Lemma Lions Magenes mixed problem Moreover nonnegative norm observation operator obtain operator P(t optimal control problem optimal cost partial differential equations present chapter proof of Theorem Proposition proved recalling regularity properties regularity results regularity theory Remark right-hand side s.c. semigroup satisfies second-order hyperbolic equations Section 9.l self-adjoint operator smooth Sobolev space solution P(t term trace regularity trace theory uniformly unique solution Verification of Assumption verified wave equation y e T>(A yields

### References to this book

Inverse Boundary Spectral Problems Alexander Kachalov,Yaroslav Kurylev,Matti Lassas No preview available - 2001 |