## Infinite Dimensional Optimization and Control TheoryThis book concerns existence and necessary conditions, such as Potryagin's maximum principle, for optimal control problems described by ordinary and partial differential equations. The author obtains these necessary conditions from Kuhn-Tucker theorems for nonlinear programming problems in infinite dimensional spaces. The optimal control problems include control constraints, state constraints and target conditions. Fattorini studies evolution partial differential equations using semigroup theory, abstract differential equations in linear spaces, integral equations and interpolation theory. The author establishes existence of optimal controls for arbitrary control sets by means of a general theory of relaxed controls. Applications include nonlinear systems described by partial differential equations of hyperbolic and parabolic type and results on convergence of suboptimal controls. |

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### Contents

Calculus of Variations and Control Theory | 3 |

Hypersonic Flow | 17 |

Optimal Control Problems Without Target Conditions | 26 |

The Minimum Principle | 84 |

The Minimum Principle for General Optimal Control Problems | 122 |

Nose Shape Problem | 151 |

Differential Equations in Banach Spaces and Semigroup Theory | 169 |

Abstract Minimization Problems in Hilbert Spaces | 251 |

Interpolation and Domains of Fractional Powers | 385 |

Linear Control Systems | 426 |

Optimal Control Problems with State Constraints | 474 |

Optimal Control Problems with State Constraints | 509 |

Spaces of Relaxed Controls Topology and Measure Theory | 603 |

Relaxed Controls in Finite Dimensional Systems | 674 |

Relaxed Controls in Infinite Dimensional Systems | 709 |

773 | |

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### Common terms and phrases

additive adjoint apply approximate arbitrary assume assumptions ball Banach space belongs boundary condition bounded called closed compact complete condition consider consists constant constraint construct contains convergence convex Corollary corresponding countably defined definition denote dense depend derivative differentiable domain element ends equality equation equicontinuous equivalent estimate Example exists extended fact Finally finite dimensional fixed follows function given hand hence holds implies independent inequality infinite instance integral interval Lebesgue Lemma limit linear maximum means metric space minimizing minimum principle norm obtain obvious operator optimal control particular problem proof proved relaxed Remark require resp respect satisfies satisfies Hypothesis semigroup separable sequence side solution strongly continuous strongly measurable subsequence subset target condition term Theorem theory topology trajectories uniformly valued variational vector yo(t zero

### Popular passages

Page 793 - Regularity and stability for the mathematical programming problem in Banach spaces.

Page 779 - The time-optimal problem for boundary control of the heat equation. Calculus of Variations and Control Theory (DL Russell, Ed.) Academic Press (1976) 305-320.

Page 779 - Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems, Appl. Math.

Page 773 - Existence of optimal controls for a class of systems governed by differential inclusions on a Banach space.

Page 773 - Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints, /. Optimization Theory & Appl.

Page 773 - Convergence of finite element approximations to state constrained convex parabolic boundary control problems, SIAM J.