## 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 |

7 | 76 |

The Minimum Principle | 84 |

3 | 92 |

5 | 101 |

8 | 107 |

Abstract Minimization Problems in Hilbert Spaces | 251 |

Abstract Minimization Problems in Banach Spaces | 310 |

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 |

The Minimum Principle for General Optimal Control Problems | 122 |

9 | 151 |

Differential Equations in Banach Spaces and Semigroup Theory | 169 |

1 | 189 |

Relaxed Controls in Infinite Dimensional Systems | 709 |

773 | |

795 | |

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### Common terms and phrases

adjoint apply approximate arbitrary assume assumptions Ay(t Banach space bounded Cauchy problem characteristic function closed compact continuous functions control set control theory convex Corollary cost functional countably defined definition denote dense differential equations Dirichlet boundary condition domain dominated convergence theorem E-valued E-weakly element ends the proof equicontinuous equivalent Erba Erca Example exists fact fixed fo(s fo(t follows Hilbert space implies inequality infinite dimensional integral equation interval Lebesgue Lebesgue point Lemma linear functional lower semicontinuous maximum measurable function metric space minimizing sequence minimum principle nonlinear norm obtain obvious optimal control optimal control problem optimal problem proof of Theorem relaxed controls resp satisfies Hypothesis semigroup solution operator strongly continuous semigroup strongly measurable subset subspace target condition topology trajectories uniformly unit ball variational equation vector y₁ yo(t zero

### Popular passages

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

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