## A Theory of Optimization and Optimal Control for Nonlinear Evolution and Singular Equations: Applications to Nonlinear Partial Differential EquationsThis research monograph offers a general theory which encompasses almost all known general theories in such a way that many practical applications can be obtained. It will be useful for mathematicians interested in the development of the abstract Control Theory with applications to Nonlinear PDE, as well as physicists, engineers, and economists looking for theoretical guidance in solving their optimal control problems; and graduate-level seminar courses in nonlinear applied functional analysis. |

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### Contents

A MAXIMAL ELEMENT PRINCIPLE FOR CONSTRAINT SETS | 1 |

Nonlinear Evolution Equations as Constraints | 11 |

GLIMla | 18 |

Elliptic Regularization | 31 |

Smoothing Operators Combined with Elliptic | 40 |

Existence of Tangent Directions | 47 |

Solving Nonlinear Evolution Equations | 55 |

Existence of Tangent Directions via GLIMII lia | 61 |

Smoothing Operators Combined with Elliptic | 140 |

OPTIMIZATION AND OPTIMAL CONTROL OF NONLINEAR | 148 |

Smoothing Operators Combined with Elliptic | 154 |

Smoothing Operators Combined with Elliptic | 162 |

GLIMIIIa | 170 |

Existence of Tangent Directions via GLIMIII | 176 |

Smoothing Operators and Elliptic Regularization | 182 |

Tangent Directions for Optimal Control | 188 |

Contents | 66 |

GLIMIIIa | 77 |

Elliptic Regularization and Convex Approximate | 85 |

Smoothing Operators and Elliptic Regularization | 95 |

Tangent Directions involving Control Functions | 101 |

SMOOTHING OPERATORS WITHOUT ELLIPTIC REGULARIZATION | 107 |

Smoothing Operators and GLIMIb | 113 |

OPTIMIZATION AND OPTIMAL CONTROL OF NONLINEAR | 120 |

Smoothing Operators Combined with Elliptic | 127 |

Existence of Tangent Directions | 134 |

Contents | 191 |

An Extremum Principle | 198 |

OPTIMIZATION AND OPTIMAL CONTROL OF NONLINEAR | 236 |

Existence of Tangent Directions for Nonlinear | 243 |

Tangent Directions for Controlled Nonlinear | 249 |

Existence of Tangent Directions for Nonlinear | 256 |

Contents | 263 |

Optimal Control with Nondifferentiable Objective | 269 |

### Other editions - View all

A Theory of Optimization and Optimal Control for Nonlinear Evolution and ... Mieczyslaw Altman No preview available - 1990 |

### Common terms and phrases

6.1 remains valid admits approximate solutions assumptions 1.A0 Banach spaces bounded linear operator C0-semigroups Cauchy problem 6.1 Cauchy sequence Chapter cone control of nonlinear convergence convex convex set Definition 1.1 dh/dt differentiable direction at x0 direction at x0,u0 elliptic regularization equality constraint equation 1.1 Euler-Lagrange equation existence of tangent existence theorems exists a constant F x,u)v following assumptions following estimates hold GLIM-II Hence hypotheses of Lemma hypotheses of Theorem imply induction assumption inequality iterative method maximal element modified linearized equation Moser nonlinear evolution equations nonlinear singular equations nonlinear singular operator obtain optimal control problem partial differential equations problem 1.1 proof follows proof of Theorem prove the existence q)tn replaced by Definition results of Section satisfy the relations scale of Banach sense of Definition suppose that conditions tangent directions Theorem 1.1 carries x e G x e V0 x0 replaced