## Optimal Control of Nonsmooth Distributed Parameter SystemsThe book is devoted to the study of distributed control problems governed by various nonsmooth state systems. The main questions investigated include: existence of optimal pairs, first order optimality conditions, state-constrained systems, approximation and discretization, bang-bang and regularity properties for optimal control. In order to give the reader a better overview of the domain, several sections deal with topics that do not enter directly into the announced subject: boundary control, delay differential equations. In a subject still actively developing, the methods can be more important than the results and these include: adapted penalization techniques, the singular control systems approach, the variational inequality method, the Ekeland variational principle. Some prerequisites relating to convex analysis, nonlinear operators and partial differential equations are collected in the first chapter or are supplied appropriately in the text. The monograph is intended for graduate students and for researchers interested in this area of mathematics. |

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### Contents

SEMILINEAR EQUATION | 29 |

VARIATIONAL INEQUALITIES | 77 |

FREE BOUNDARY PROBLEMS | 122 |

Copyright | |

1 other sections not shown

### Common terms and phrases

adapted penalization adjoint admissible pair approximate problem argument assume Banach space Barbu 12 boundedness Ch.I Chapter coercive compact consider continuous control constraints control problems governed convergence convex function corresponding cost functional D.Tiba deduce defined definition denote equivalent estimates existence Fenchel conjugate finite Gateaux differentiable given governed by variational grad Hilbert space hypothesis implies indicator function infl integrate over 0,t J.L.Lions least one optimal Lemma limit limsup linear locally Lipschitzian lower semicontinuous mapping maximal monotone graph maximal monotone operator Minimize Moreover multiply obtain Obviously optimal control optimal pair partial differential equations problem 3.1 Proof Let properties regularity results Remark saddle function satisfies semigroup sequence shows Sobolev embedding theorem solution of 5.2 strongly in C(0,T;H subdifferential subsequence sufficiently small Theorem Tiba two-phase Stefan problems unique solution V.Barbu variational inequalities weak solution weak topology weakly xi xi yields yo(x Yosida approximation