Martínez-Frutos, Jesús; Periago Esparza, Francisco Optimal control of PDEs under uncertainty. An introduction with application to optimal shape design of structures. (English) Zbl 1426.49001 SpringerBriefs in Mathematics; BCAM SpringerBriefs. Cham: Springer; Bilbao: BCAM – Basque Center for Applied Mathematics (ISBN 978-3-319-98209-0/pbk; 978-3-319-98210-6/ebook). xix, 123 p. (2018). In the so called “deterministic theory” of PDEs the results as: continuous dependence on the initial data, on the parameters, on the boundary conditions, different kind of stabilities are motivated by the uncertainty of the input data. According to the authors of this book “the book aims at introducing graduate students and researchers to the basic theoretical and numerical tools which are being used in the analysis and numerical resolution of optimal control problems constrained by PDEs with random inputs”. The motivation of this type of approach is: “more realistic optimal control problems should account for uncertainties in their mathematical formulations”. For accomplish the above purpose in Chapter 3 of the book, the random Laplace-Poisson equation, the random heat equation, and the random Bernoulli-Euler beam equation are considered and for them the existence of the optimal control problem is presented. In Chapter 4, the numerical resolution of robust optimal control problem is developed, and in Chapter 5, the numerical resolution of risk avers optimal control is presented. In Chapter 6, mainly the existence of the optimal shape and the numerical approximation of the shape is developed. In Chapter 7, miscellaneous topics and open problems are presented. This book is addressed to graduate students and researches in engineering, physics and mathematics who are interested in optimal control and optimal design using random partial differential equations. Reviewer: Stefan Balint (Timişoara) Cited in 17 Documents MSC: 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control 49J20 Existence theories for optimal control problems involving partial differential equations 49Q10 Optimization of shapes other than minimal surfaces 49K20 Optimality conditions for problems involving partial differential equations 49M25 Discrete approximations in optimal control Keywords:optimal control problems; uncertainties; random Laplace-Poisson equation; random heat equation; random Bernoulli-Euler beam equation; optimal shape; numerical approximation; optimal design Software:DAKOTA; FERUM; Sparse Grid Interpolation; compliance; Allaire_Scilab; UQLab; OpenTURNS; Burkardt; Allaire_FreeFem; Matlab PDFBibTeX XMLCite \textit{J. Martínez-Frutos} and \textit{F. Periago Esparza}, Optimal control of PDEs under uncertainty. An introduction with application to optimal shape design of structures. Cham: Springer; Bilbao: BCAM -- Basque Center for Applied Mathematics (2018; Zbl 1426.49001) Full Text: DOI