Course teached as: B018755 - ANALISI SUPERIORE Second Cycle Degree in MATHEMATICS Curriculum APPLICATIVO
Teaching Language
Italian
Course Content
- Perspectives on linear Partial Differential Equations (PDE):
i) the Cauchy problem, inverse problems;
ii) initial and boundary value problems for evolutionary PDE as differential equations (and systems) in infinite dimensional spaces; introduction to the theory of operator semigroups.
- Some results and methods for the study of inverse and control problems.
- Some tools of modern mathematical analysis: Carleman estimates, boundary regularity and observability estimates.
- L.C. Evans, Partial differential equations, Second edition.
Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp.
- F. John, Partial differential equations. Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991. x+249 pp.
- J. Zabczyk, Mathematical control theory: An introduction. Second edition. Systems & Control: Foundations & Applications. Birkhauser/Springer, Cham, 2020. xxvi+336 pp.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp.
- A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and control of infinite dimensional systems. Second edition. Systems & Control: Foundations & Applications. Birkhauser Boston, Inc., Boston, MA, 2007. xxviii+575 pp.
In addition: notes by the instructors; online materials (freely available) suggested by the instructors
Learning Objectives
Knowledge objectives: acquisition of knowledge of the principal topics and questions of the course (see "Extended program")
Competence objectives: understanding of language and methods for the study of inverse and control problems
Skills acquired at the end of the course: ability to understand and/or formalize simple inverse and control problems, to propose methods for their resolution, to interpret the relative results. Being able to access the scientific literature for in-depth study.
Prerequisites
Fundamentals of the theory of Ordinary Differential Equations. Functional Analysis. Normed spaces and continuous linear maps. Hahn-Banach theorem. Banach spaces. Hilbert spaces. Differential calculus in R^n. Theory of Lebesgue measure. L^p spaces. Holder spaces. Convolution. Fourier transform. Definition and basic properties of Sobolev spaces. Harmonic functions.
Teaching Methods
Lectures and discussion of exercises/problems in the classroom. Seminar activity on material provided by the instructors
Further information
9 CFU, that means: 225 hours (students' total workload), 72 hours (classes)
Type of Assessment
Oral examination: more specifically, a seminar on a topic agreed with each and every student
Course program
Detailed programme:
MATHEMATICAL CONTROL THEORY. Introduction. Linear control systems in finite dimensional spaces. Controllability, stability (and stabilizability), optimal control: definitions, a few relevant results.
The infinite-dimensional context: the interconnection between the methods of functional analysis and purely partial differential equations (PDE) methods.
THE THEORY of OPERATOR SEMIGROUPS. Strongly continuous semigroups in Banach spaces, the semigroup's generator. Generation theorems: Hille-Yosida theorem, Lumer-Phillips theorem. Analytic semigroups. Groups.
Interpretation of the properties of admissibility and controllability in terms of inequalities (direct and inverse ones, respectively).
THE CAUCHY PROBLEM FOR PDE.
a) Analytic solutions. The Cauchy-Kovalevskaya theorem. Holmgren's theorem.
b) Well-posed problems in the sense of Hadamard.
c) Uniqueness and continuous dependence for equations with non-analytic coefficients. Stability estimates for second order elliptic equations. Introduction to the method of Carleman estimates. Three spheres inequality.
EXAMPLES OF INVERSE PROBLEMS. Tomography. Backward problem for the heat equation. The inverse conductivity problem: formulation of the problem. The Dirichlet-to-Neumann map: definition, its main properties.