ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Marco Di Francesco

Aims:

• Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form.

Content:

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
• Introduction to the theory of unbounded operators. Linear differential operators. Applications.
• Introduction to infinite-dimensional differential calculus and variational methods.

Pre-requisites:

Basic calculus and analysis in several variables, linear algebra.

Reading list:

• John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
• H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Corrado Lattanzio

Aims:

Students will know basic of properties (existence, uniqueness, etc.) and techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic PDEs (conservation laws, heat equation, Laplace equation, wave equation).

Content:

Semilinear first order PDE's. Method of characteristics. Partial differential equations of second order. Classification, canonical forms. Well posed problems, IBV problems. The heat equation. Derivation, maximum principle, fundamental solution. Laplace's and Poisson's equations. Maximum principle, fundamental solution and Green's functions. The wave equation. One dimensional equation, fundamental solution and D'Alembert formula. Fundamental solution in three dimensions and strong Huygens' principle, Kirchoff’s formula. Method of descent and solution formula in two dimensions.

Additional info:

http://people.disim.univaq.it/~corrado/

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Alessandro D'Innocenzo

Aims:

The course provides the basic methodologies for modeling, analysis and controller design for continuous-time linear time-invariant systems.

Content:

• Frequency domain models of Linear Systems: Laplace Transform, Transfer Function, Block diagrams.
• Time domain models of Linear Systems:State space representation. BIBO stability.
• Control specifications for transient and steady-state responses. Polynomial and sinusoidal disturbances rejection.
• The Routh-Hurwitz Criterion. PID controllers.
• Analysis and controller design using the root locus.
• Analysis and controller design using the eigenvalues assignment: controllability, observability, the separation principle.
• Reference inputs in state space representations.
• Controller design using MATLAB.
• Advanced topics in control theory.

Pre-requisites:

Mathematical analysis.

Reading list:

• R. C. Dorf, R. H. Bishop. "Modern Control Systems". Prentice Hall, 12th Edition, 2008 (Main textbook)
• G. F. Franklin, J. D. Powell, A. Emami-Naeini. "Feedback control of dynamic systems". Prentice Hall, 4th Edition, 2002
• G. F. Franklin, J. D. Powell, M. L. Workman. "Digital Control of Dynamic Systems". Addison-Wesley, 3rd Edition, 1998
• K. Ogata. "Discrete-Time Control Systems". Prentice Hall, 2nd Edition, 1995A. Isidori. "Sistemi di controllo". Siderea, 1996
• A. Isidori. "Sistemi di controllo". Siderea, 1996

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Bruno Rubino

Aims: The course is intended to introduce and develop an understanding of the concepts in nonlinear dynamical systems and bifurcation theory, and an ability to analyze nonlinear dynamic models of physical systems. The emphasis is to be on understanding the underlying basis of local bifurcation analysis techniques and their applications to structural and mechanical systems.

Content: Review of: first-order nonlinear ODE, first-order linear systems of autonomous ODE. Local theory for nonlinear dynamical systems: linearization, stable manifold theorem, stability and Liapunov functions, planar non-hyperbolic critical points, center manifold theory, normal form theory. Global theory for nonlinear systems: limit sets and attractors, limit cycles and separatrix cycles, Poincaré map. Hamiltonian systems. Poincaré-Bendixson theory. Bifurcation theory for nonlinear systems: structural stability, bifurcation at non-hyperbolic equilibrium points, Hopf bifurcations, bifurcation at non hyperbolic periodic orbits. Applications.

Pre-requisites: Ordinary differential equations

Reading list: Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001

ECTS Credits: 3   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Donatella Donatelli

Aims:

The aim of the course is to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to the mathematical modeling of fluid dynamic type.

At the end of the course students will be able to understand constitutive models for fluids and to be able to solve simple boundary value problems for fluids and solids.

Content:

• Derivation of the governing equations: Euler and Navier-Stokes
• Eulerian and Lagrangian description of fluid motion; examples of fluid flows
• Vorticity equation in 2D and 3D
• Dimensional analysis: Reynolds number, Mach Number, Frohde number.
• From compressible to incompressible models
• Example of fluid models in various fields

Pre-requisites:

Basic notions of functional analysis and multi variable calculus,  standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

ECTS Credits: 3   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   

Aims:

Students will reach a basic level of both written and spoken Italian (A1 level according to CEFR), and will acquire a smattering of Italian culture

Content:

• Greetings and introductions.
• Expressing likes and dislikes.
• Talking about daily activities.
• Understanding and using everyday expressions as well as basic phrases related to daily needs (buying something, asking for directions, ordering a meal).
• Interacting in a very simple way about known topics (family, nationality, home, studies).
• Italian gestures.
• Italian geography.
• Introduction to the most important Italian cities.
• Italian food.

Reading list:

Nuovo Espresso 1, by Luciana Ziglio and Giovanna Rizzo, published by Alma Edizioni, 2014, ISBN: 978-8861823181.