# Course Details

#### Name:

**Mathematical modelling and optimization / Mathematical modelling and optimization**

##### Type:

##### Modules:

### Basic information

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##### Sector:

##### Credits:

*Master Degree in Mathematical Engineering:*18 Ects (b)

*Master Degree in Mathematics:*6 Ects (b)

##### Term:

*Module Advanced Analysis 1:*1° semester

*Module Modelling and control of networked distributed systems:*1° semester

*Module Process and Operations Scheduling:*1° semester

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##### Language:

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### Course Objectives

**Module Advanced Analysis 1:** Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics,
as Sobolev Spaces, distributions.
Application of this knowledge to a variety of topics, including the basic equations of mathematical physics
and some current research topics about linear and nonlinear partial differential equations.**Module Modelling and control of networked distributed systems:** The aim of this course is to provide basic knowledge of the analysis and design of dynamic multiagent networks. **Module Process and Operations Scheduling:** Train the students in recognizing machine scheduling problems, classify them in terms of computational complexity and solve them by heuristic, approximation or exact algorithms.

### Course Content

##### Module Advanced Analysis 1

- Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.
- Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).
- Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo-Nirenberg-Sobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.
- Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.
- First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.

##### Module Modelling and control of networked distributed systems

- Introduction to graph theory: graphs; matrices representation; algebraic and spectral graph theory; graph symmetries.
- The agreement protocol - the static case: undirected and directed networks; agreement and markov chains; the Factorization Lemma.
- The agreement protocol - Lyapunov and LaSalle: agreement via Lyapunov functions, agreement over switching digraphs, edge agreement, generalizations to nonlinear systems.
- Formation Control: formation specification-shapes and relative states; shape based control; relative state based control, dynamic formation selection, assigning roles.
- Mobile Robots: Cooperative robotics; weighted graph based feedback; dynamic graphs; formation control revisited; the coverage problem.

##### Module Process and Operations Scheduling

- Elements of a (deterministic) scheduling problem, examples of practical applications
- Classification of scheduling problems
- Integer Linear Programming formulations
- Single machine scheduling: computational complexity, heuristic and exact algoritms
- Parallel machine scheduling: exact, heuristic and approximation algorithms
- Relationships with basic Combinatorial Optimization problems
- Optimization problems in Project Scheduling
- Job Shop scheduling: formulations, heuristic and exact algorithms

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

##### Module Process and Operations Scheduling

- Acquire knowledge of Machine Scheduling problems, their classification in terms of computational complexity and algorithmic techniques developed for their solution. Acquire the fundamentals of optimization methods for project management.
- Acquire the ability to recognize Machine Scheduling problems in different application contexts, such as computer science, industrial engineering and management, and to identify effective solution paradigms.
- Acquire autonomy in modeling and algorithmic choices for complex problems related to scheduling and project management.
- Being able to hold a conversation and to read texts on topics related to the modeling of scheduling problems and the evaluation of algorithms for their solution
- Acquire skills upgrading flexible knowledge and skills in the field of scheduling problems that arise in various areas, such as computer science, industrial engineering and management

### Prerequisites and Learning Activities

**Module Advanced Analysis 1:** Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.**Module Modelling and control of networked distributed systems:** Linear Algebra. Linear control systems. Stability theory for linear control systems.**Module Process and Operations Scheduling:** basic elements of computational complexity, linear programming and network flows

### Teaching Methods

**Language**: Italian
**Module Advanced Analysis 1:** Lectures**Module Modelling and control of networked distributed systems:** Lectures and exercises.**Module Process and Operations Scheduling:** standard lessons and exercise sessions

### Assessment Methods and Criteria

**Module Advanced Analysis 1:** Oral exam**Module Modelling and control of networked distributed systems:** Written and oral exam**Module Process and Operations Scheduling:** a paper test concerning with theoretical or computational exercises; an oral test, accessible only with a sufficient grade at the paper test, about general machine scheduling theoretical issues

### Textbooks

##### Module Advanced Analysis 1

- G. Gilardi,
**Analisi 3**. McGraw–Hill. - V.S. Vladimirov,
**Equations of Mathematical Physics**. Marcel Dekker, Inc.. - C.M. Dafermos,
**Hyperbolic Conservation Laws in Continuum Physics**. Springer. - L.C. Evans,
**Partial Differential Equations**. AMS. - M.E. Taylor,
**Partial Differential Equations, Nonlinear equations**. Springer. - H. Brezis,
**Sobolev Spaces and Partial Differential Equations**. Springer.

##### Module Modelling and control of networked distributed systems

- M. Mesbahi and M. Egerstedt,
**Graph Theoretic Methods in Multiagent Networks**. Princeton University Press. 2010. http://press.princeton.edu/titles/9230.html

##### Module Process and Operations Scheduling

- Michael Pinedo,
**Scheduling Theory, Algorithms, and Systems**. Prentice Hall.

### Online Teaching Resources

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### Recent teaching material

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### Course page updates

This course page is available (with possible updates) also for the following academic years:*Course information last updated on: 02 novembre 2016, 16:02*