Dynamical Systems and Optimization

Code:

EMG0017

Acronym:

SDO

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 2S

Active?	Yes
Responsible unit:	Mining Engineering Department
Course/CS Responsible:	Bachelor in Mining and Geo-Environmental Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LCEEMG	13	Plano de estudos oficial a partir de 2008/09	2	-	6	56	162

Teaching language

Suitable for English-speaking students
Obs.: Aulas em inglês sempre que se verifique a presença de estudantes não falantes da lingua portuguesa.

Objectives

The place occupied by the course in the study plans dictate its role in establishing the linking between mathematical analysis and some specific matters which need a strong mathematical background. The course is divided in two parts, to know: 1st part – Dynamical Systems; 2nd part – Linear Optimization. The main objectives of the 1st part are: Knowledge: - To recall the concepts of function, independent and dependent variables; - To acquire the concept of dynamic system; - To classify dynamic systems from the ordinary differential equation and the initial conditions; - To operate with integral transformations. Comprehension: - To review the meaning of ordinary differential; - To identify order of a dynamical system; - To make the distinction between linear systems and non linear systems, continuous and discrete; - To describe the methods applicable to the resolution of each differential equation; - To translate mathematically a 1st order kinetic. Application: - To interpret the physical meaning of the solution of a differential equation; - To use CAS (computer algebra systems) to implement and solve differential equations; - To operate with abstract complex entities such as integral transformations. The main objectives of the 2nd part are: Knowledge: - To learn the concept of objective function, equality and inequality; - To define the feasible region. Comprehension: - To identify problems of linear optimization; - To identify the primal problem and; - To write the dual problem; - To discuss the solution of a linear programming problem; - To formulate mathematically simple problems (2 or 3 variables) of linear programming. Application: - To apply the Simplex algorithm to solve linear optimization problems; - To use the spreadsheet to obtain the solution of a problem. Analysis: - To interpret and criticize the obtained solutions for a linear optimization problem; - To test the behavior of the obtained solutions varying the recess variables; - To analyze how the optimal solution varies as a consequence of small variations in the problem data.

Learning outcomes and competences

1ª parte:

Dominar o conceito de sistema dinâmico; classificar os sistemas dinâmicos a partir da equação diferencial e das condições iniciais; operar com transformações integrais; fazer a destrinça entre sistemas lineares de não lineares, contínuos de discretos; traduzir matematicamente um processo cinético de 1ª ordem; interpretar em termos físicos a solução de uma equação diferencial; usar manipuladores algébricos na implementação e resolução de equações diferenciais;

2ª Parte

Dominar o conceito de função objectivo, ligações e constrições; definir praticável; identificar problemas de optimização linear; identificar o problema primal e escrever o correspondente problema dual; interpretar e discutir a solução de um problema de programação linear; formular matematicamente problemas simples (2 e 3 variáveis) de programação linear; aplicar o algoritmo de Simplex na resolução de problemas de optimização linear; usar a folha de cálculo na obtenção da solução do problema.

Working method

Presencial

Program

1st Part – Dynamical Systems 1- Introduction – 1st order differential equations; 2- 1st order dynamical systems : 2.1- Kinetics processes – radioactive decay; 2.2- Chemical reactions – 1st order kinetics; 2.3- Population growth – logistic law ; 3- High order continuous systems: 3.1- Linear differential equation of order n with constant coefficients; 3.2- Example of an invariant linear system of 2nd order – free oscillations; 4- Physical interpretation of differential equations: 4.1- 1st order invariant systems – direction field, critical points and stability; 4.2- 2nd order linear systems – phase space trajectory, proper vectors and values, critical points and stability; 5- Convolution and Dirac pulse; 6- Laplace transform: 6.1- Integral transformations, origin and image; 6.2- Properties of Laplace transform; 6.3- Inversion of Laplace transform – decomposition in simple fractions; 6.4- Application of Laplace transform to the resolution of differential equations. 2nd Part – Linear Optimization 1- Introduction, basic concepts; 2- Linear programming: 2.1- The meaning of a linear model; 2.2- Problems formulation; 2.2- The Simplex algorithm; 2.3- The dual problem; 2.4- Sensitive analysis.

Mandatory literature

António Fiúza; Dinâmica de Sistemas, 2000

Complementary Bibliography

Shepley L. Ross; Introduction to ordinary differential equations. ISBN: 0-471-09881-7
Frederick S. Hillier, Gerald J. Lieberman; Introduction to operations research. ISBN: 0-0-07-100492-0
Jaime Enrique Villate Matiz; Introdução aos sistemas dinâmicos. ISBN: 972-99396-0-8

Teaching methods and learning activities

Theoretical-practical classes will be distributed weekly over two two-hour periods on different days. The first hour of each TP class will be used for oral presentation of the contents to students, the second hour will be occupied with applications of the subjects taught, with emphasis on the diversification of implementation methods - use of computers (algebraic manipulators), solving exercises with a volunteer (student or teacher) working on the board, and the remaining students working in place under the supervision and monitoring of the teacher. Exercise statements are previously available on online platforms (moodle and sigarra).

Software

Excel (solver)
WxMaxima

keywords

Technological sciences > Engineering > Systems engineering > Systems theory

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	10,00
Teste	90,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	112,00
Frequência das aulas	50,00
Total:	162,00

Eligibility for exams

To obtain attence, the student cannot exceed the limit number of absences - 25% of the number of classes given.

Calculation formula of final grade

The first call grade will be the average of the distributed evaluation components.
N1 = 0.45 * T1 + 0.45 * T2 + 0.1 * P
Being T1 and T2 mandatory tests; P stands for class participation and homework.
Students whose 1st call grade (N1) is less than 10 will be subjected to a second call exam.