Dynamical Systems and Optimization

Code:

EMG0017

Acronym:

SDO

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 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	27	Plano de estudos oficial a partir de 2008/09	2	-	6	56	162

Teaching language

Suitable for English-speaking students

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; 7- Discrete systems: 7.1- 1st and 2nd order linear difference; 7.2- Examples of 1st order linear discrete systems – an econometric model and Markov processes. 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
Jaime Villate; Introdução aos Sistemas Dinâmicos - Uma abordagem prática com Maxima, 2007. ISBN: 972-99396-0-8

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

Teaching methods and learning activities

Theoretical lessons will be distributed weekly of the following form: two hours for verbal exposition of the contents to the students; one hour in computers room implementing problems in computer (maxima and spreadsheet). The practical lessons reserved for the resolution of exercises.

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	108,00
Frequência das aulas	54,00
Total:	162,00

Calculation formula of final grade

A nota da primeira chamada será a média das componentes de avaliação distribuida.
N1=0,45*T1+0,45*T2+0,1*P
Sendo T1 e T2 testes OBRIGATÓRIOS; P representa a participação nas aulas e trabalhos para casa.
Os alunos cuja nota da 1ª chamada (N1) seja inferior a 10, submeter-se-ão a exame de recurso.