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Calculus of Variations and Optimal Control 2,1 Examples of problems addressed by the course. 2 Background on finite and infinite dimensional optimization. 3 Calculus of variations,4 Pontryagin s Principle,5 The Hamilton Jacobi Bellman equation. 6 The Linear Quadratic problem,7 Optimal Control Problems in Discrete time. 8 Numerical Methods, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 3,1 Examples of problems addressed by the course.

Objective Introduce a novel class of optimization problems that are solved. with respect to infinite dimensional variables Problems of Calculus of. Variations and Optimal Control,Refs L2012 26 31, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 4,A classical problem The brachistochrone curve. What is the shape of the curve that connects points A and B such data a point. mass under the force of gravity alone slides frictionless from A to B in. minimum time,Which function y x,minimizes the,travel time between. P mg A and B, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 5, Computing the travel time assuming y x 0 x x 2 known.

Without friction the increase of kinetic energy is equal to the loss of potential. energy and mv 2 mgy or, Let be the arclength From Pythagoras theorem we get the kinematics. dt dt dx dt dt, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 6,Energy balance. Kynematics,Eliminate by equating the r h s,2 gy x 1 y x. J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 7,2 gy x 1 y x.

Traveling time is obtained by integration, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 8, If we know the function we can compute the travel time. J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 9, For instance if the path to follow is a straight line between A and B. y2 A 0 0 x,y x x with x2,The travel time for the rectilinear path is. 1 2 12 1 2 1 2, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control.

Calculus of Variations and Optimal Control 10, If we want to compare the travel time for the rectilinear path with the one of. another curve for instance an arc of circle we can do it and decide which. one leads to the pastest path, However the point is that we don t know the shape of the optimal curve. We want to optimize with respect to the curve and this is an infinite. dimensional problem because it depends on the position of the points on the. curve that are infinitely many, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 11,The expression. defines the functional to minimize J y T R,To each differentiable function y x defined.

on 0 x2 that satisfies the boundary A,conditions y 0 0 and y x2 y2. if associates a real number the,travel time B, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 12, The Brachistochrone problem was published in 1 January. 1667 by Johann Bernouilli as a challenge to the scientific. community Nothing is more attractive to intelligent persons. than an honest problem that challenges them and which. solution brings fame and stays as a lasting monument. 60 years before Galileo new already that the minimum time. trajectory could not be a straight line although he thought. erroneously that it was a circumference arc, This challenge was tackled by six of the most brilliant minds of the time His elder. broither Jacob Leibniz Tschirnhaus l Hopital and Newton who published his solution. anonymously, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control.

Calculus of Variations and Optimal Control 13, An historical perspective with technical content of the Brachistochrone. problem and of its relations with Optimal Control may be seen in. Sussmann H J e J C Willems 1997 300 Years of Optimal Control From. the Brachystochrone to the Maximum Principle IEEE Control Systems. 17 3 32 44, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 14,A machine to exemplify the. brachistochrone Museu Pombalino,de F sica da Universida de Coimbra. The challenge of J Bernouilli as,published in Acta Eroditorum.

J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 15, What is the relation between ancient Rome poetry the Phoenicians and. Control and Estimation theory, J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control. Calculus of Variations and Optimal Control 10 J Miranda Lemos IST DEEC Sci Area of Systems Decision and Control If we want to compare the travel time for the rectilinear path with the one of another curve for instance an arc of circle we can do it and decide which one leads to the pastest path

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