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Classical Problems in Calculus of Variations and Optimal Control

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Classical Problems in Calculus of Variations and Optimal Control Classical Problems in Calculus of Variations and Optimal Control Stephen Lamb Supervised by Lyle Noakes The University of Western Australia Vacation Research Scholarships are funded jointly by the Department of Education and Training and the Australian Mathematical Sciences Institute. 1 Introduction Calculus of variations (COV) is a field of mathematics that deals with finding the extremals of Lagrangian functions defined by functionals, in an attempt to find optimal solutions. Although the subject has a long and rich history, current research in the field is still producing new results. Op- timal control is the generalisation of the calculus of variations. My project focuses on the classical problem of COV, and how the tools of optimal control can be used to simplify results, and even produce results where COV was too blunt an object. My project started with a brief study of some classical problems in COV and then I started generating the differential equations that we solve to answer the problems. These include such problems as the brachistochrone and the catenary. The next task was to learn about optimal control and its link to COV. Once able to write out our control problem, I needed a similar tool to solve for extremals as above, which in optimal con- trol is the Pontryagin Maximum Principle. The most crucial part of the project was about learning and understanding how to use the Pontryagin Maximum Principle to solve the optimal control problems. We then extended it from Rn to other smooth manifolds using Riemannian geometry. We used our framework of optimal control to solve two very important classical problems: geodesics on Riemannian manifolds and elastica curves. Lastly, after generating solutions to these problems for different spaces, we had to have a method of solving these boundary value problems numerically, for problems where no closed form solution exists. The study of classical problems in calculus of variations and optimal control has provided me with a good foundation for further study into differential geometry, optimisation methods and functional analysis. It has also given good insight into the topics of topology, Riemannian geometry and anal- ysis. Although I have aquired a deeper level of understanding through this research project, I am left with far more questions to answer. Future work into the topic involves study of dependence of these classical problems to assumed conditions. e.g. The study of the brachistochrone under different gravitational field conditions or the catenary under a changing viscocity of air as a function of height. Further research into geodesics phrased using optimal control could involve looking into the geodesics on manifolds that aren't as nice and symmetric as the ones chosen. Also research into the existence of abnormal geodesics in sub-Riemannian manifolds would yield enlightening results. For the problems of elastic curves, future work could involve the study of elastic curves in other riemannian manifolds. Lastly further investigation into other numerical solving methods for solving these problems, aiming at reducing computing cost could prove a good idea. 2 2 Calculus of Variations To understand what calculus of variations is, and in turn what optimal control is, we require understanding of the lagrangian function and how to determine extremals from it. Definition 2.1. The Lagrangian (L) is an energy function defined on the Tangent bundle, that maps onto the space of real numbers. We can write L as L(q(t); q_(t)) L : TM ! R Calculus of variations is concerned with finding the extremals of functions defined by: Z t2 J(t) = L(q(t); q_(t))dt; (1) t1 where extremals are the critical points of the Lagrangian. Theorem 2.2. If q is an extremal of the Lagrangian, then it must satisfy the Euler-Lagrange equation: d @L @L = (2) dt @q_ @q Proof. Consider the system defined above by (1), and let's assume that q is an extremal. Without loss of generality, we will assume that q is a minimising function. Hence we can conclude that: L(q(t); q_(t)) < L(q(t) + η(t); q_(t)η_(t)) Therefore if we take the derivative with respect to , the following result is obtained: d d Z t2 J(t)j=0 = L(q(t) + η(t); q_(t) + η_(t))dtj=0 = 0 d d t1 Z t2 @L @L =) η + η_ dt = 0 t1 @q @q_ Using integration by parts on the second term above, we get the following: Z t2 @L d @L =) η − η dt = 0 t1 @q dt @q_ Factoring out η we get the Euler-Lagrange equation: d @L @L = dt @q_ @q 3 2.1 Classical Problems in COV 2.1.1 Brachistochrone The Brachistochrone is a classical problem that deals with finding a curve between two points: (x0; y0) and (x1; y1) with x0 < x1 and y0 < y1, such that the time taken for a bead to go along the curve between the two points on the curve is minimal. Time is the variable that is going to be minimised. Let the total length of the curve be L. Using Newtons equations of motion, we can derive the equation for total time taken to traverse the curve: Z 0 ds T (y) = ds (3) L v(s) (Where v(s) is the velocity and ds is the arclength) Using the conservation of energy principle, we can define this expression in terms of y(x). KE + PE = E = constant 1 E = mgy(x) + mv(x)2 (4) 2 Rearranging for v(x), we get the following: r 2(E − mgy(x)) v(x) = (5) m Thus using (3), we get the following result: x p Z 1 1 +y _2 T (y) = q dx (6) x0 2(E−mgy(x)) m With y(x0) = y0, and y(x1) = y1 1 2E−2gmy(x) Simplifying this expression with the substitution: z = 2g m we get the following func- tional: r Z x1 1 +z _2 J(z) = p2g dx (7) x0 z Using the Euler-Lagrange equation, we derive the following differential equation: z(1 +z _2) = c (8) Using the substitution z = tan(θ), then 1+z _2 = sec(θ). Hence the curve that satisfies the conditions of the brachistochrone curve is the parametric equation of a cycloid: y(θ) = d1(1 + cos(2θ)) (9) x(θ) = d2 − d1(2θ + sin(2θ)) (10) 4 π Figure 1: Brachistochrone curve plotted between fixed points (− 2 ; 1) and (0; 0) This problem can be extended and studied in many ways for various reasons. As such, some of these include having a forcing/damping system, or even considering a non-constant gravitational field, dependent on the y(x). Harry H. Denman goes into other possible solutions to the brachistochrone problem by varying fields. See [2] for more. 5 2.1.2 Tautochrone The tautochrone problem questions finding a curve such that under constant acceleration, a particle placed anywhere on the curve will take the same amount of time to reach the bottom of the curve, no matter its start position. In a constant gravitational field, it can be easily shown that the brachistochrone curve satisfies such a property. [2] 2.1.3 Catenery The catenary is a sometimes refered to as the hanging chain problem. It seeks to determine the shape of a curve made between two fixed points in a constant gravitational field. In practice it is the curve made when hanging a chain/wire between two fixed points, making a curve called the catenary curve, who's name originates from the problem. The problem can also be described as a curve that minimises gravitational potential energy, which we take advantage of to derive our answer. We start by defining the functional to minimise: Z L PE = m g y(s) ds (11) 0 (where PE is the potential energy of the whole curve, L is the length of the curve and s is the arclength of the curve) r 2 2 2 2 dy Using the fact that ds = dx + dy =) ds = 1 + dx dx, we can rewrite the above equation as: s Z L dy 2 PE = m g y(x) 1 + dx (12) 0 dx Now we can employ the Euler-Lagrange equation above. Hence any extremal of the above equation must satisfy the following: d @P E @P E − = 0 dx @y_ @y Simplifying the above equation gives the following differential equation y y_ p y_ − y 1 +y _2 = Constant p1 +y _2 y2 = D2 (13) 1 +y _2 1 We ignore the trivial case and only consider D 6= 0 to get: s y2 y_ = 2 − 1 (14) D1 Separating variables and integrating we get the follwoing equation: p 2 2 ! y + y − D1 x = D1 ln + D2 (15) D1 6 Rearanging the above equation, we get the solution of the catenary curve in terms of y as: x − D2 y = D1 cosh (16) D1 which is graphed below. (D1 and D2 are constants) Figure 2: Plot of equation (16) - Catenary curve/Hanging chain This curve looks similar to the parabolic curve, however the two are very different. Catenary curves find themselves in nature and also lend themselves to building the perfect arches for buildings. 2.1.4 Isoperimetric Problem The isoperimetric problems set to solve constrained variational problems by first transforming them into unconstrained problems using Lagrange multipliers. These problems allow solutions of problems constrained to have extremals exist on curves or other constraints. These are very interesting problems, however further research is needed to fully understand the importance of Lagrange multipliers. 2.1.5 Minimal Surfaces Minimal surfaces are objects on spaces that have zero mean curvature (See [1] for definition).
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