Notes | Mechatronic Systems and Laboratory

3.1 Introduction

We are concerned with the problem of finding minima (maxima) of a functional $F : ℱ \to ℝ$ , where $ℱ$ is a subset of a (normed) linear space $𝒳$ of real-valued (or real-vector-valued) functions. The formulation of a problem of the calculus of variations requires two steps: the specification of a performance criterion is discussed and the statement of physical constraints that should be satisfied.

Remark 3.1: Performance Criterions

A performance criterion $F$ , also called cost functional or simply cost must be specified for evaluating the performance of a system quantitatively. The typical form of $F$ (also called Lagrange problem of the calculus of variations) is

ℱ [x] : = \int_{t_{1}}^{t_{2}} f (t, x (t), \dot{x} (t)) d t

where $t \in ℝ$ is the real or independent variable, usually called time, $x (t) \in ℝ^{n_{x}}$ , $n_{x} \geq 1$ , is a real vector variable, usually called the phase variable. The functions $x (t) = (x_{1}, \dots, x_{n_{x}})$ $t_{1} \leq t \leq t_{2}$ are generally called trajectories or curves; $\dot{x} (t) \in ℝ^{n_{x}}$ stands for the derivative of $x (t)$ with respect to time; and $f : ℝ \times ℝ^{n_{x}} \times ℝ^{n_{x}} \to ℝ$ is a real-valued function, called a Lagrangian function or, briefly, a Lagrangian ^a , the function $f$ measures the instantaneous cost and is sometimes called running cost. Overall, we may thus think of $ℱ (x)$ as dependent on an real-vector-valued continuous function $x (t) \in 𝒳$ . A slightly different performance criterion is:

F [x] = φ (t_{1}, x (t_{1}), t_{2}, x (t_{2})) + \int_{t_{1}}^{t_{2}} f (t, x (t), \dot{x} (t)) d t

an additional function $φ$ weights the endpoints of the phase trajectory ${t_{1}, x (t_{1})}$ and ${t_{2}, x (t_{2})}$ . This problem is called Bolza problem. In the same way one can define a third type of cost functional known as Mayer problem ^b :

F [x] = ψ (t_{1}, x (t_{1}), t_{2}, x (t_{2}))

It can be shown that all these formulations are equivalent that means that can be transformed into one another via a variable transformation.

Definition 3.1: Calculus of Variation Basic Problem

The basic problem of Calculus of Variation is defined as:

\begin{matrix} \min_{x (t)} & F [x] = \int_{t_{1}}^{t_{2}} f (t, x (t), \dot{x} (t)) d t \\ s . t . & x \in ℱ \subset 𝒳 \end{matrix}

For example we can take as $𝒳 = C^{1} ([t_{1}, t_{2}])$ the space of real continuously differentiable functions on $[t_{1}, t_{2}]$ and as $ℱ$ a subset of this function space $ℱ : = {x \in 𝒳 such that x (t_{1}) = x_{1}}$ that is the space of continuously differentiable functions that is equal to a fixed $x_{1}$ at $t_{1}$ . Enforcing constraints in the optimization problem reduces the set of candidate functions, i.e., not all functions in $𝒳$ are allowed. This leads to the following:

Definition 3.2: Admissible Trajectory

A trajectory $x$ ^a ^a Again we emphasize the fact that a trajectory $x \in 𝒳$ is a real or vector valued function, we use both the notation $x$ or $x (t)$ in a real linear function space $𝒳$ is said to be an admissible trajectory provided that it satisfies all the physical constraints (if any) along the interval $[t_{1}, t_{2}]$ . The set $𝒟$ of admissible trajectories is defined as

𝒟 : = {x \in 𝒳 : x admissible}

Typically, the functions $x (t)$ are required to satisfy conditions at their end-points. Problems of the calculus of variations having end-point constraints only, are often referred to as free problems of the calculus of variations. A great deal of boundary conditions is of interest. The simplest one is to enforce both end-points fixed, e.g., $x (t_{1}) = x 1$ and $x (t_{2}) = x 2 .$ Then, the set of admissible trajectories can be defined as:

𝒟 : = {x \in 𝒳 : x (t_{1}) = x 1, x (t_{2}) = x 2} .

In this case, we may say that we seek for trajectories $x (t) \in 𝒳$ joining the fixed points $(t_{1}, x 1)$ and $(t_{2}, x 2)$ . Alternatively, we may require that the trajectory $x (t) \in 𝒳$ join a fixed point $(t_{1}, x 1)$ to a specified curve $Γ : x = g (t), t_{1} \leq t \leq T$ . Because the final time $t_{2}$ is now free, not only the optimal trajectory $x (t)$ shall be determined, but also the optimal value of $t_{2}$ . That is, the set of admissible trajectories is now defined as

𝒟 : = {(x, t_{2}) \in 𝒳 \times [t_{1}, T] : x (t_{1}) = x 1, x (t_{2}) = g (t_{2})}

Besides bound constraints, another type of constraints is often required:

F_{k} [x] : = \int_{t_{1}}^{t_{2}} f_{k} (t, x (t), \dot{x} (t)) d t = C_{k} k = 1, \dots, m

for $m \geq 1$ lagrangian functionals $F_{1}, \dots, F_{m}$ with lagrangian functions $f_{1}, \dots, f_{m}$ . These constraints are often referred to as isoperimetric constraints. Similar constraints with $\leq$ sign are sometimes called comparison functionals. Finally, further restriction may be necessary in practice, such as requiring that $x (t) \geq 0$ for all or part of the optimization horizon $[t_{1}, t_{2}]$ . More generally, constraints of the form

\begin{aligned} Ψ (t, x (t), \dot{x} (t)) & \leq 0 \\ Φ (t, x (t), \dot{x} (t)) & = 0 \end{aligned}

for $t$ in some interval $I \subset [t_{1}, t_{2}]$ are called constraints of the Lagrangian form, or path constraints. A discussion of problems having path constraints is deferred until the following chapter on optimal control. We now give a series of examples of classical problems of the Calculus of Variations whose admissible function spaces present some of the constraints discussed previously. The most famous and maybe first problem of the calculus of variations is the Brachistochrone Problem below.

Example 3.1 (Brachistochrone Problem). Consider the problem of finding the curve $x (ξ), ξ_{A} \leq ξ \leq ξ_{B},$ in the vertical plane $(ξ, x),$ joining given points $A = (ξ_{A}, x_{A})$ and $B = (ξ_{B}, x_{B}), ξ_{A} < ξ_{B}, x_{A} < x_{B},$ and such that a material point sliding along $x (ξ)$ without friction from $A$ to $B$ , under gravity and with initial speed $v_{A} \geq 0,$ reaches $B$ in a minimal time (see Figure 3.1 ). This problem was first formulated then solved by Johann Bernoulli, more than 300 years ago!

The objective function $F [x]$ is the time required for the point to travel from $A$ to $B$ along the curve $x (ξ)$

F [x] = \int_{ξ_{A}}^{ξ_{B}} d t = \int_{ξ_{A}}^{ξ_{B}} \frac{d s}{v (ξ)}

where $s$ denotes the arc length of $x (ξ),$ defined by $d s = \sqrt{1 + ẋ {(ξ)}^{2}} d ξ,$ and $v,$ the velocity along $x (ξ)$ . Since the point is sliding along $x (ξ)$ without friction, energy is conserved,

\frac{1}{2} m (v {(ξ)}^{2} - v_{A}^{2}) + m g (x (ξ) - x_{A}) = 0

with $m$ being the mass of the point, and $g$ , the gravity acceleration. That is, $v (ξ) =$ $\sqrt{v_{A}^{2} - 2 g (x (ξ) - x_{A})},$ and

F [x] = \int_{ξ_{A}}^{ξ_{B}} \sqrt{\frac{1 + ẋ {(ξ)}^{2}}{v_{A}^{2} - 2 g (x (ξ) - x_{A})}} d ξ

The Brachistochrone problem thus formulates as a Lagrange problem of the calculus of variations.

Figure 3.1:: The Brachistochrone Problem

We note that this problem is a free problem since we have assigned endopoints $A = (ξ_{A}, x_{A})$ and $B = (ξ_{B}, x_{B})$ were given. Another classical problem of the calculus of variations with Lagrange cost functional is the catenary problem.

Example 3.2 (Hanging rope with fixed length). Given a rope of length $L$ attached at two poles $A = (x_{A}, y_{A})$ and $B = (x_{B}, y_{B})$ find the function $y (x)$ that describes the shape assumed by the rope under the action of gravity. Using the infinitesimal arc length $d s$ we can express the constraint enforcing the length $L$ of the rope as an isoperimetric constraint. That is:

F_{1} [y] = \int_{x_{A}}^{x_{B}} d s = \int_{x_{A}}^{x_{B}} \sqrt{1 + ẏ^{2}} d x = L

¹ ¹ Note that in this case x is the independent variable and ẏ = d y d x

Therefore the admissible space of functions $𝒟$ is a subset of a general linear function space $𝒳$ that is defined as:

𝒟 : = {y \in 𝒳 : y (x_{A}) = y_{A}, y (x_{B}) = y_{B}, F_{1} [y] = L}

Since we are seeking for a static equilibrium condition, we look for a function $y$ that minimzes the potential energy. In the continuous setting the potential energy of the rope can be expressed as:

F [x] = V [x] = \int_{M} y (x) g d m = \int_{x_{A}}^{x_{B}} y g ρ d s = ρ g \int_{x_{A}}^{x_{B}} y \sqrt{1 + ẏ^{2}} d x

Therefore the catenary problem is:

\begin{aligned} \min_{y (x)} & F [y] = ρ g \int_{x_{A}}^{x_{B}} y \sqrt{1 + ẏ^{2}} d x \\ s . t . & y \in 𝒟 \end{aligned}

pict — Figure 3.2:: The Catenary Problem

A similar problem to the catenary one is Dido’s isoperimetric problem that, according to Virgil’s Aeneid, dates back to the foundation of Carthago around 850 B.C. Dido’s problem is to find the arc that given the endpoints and the length is able to maximize the area.

Example 3.3 (Dido’s Problem). Given a length $L$ , find the curve of length L with fixed endpoints that maximize the area between the curve and the x-axis. The isoperimetric constraint is again:

G [y] = \int_{x_{A}}^{x_{B}} d s = \int_{x_{A}}^{x_{B}} \sqrt{1 + ẏ^{2}} d x = L

While the cost functional to maximize is:

F [y] = \int_{x_{A}}^{x_{B}} y d x