Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Mathematics for IIT JAM, CSIR NET, UGC NET

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Mathematics : Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

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1 The second variation

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev be a nonlinear functional, with x(a) = A and x(b) = B fixed. As usual, we will assume that F is as smooth as necessary.
The first variation of J is

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

where h(t) is assumed as smooth as necessary and in addition satisfies h(a) = h(b) = 0. We will call such h admissible.

The idea behind finding the first variation is to capture the linear part of the J [x]. Specifically, we have

J[x + εh] = J [x] + εδJx[h] + o(ε),

where o(ε) is a quantity that satisfies

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

The second variation comes out of the quadratic approximation in ε,

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

It follows that

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

To calculate it, we note that

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Applying the chain rule to the integrand, we see that

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

where the various derivatives of F are evaluated at Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev Setting ε = 0 and inserting the result in our earlier expression for the second variation, we obtain

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Note that the middle term can be written as Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRevUsing this in the equation above, integrating by parts, and employing h(a) = h(b) = 0, we arrive at

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

2 Legendre’s trick

Ultimately, we are interested in whether a given extremal for J is a weak (relative) minimum or maximum. In the sequel we will always assume that the function x(t) that we are working with is an extremal, so that x(t) satisfies the Euler-Lagrange equation, Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRevmakes the first variation δJx[h] = 0 for all h, and fixes the functions Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

To be definite, we will always assume we are looking for conditions for the extremum to be a weak minimum. The case of a maximum is similar.
Let’s look at the integrand Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev It is generally true that a function can be bounded, but have a derivative that varies wildly. Our intuition then says that Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev  is the dominant term, and this turns out to be true. In looking for a minimum, we recall that it is necessary that δ2Jx [h] ≥ 0 for all h. One can use this to show that, for a minimum, it is also necessary, but not sufficient, that P ≥ 0 on [a, b]. We will make the stronger assumption that P > 0 on [a, b]. We also assume that P and Q are smooth.

Legendre had the idea to add a term to δ2J to make it nonnegative. Specifically, he added Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev to the integrand in (1). Note that Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev Hence, we have this chain of equations,

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

where we completed the square to get the last equation. If we can find w(t) such that

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

then the second variation becomes

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Equation (4) is called a Riccati equation. It can be turned into the second order linear ODE below via the substitution Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

which is called the Jacobi equation for J . Two points t = α and Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRevNecessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev are said to be conjugate points for Jacobi’s equation if there is a solution u to (6) such that Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev between α and Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev and such that Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

When there are no points conjugate to t = a in the interval [a, b], we can construct a solution to (6) that is strictly positive on [a, b]. Start with the two linearly indepemdent solutions u0 and u1 to (6) that satsify the initial conditions

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Since there is no point in [a, b] conjugate a, u0 (t) ≠ 0 for any a < t ≤ b. In particular, since u˙ 0 (a) = 1 > 0, u(t) will be strictly positive on (a, b]. Next, because u1(a) = 1, there exists t = c, a < c ≤ b, such that u1(t) ≥ 1/2 on [a, c]. Moreover, the continuity of u0 and u1 on [c, b] implies that minc≤t≤b u0(t) = m0 > 0 and minc≤t≤b u1(t) = m1 ∈ R. It is easy to check that on [a, b],

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

and, of course, u solves (6).

This means that the substitutuion Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev yields a solution to the Riccati equation (4), and so the second variation has the form given in (5).
It follows that δ2Jx[h] ≥ 0 for any admissible h. Can the second variation vanish for some h that is nonzero? That is, can we find an admissible h ≡ 0 such that δ2Jx[h] = 0? If it did vanish, we would have to have

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

and, since P > 0, this implies that Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev This first order linear equation has the unique solution,

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

However, since h is admissible, h(a) = h(b) = 0, and so h(t) ≡ 0. We have proved the following result.

Proposition 2.1. If there are no points in [a, b] conjugate to t = a, the the second variation is a positive definite quadratic functional. That is, δ2Jx [h] > 0 for any admissible h not identical ly 0.

3 Conjugate points

There is direct connection between conjugate points and extremals. Let x(t, ε) be a family of extremals for the functional J depending smoothly on a parameter ε. We will assume that x(a, ε) = A, which will be independent of ε. These extremals all satisfy the Euler-Lagrange equation

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

If we differentiate this equation with respect to ε, being careful to correctly apply the chain rule, we obtain

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Cancelling and rearranging terms, we obtain

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Set ε = 0 and let u(t) Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRevObserve that the functions in the equation above, which is called the variational equation, are just Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev and Q = Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Consequently, (7) is simply the Jacobi equation (6). The difference here is that we always have the initial conditions,

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

We remark that if u˙ (a) = 0, then u(t) ≡ 0.

What do conjugate points mean in this context? Suppose that Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev is conjugate to t = a. Then we have

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

which holds independently of how our smooth family of extremals was constructed. It follows that at Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev, we have Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev. Thus, the family either crosses again at Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev or comes close to it, accumulating to order higher than ε there.

4 Sufficient conditions 

A sufficient condition for an extremal to be a relative minimum is that the second variation be strongly positive definite. This means that there is a c > 0, which is independent of h, such that for all admissible h one has

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

where H1 = H1[a, b] denotes the usual Sobolev space of functions with distributional derivatives in L2 [a, b].
Let us return to equation (2), where we added in terms depending on an arbitrary function w. In the integrand there, we will add and subtract Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev where σ is an arbitary constant. The only requirement for now is that 0 < σ < mint∈[a,b] P (t). The result is

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

For the first integral in the term on the right above, we repeat the argument that was used to arrive at (5). Everything is the same, except that P is replaced by P − σ. We arrive at this:

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

We continue as we did in section 2. In the end, we arrive at the new Jacobi equation,

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

The point is that if for the Jacobi equation (6) there are no points in [a, b] conjugate to a, then, because the solutions are continuous functions of the parameter σ, we may choose σ small enough so that for (9) there will be no points conjugate to a in [a, b]. Once we have fouund σ small enough for this to be true, we fix it. We then solve the corresponding Riccati equation and employ it in (8) to obtain

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Now, for an admissble h, it is easy to show that Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRevthat we have

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Consequently, we obtain this inequality:

Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

which is what we needed for a relative minimum. We summarize what we found below.

Theorem 4.1. A sufficient condition for an extremal x(t) to be a relative minimum for the functional Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev where x(a) = A and x(b) = B , is that P (t) Necessary and Sufficient Conditions for Extrema - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev and that the interval [a, b] contain no points conjugate to t = a.

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