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Ordinary Differential Equations

1.1. Introduction to the theory of ordinary differential equations

1.1.1. Existence-uniqueness of first-order ordinary differential equations

The most important result from the theory of ordinary differential equations ensures the existence and uniqueness of solutions to equations of the form

where

Here I is an open interval of R containing tInit, and O is an open set of RD containing yInit. The variable t ? I is called the time variable, and the variable y is referred to as the state variable. When the function f depends only on the state variable, equation [1.1] is said to be autonomous. We say that a function t ? y(t) is a solution of [1.1] if

1. - y is defined on an interval J that contains tInit and is included in I;
2. - y(tInit) = yInit and for all t ? J, y(t) ? O;
3. - y is differentiable on J and for all t ? J, y´(t) = f(t, y(t)).

THEOREM 1.1 (Picard-Lindelöf1).- Assume that f is a continuous function on I × O and that for every (t?, y?) ? I × O, there exist ?? > 0 and L? > 0, such that B(y?, ??) ? O, [t? - ??, t? + ??] ? I, and if y, z ? B(y?, ??) and |t - t?|= ??, then we have

Then for every (t?, y?) ? I × O, there exist r? > 0 and h? > 0, such that if |yInit - y?| = r? and |tInit - t?| = r?, the problem [1.1] has a solution y :]tInit - h?, tInit + h?[ O, which is a class C1 function.

This solution is unique in the sense that if z is a function of class C1 defined on ]tInit - h?, tInit + h?[ satisfying [1.1], then y(t) = z(t) for all t ?]tInit - h?, tInit + h?[.

Finally, if f is a function of class Ck on I × O, then y is a function of class Ck+1.

This statement calls for a number of comments, which we present in detail here.

1. 1) Theorem 1.1 assumes that the function f satisfies a certain regularity property with respect to the state variable, this property is stronger than mere continuity: f must be Lipschitz continuous in the state variable, at least locally.2 In particular, note that if f is a function of class C1 on I × O, then it satisfies the assumptions of theorem 1.1. This regularity hypothesis cannot be completely ruled out. However, we will see later that it can be relaxed slightly.
2. 2) Theorem 1.1 only defines the solution in a neighborhood of the initial time tInit. Once the question of existence-uniqueness is settled, it can be worthwhile to take interest in the solution's lifespan: is the solution only defined on a bounded interval, or does it exist for all times? We will see that the answer depends on estimates that can be established for the solution of [1.1].

The "classic proof" for the Picard-Lindelöf theorem is based on a fixed point argument that requires the following statement.

THEOREM 1.2 (Banach theorem).- Let E be a vector space with norm ?·?, for which it is assumed that E is complete. Let be a strict contraction mapping, that is to say, such that there exists 0 < k < 1, which satisfies the following inequality for all x, y ? E:

Then, has a unique fixed point in E.

PROOF.- Let us begin by establishing uniqueness, assuming existence: if x and y satisfy and , then , which implies that x = y because 0 < k < 1. In order to show existence, let us examine the sequence defined iteratively by , starting at any x0 ? E. We have

Since 0 < k < 1, the series converges. It follows that the sequence (xn)n?N is Cauchy in the complete space E. So, it has a limit x and by continuity of the mapping , we obtain .

PROOF OF THEOREM 1.1.- The proof of theorem 1.1 is based on a functional analysis argument: the subtle trick works with a vector space whose "points" are functions. In this case, it is important to distinguish between:

1. - the function t ? y(t), which is a point in the functional space (here C0([tInit, T [; RD), for example);
2. - and its value y(t) for a fixed t, which is a point in the state space RD.

We will justify theorem 1.1 in the case where f is globally Lipschitz with respect to the state variable: we assume that f is defined on R × RD and that there exists an L > 0, such that for all x, y ? RD and any t ? R, we have

This technical limitation is important, but enables us to only focus on the key elements of the proof. A proof of the general case can be found in [BEN 10] or [ARN 88, Chapter 4], and later we present a somewhat different approach, which starts from the perspective of numerical approximations. The starting point of the proof is to integrate [1.1] to obtain

such that the solution y of [1.1] is interpreted as a fixed point of the mapping

We will see that this point of view, which transforms from a differential equation to an integral equation (the formulation of [1.3]), is also useful for finding numerical approximations to the solutions of [1.1]. It is now necessary to construct a functional space and a norm in order for to be a contraction. Thus, the sequence defined by y0 given in C0(R; RD) and yn+1 = (yn) will converge to a fixed point of , which will be the solution to [1.1] (Picard method). We only focus on time t = tInit. We introduce the auxiliary function

and we set

with M > 0 that remains to be defined. We denote the subspace of functions z ? C0([tInit, 8[; RD), such that ?z? < 8 as . With norm?·?, this space is complete. If , with z ? , we can write

and deduce that . Indeed, this function t ? y(t) is continuous and satisfies

for every t = tInit. Finally, using integration by parts, we calculate

Thus, we have ?y? < 8. Similarly, we have

By choosing M > L, the mapping appears as a contraction in the complete space . The Banach theorem ensures the existence and uniqueness of a fixed point, since the relation proves that t ? y(t) is a continuous and even C1 function because f is continuous. We can easily adapt the proof in order to expand the resulting solution to time t = tInit. Interestingly, by assuming f is globally Lipschitz continuous with respect to the state variable, see [1.2], it has been possible to directly show that the solution is defined for any time. This fact is important and should be justified on its own.

THEOREM 1.3 (Picard-Lindelöf theorem, assuming global Lipschitz continuity).- Let f be a continuous function defined on R × RD, which satisfies [1.2]. Then, for every yInit ? RD, the equation [1.1] has a unique solution y of class C1 defined on R.

Let us now return to the comments for theorem 1.1 on the regularity of the function f. First, in problem [1.1], if we interpret the equation as [1.3], the assumption that f is continuous in the time variable might be weakened; it suffices to assume integrability. For example, the proof for theorem 1.3 can be slightly modified in order to justify the existence and uniqueness of a fixed point of the mapping in a space of continuous functions on [tInit, 8[ by assuming that there exists t ? L(t), a function locally integrable on [tInit, 8[, such that for every t = tInit, and all x, y ? RD, we have

This hypothesis generalizes [1.2] (which amounts to the case of L(t) = L). We therefore...