Exponential Dichotomies by Ekeland’s Variational Principle

Abstract

We give a new proof for an enhanced version, due to Coppel, of the well known functional characterization of exponential dichotomies. The approach is especially new for scalar equations and is based on the Ekeland Variational Principle, that replaces the role of the Closed Graph Theorem in the modern style proofs: this fact allows to recover the spirit of the Perron original work on stability, providing a simple proof with a clear dynamical meaning.

Appendices

Ekeland’s Variational Principle

The following theorem was stated and proved by Ekeland in [6].

Theorem A.1

([6]) Let X be a complete metric space and $\varphi :X\rightarrow {\mathbb {R}}$ a lower semi-continuous function with $L=\inf _X \varphi >-\infty$. For every $\varepsilon , \lambda >0$ and $u\in X$ with $\varphi (u)\le L+\varepsilon$ there exists $v\in X$ such that:

• (1) $\varphi (v)\le \varphi (u)$

• (2) $d(u,v)\le \lambda$

• (3) $\varphi (w) > \varphi (v) - (\varepsilon /\lambda )d(w,v)$ for every $w\not = v$

A similar result holds for upper semi-continuous functions, with the greatest lower bound replaced by the least upper bound. Here we are only interested in a well known corollary of the Ekeland Variational Principle, which we prove hereafter just for the reader convenience. Suppose that:

$$\begin{aligned} X=J\quad \hbox {is a } \textit{ closed } \text { interval of }{\mathbb {R}}\end{aligned}$$

and $\varphi$ is continuous on J and differentiable in the interior of J. Suppose moreover that, for reasons which are external to the theorem, we know that the point v in the statement fulfills:

$$\begin{aligned} v\not \in \partial J\;. \end{aligned}$$

(A.1)

Setting $\lambda =\sqrt{\varepsilon }$, the claim (3) reads:

$$\begin{aligned} \frac{\varphi (w)-\varphi (v)}{|w-v|}\ >\ -\sqrt{\varepsilon }\qquad \forall w\not =v\;. \end{aligned}$$

Passing to the limit as $w\rightarrow v^\pm$ we get:

$$\begin{aligned} -\sqrt{\varepsilon }\ \le \ D^+\varphi (v)\qquad \qquad D^-\varphi (v)\ \le \ \sqrt{\varepsilon } \end{aligned}$$

(A.2)

Summing up, claim (3) implies:

$$\begin{aligned} |\varphi '(v)| \le \sqrt{\varepsilon }\;. \end{aligned}$$

When on the contrary $v\in \partial J$, only one of the two estimates (A.2) holds true and the smallness of the derivative (admitting that it does exist) is no longer guaranteed. Notice that (A.1) is automatically satisfied for small values of $\varepsilon$ when, for instance:

$$\begin{aligned} \inf _J\varphi \ <\ \inf _{\partial J}\varphi \;. \end{aligned}$$

A similar condition is actually the typical assumption in the applications, also for higher dimensional X’s: see [6] or [5].

Exponential Dichotomy for Block Triangular Equations

Assume that:

$$\begin{aligned} A, b, c\;\in \; BC({\mathbb {R}}^+) \end{aligned}$$

where A(t) is a $N\times N$ matrix, c(t) a N vector and b(t) a scalar, for every $t\ge 0$. Then consider the lower block-triangular equation:

$$\begin{aligned} \left( \begin{array}{c} {\dot{x}} \\ {\dot{z}} \\ \end{array} \right) \ =\ \left( \begin{array}{cc} A(t) &{} 0 \\ c(t)^T &{} b(t) \\ \end{array} \right) \left( \begin{array}{c} x \\ z \\ \end{array} \right) \end{aligned}$$

(B.1)

together with its block-diagonal equations:

$$\begin{aligned} {\dot{x}}= & {} A(t) x \end{aligned}$$

(B.2)

$$\begin{aligned} {\dot{z}}= & {} b(t) z \end{aligned}$$

(B.3)

The following result is used in Section 4, and precisely in the proof of Theorem 4.1.

Lemma B.1

If equations (B.2)–(B.3) have separately an exponential dichotomy on ${\mathbb {R}}^+$, then Eq. (B.1) has also an exponential dichotomy on ${\mathbb {R}}^+$.

The converse is also true, but is not of interest here. The result is known in the literature and the quickest proof probably consists in reducing the non-trivial off-diagonal block by means of the change of variables:

$$\begin{aligned} x = \varepsilon \tilde{x}\qquad \quad z = \tilde{z} \end{aligned}$$

and then using the roughness property of the exponential dichotomy. This property is sometimes proved via the functional characterization expressed in Theorem 1.1, which we want to avoid, but a direct proof is available in Coppel’s book [3]. In any case, the point is that roughness is a much deeper property than Lemma B.1 and, of course, much more difficult to prove. In order to keep reasonably simple our way to Theorem 4.1, we give a direct proof of the lemma in this appendix. However, it has to be mentioned that a direct proof of a much more general result is also available in the beautiful paper [1, 10], where Battelli and Palmer investigate the relationships between the exponential dichotomy of a block-triangular system and the corresponding block-diagonal one: our proof is just a specialized version of their proof.

Proof

To fix notations, we denote X(t) the principal solution of (B.2), that is the only fundamental solution satisfying $X(0) = I$, and assume that the corresponding projector P and dichotomy constants $K, \alpha$ are those involved in the inequalities (2.2) for $J={\mathbb {R}}^+$. Concerning the exponential dichotomy of the scalar equation (B.3) we set:

$$\begin{aligned} \widehat{b}(t) = \int _0^t b(\tau ) \end{aligned}$$

and assume the existence of constants $H,\beta >0$ such that:

$$\begin{aligned} e^{\widehat{b}(t)-\widehat{b}(s)}\ \le \ H\, e^{\beta |t-s|}\qquad \qquad \left\{ \begin{array}{c} \hbox {for every}\ \ t\ge s\ge 0 \\ \hbox {or} \\ \hbox {for every}\ \ s\ge t\ge 0 \end{array} \right. \end{aligned}$$

(B.4)

the upper alternative describing the stable case, while the unstable one is given by the lower alternative.

Let us now consider the triangular equation (B.1). A direct computation shows that the principal solution and its inverse are respectively written as:

$$\begin{aligned} Y = \left( \begin{array}{cc} X &{} 0 \\ e^{\widehat{b}} u^T &{} e^{\widehat{b}} \\ \end{array} \right) \qquad \qquad \quad Y^{-1} = \left( \begin{array}{cc} X^{-1} &{} 0 \\ - u^T X^{-1} &{} e^{-\widehat{b}} \\ \end{array} \right) \end{aligned}$$

where we set:

$$\begin{aligned} u(t)^T = \int _0^t e^{-\widehat{b}(\tau )} c(\tau )^T X(\tau )\, d\tau \;. \end{aligned}$$

Our goal is to find a projector Q on ${\mathbb {R}}^{N+1}$ and two constants $L\,,\gamma >0$ such that the two exponential estimates:

$$\begin{aligned} \left\| Y(t) Q Y(s)^{-1} \right\|\le & {} L\, e^{-\gamma (t-s)} \quad \forall t\ge s \ge 0 \end{aligned}$$

(B.5)

$$\begin{aligned} \left\| Y(t) (I-Q) Y(s)^{-1} \right\|\le & {} L\, e^{-\gamma (s-t)}\quad \forall s\ge t\ge 0 \end{aligned}$$

(B.6)

are satisfied. Recall from Sect. 2 that the dynamics only determines $\hbox {Im}\,Q$ as the subspace of the initial conditions $y_0 = (x_0, z_0)$ giving rise to solutions of (B.1) that are bounded in ${\mathbb {R}}^+$, and that we may take as Q any projector having this image.

Let us then compute the bounded solutions of (B.1). The equation for the x component coincides with (B.2) and its bounded solutions read $x(t) = X(t)P x_0$ with $x_0\in {\mathbb {R}}^N$. There remains to understand when the solutions of the equation for the z component, namely:

$$\begin{aligned} z(t)\ =\ e^{\widehat{b}(t)} u(t)^T P x_0 + e^{\widehat{b}(t)}z_0 \end{aligned}$$

(B.7)

are bounded too. If Eq. (B.3) is stable, then the second term in the above expression for z(t) is bounded for every $z_0\in {\mathbb {R}}$. Moreover, the first term can be estimated as follows:

$$\begin{aligned} \begin{array}{ll} \left| e^{\widehat{b}(t)} u(t)^T P x_0\right| \ &{}=\displaystyle \ \left| \int _0^t e^{\widehat{b}(t)-\widehat{b}(\tau )} c(\tau )^T X(\tau )P x_0\, d\tau \right| \\ &{}\le \displaystyle \ \Vert c\Vert _\infty \, \Vert XP x_0\Vert _\infty \, \int _0^t H e^{-\beta (t-\tau )}\, d\tau \ \le \ \Vert c\Vert _\infty \, \Vert XP x_0\Vert _\infty \,\frac{H}{\beta } \end{array} \end{aligned}$$

for every $t\ge 0$ and then is also bounded in ${\mathbb {R}}^+$ for every $x_0\in {\mathbb {R}}^N$.

On the contrary when Eq. (B.3) is unstable, one has $\widehat{b}(+\infty )=+\infty$. We can check the existence of a unique value $z_0$ so that expression in (B.7) is bounded. Let us observe that the function in (B.7) is a solution of

$$\begin{aligned} z'=b(t)z+c(t)^TX(t)Px_0 \end{aligned}$$

This linear differential equation has bounded coefficients since: $\vert X(t)P\vert \le k$ by the exponential dichotomy of (B.2) and, the corresponding homogeneous equation has an unstable exponential dichotomy. This bounded solution has the expression

$$\begin{aligned} z(t)=-\int _t^{+\infty } e^{\widehat{b}(t)-\widehat{b}(\tau )} c(\tau )^T X(\tau )P x_0\, d\tau , \end{aligned}$$

therefore

$$\begin{aligned} z_0=-\int _0^{+\infty } e^{\widehat{b}(t)-\widehat{b}(\tau )} c(\tau )^T X(\tau )P x_0\, d\tau , \end{aligned}$$

and $z_0=-\xi ^T x_0$, where

$$\begin{aligned} \xi ^T:=\int _0^{+\infty } e^{-\widehat{b}(\tau )} c(\tau )^T X(\tau )P\, d\tau . \end{aligned}$$

(B.8)

Actually, the previous conclusion can be reversed. Indeed, if we take z(t) as in (B.7), with $z_0=-\xi ^T x_0$ and $\xi$ as in (B.8), then:

$$\begin{aligned} \begin{array}{ll} |z(t)| &{}=\displaystyle \left| -\int _t^{+\infty } e^{\widehat{b}(t)-\widehat{b}(\tau )} c(\tau )^T X(\tau )P x_0\, d\tau \right| \\ &{}\le \displaystyle \Vert c\Vert _\infty \, \Vert XP x_0\Vert _\infty \, \int _t^{+\infty } H e^{-\beta (\tau -t)}\, d\tau \ \le \ \Vert c\Vert _\infty \, \Vert XP x_0\Vert _\infty \,\frac{H}{\beta } \end{array} \end{aligned}$$

for every $t\ge 0$ and hence z(t) is bounded in ${\mathbb {R}}^+$. Summing up, the initial data of the bounded solutions are:

$$\begin{aligned} \left( \begin{array}{c} Px_0 \\ z_0 \\ \end{array} \right) = \left( \begin{array}{cc} P &{} 0 \\ 0^T &{} 1 \\ \end{array} \right) \left( \begin{array}{c} x_0 \\ z_0 \\ \end{array} \right) \qquad \hbox {or}\qquad \left( \begin{array}{c} Px_0 \\ -\xi ^T x_0 \\ \end{array} \right) = \left( \begin{array}{cc} P &{} 0 \\ -\xi ^T &{} 0 \\ \end{array} \right) \left( \begin{array}{c} x_0 \\ z_0 \\ \end{array} \right) \end{aligned}$$

depending on the stable or unstable character of Eq. (B.3). Since the identity $\xi ^TP=\xi ^T$ holds true, the two matrices involved in the above formula are idempotent: thus they define the projector Q we were looking for.

There only remains to prove that (B.5)–(B.6) are satisfied for some suitable choice of $L, \gamma >0$. Now the computations are the same as paper [1], Theorem 2 which is a generalized version of this result. $\square$