Local Exponential Decay Toward Blocked Integral Surfaces

Abstract

Suppose Σ is an n-dimensional integral surface in Y, that is, an n-dimensional manifold without boundary that is positively invariant. Let, for each u ∈Σ P(u) denote the projector on the tangent space T _u (Σ) to Σ at u. Let us assume that the surface is blocked in the sense that

$$\lambda \left( {P\left( u \right)} \right) > \frac{{{\lambda _n} + {\lambda _{n + 1}}}}{2}{\text{ for all }}u$$

((7.1))

and that λ _n = Λ _m which satisfies condition (3.13). Let us consider u _o ∈ H and assume that the distance between u_o and Σ is attained at some u₁ ∈ Σ Then, clearly P(u₁)(u₁ − u₁)= 0. Let us consider the trajectories S(t)u_o, S(t)u₁. Their difference w(t) = S(t)u_o − S(t)u₁ satisfies (4.1). Denoting Λ(t) = (Aw(t), w(t))|w(t)|², we have as in Chapter 4:

$$\frac{d}{{dt}}|w\left( t \right){|^2}\left( {{k_4}\Lambda \left( t \right) - {k_7}} \right)|w{|^2} \leqslant 0.$$

((7.2))