Exact linear lumping in abstract spaces

Zoltán RÓZSA and János TÓTH

Department of Analysis,

Budapest University of Technology and Economics Budapest, Egry J. u. 1., H-1111 HUNGARY

Contents

1 Introduction 2 2 Basic notions 3 3 Lumpability 4 4 Invariant sets 5 5 Examples 6 6 Acknowledgements 9

1 Introduction

• Lumping in infinite dimensional abstract spaces • Apply it to partial differential equation system • Functional analysis

• Sufficient condition for exact lumpability

• Examples

2 Basic notions

X, X̂ Hilbert-spaces W ⊆ X : W ≅ X̂ K ∈ L(X) u : ℝ → X is a solution of

u̇ = Ku.  (1)

Statement 1

\begin{matrix}k, l : X → X̂ \\ ∀u : R → X k ∘ u = l ∘ u \\ ∪_{u}Range(u) = X \end{matrix} \} ⇒ k = l

Definition 1 Exact lumping M : X → X̂ bounded, linear Range(M) = X̂. K̂ ∈ L(X̂)

\begin{matrix}u̇ = Ku \\ û := Mu \end{matrix} \} ⇒ ˙ û = K̂û

Lumped equation:

˙û = K̂û  (2)

3 Lumpability

Definition 2 Generalized inverse of M M ∈ L(X, X̂) \overline{M} ∈ L(X̂, X) :

M\overline{M} = IX̂

Lemma 1 M ∈ L(X, X̂), Range(M) = X̂ ⇒

\overline{M} = M^{∗}(MM^{∗})^{−1}

Lemma 2 W ⊂ X, K ∈ L(X) W is invariant for K ⇔ W^{⊥} is invariant for K^{∗}

Theorem 1 Equivalent statements

a) (1) is lumpable to (2) by M
b) MK = K̂M
c) MK = MK\overline{M}M
d) K(Ker(M)) ⊆ Ker(M)
e) Ker(M)^{⊥} is invariant for K^{∗}

Theorem 2 (1) is lumpable to (2) by M ⇒

MK\overline{M} = MKM̃

Remark 1 Invariant subspaces of X under K are to be determined.

4 Invariant sets

Definition 3 Induced dynamical system ψ : R × X → X, t ↦ ψ(t, u_{0}) solution of (1) and u(0) = u_{0} ψ_{t} := ψ(t, .)

Theorem 3 The flow induced by (2)

ψ̂_{t} = Mψ_{t} ∘ \overline{M},  ψ̂_{t} ∘ M = Mψ_{t}

Definition 4 Invariant sets under ψ_{t} S ⊂ X ψ_{t}(S) ⊂ S for all t ∈ R.

Theorem 4 Invariant sets of (1) are lumped into invariant sets of (2).

Definition 5 Equilibrium K ∈ L(X) u^{∗} ∈ X

Ku^{∗} = 0

Theorem 5 Equilibria are lumped into equilibria.

Theorem 6 Periodic solutions of (1) are lumped into periodic solutions of (2).

Theorem 7 Lumped spectrum X, X̂ Hilbert-spaces, X̂ ⊂ X M ∈ L(X, X̂), Range(M) = X̂ A ∈ L(X̂), B ∈ L(X).

AM = MB ⇒ σ(A) ⊂ σ(B)

5 Examples

Example 1 Reaction A + U ⟶ 2U + 2V V ⟶ U u(t, x) := [U](t, x) v(t, x) := [V](t, x) u, v : R^{+} × [0, 1] → R

∂_{0}u = u + v, ∂_{0}v = 2u − v

Ω := [0, 1] u, v : R^{+} → L^{2}(Ω)

u̇ = u + v, v̇ = 2u − u

w = ( \begin{matrix}u \\ v \end{matrix} ) K = ( \begin{matrix}1 & 1 \\ 2 & −1 \end{matrix} )

ẇ = Kw

X = L^{2}(Ω) × L^{2}(Ω) X̂ = L^{2}(Ω)

M = ( 2 \sqrt{3} − 1 ) K̂ = \sqrt{3}

˙ŵ = \sqrt{3}ŵ.

Example 2 Reaction-Diffusion Ω := [−π, π] u, v : R^{+} → L^{2}(Ω)

u̇ = Δu + u + v v̇ = Δv + 2u − u

w := ( \begin{matrix}u \\ v \end{matrix} ) Kw = (Δu + u + v, Δv + 2u − v)

ẇ = Kw

X := L^{2}(Ω) × L^{2}(Ω) X̂ := L^{2}(Ω)

V := \{v|v = ( \begin{matrix}x ↦ a_{0} + ∑_{k=1}^{∞} (a_{2k} cos(2kx) + b_{2k} sin(2kx)) \\ x ↦ c_{0} + ∑_{k=1}^{∞} (c_{2k} cos(2kx) + d_{2k} sin(2kx)) \end{matrix} ) \}

M ( \begin{matrix}x ↦ a_{0} + ∑_{n=1}^{∞} (a_{n} cos(nx) + b_{n} sin(nx)) \\ x ↦ c_{0} + ∑_{n=1}^{∞} (c_{n} cos(nx) + d_{n} sin(nx)) \end{matrix} ) :=

x ↦ ∑_{k=1}^{∞} c_{2k+1} cos(2kx) + d_{2k+1} sin(2kx) +

+a_{2k+1} cos((2k + 1)x) + d_{2k+1} sin((2k + 1)x)

M ∈ L(X, X̂) V = Ker(M)

M^{∗} (a_{0} + ∑_{n=1}^{∞} a_{n} cos(nx) + b_{n} sin(nx)) =

∑_{k=1}^{∞} (a_{2k−1} cos((2k − 1)x) + b_{2k−1} sin((2k − 1)x))

2a_{0} cos(x) + ∑_{k=1}^{∞} (a_{2k} cos((2k + 1)x) + b_{2k} sin((2k + 1)x))

\overline{M} = M^{∗}(MM^{∗})^{−1} ŵ := x ↦ (a_{0} + ∑_{n=1}^{∞} a_{n} cos(nx) + b_{n} sin(nx))

\overline{M}ŵ(x) =

∑_{k=1}^{∞} (a_{2k−1} cos((2k − 1)x) + b_{2k−1} sin((2k − 1)x))

a_{0} cos(x) + ∑_{k=1}^{∞} (a_{2k} cos((2k + 1)x) + b_{2k} sin((2k + 1)x))

K̂ŵ(x) = MK\overline{M}ŵ(x) =

( − 2a_{0} + 2a_{1} + a_{0} cos(x) + ∑_{k=1}^{∞} α_{2k} cos((2k)x) + β_{2k} sin((2k)x) +

α_{2k+1} cos((2k + 1)x) + β_{2k+1} sin((2k + 1)x))

\begin{aligned}α_{2k+1} & := & a_{2k} + a_{2k+1} − (2k + 1)^{2}a_{2k+1} \\ β_{2k+1} & := & b_{2k} + b_{2k+1} − (2k − 1)^{2}b_{2k+1} \\ α_{2k} & := & 2a_{2k+1} − a_{2k} − (2k + 1)^{2}a_{2k} \\ β_{2k} & := & 2b_{2k+1} − b_{2k} − (2k + 1)^{2}b_{2k} \end{aligned}

6 Acknowledgements

The authors thank Professor M. Horváth and Z. Sebestyén for their help, and the French Ministry for Territorial Planning and Environment BCRD AB2001, and for the National Scientific Research Foundation (No. T037491) for partial financial support.