On perturbations in the leading coefficient matrix of time-varying index-1 DAEs
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1 On perturbations in the leading coefficient matrix of time-varying index-1 DAEs Institute of Mathematics, Ilmenau University of Technology Darmstadt, March 27, 2012 Institute of Mathematics, Ilmenau University of Technology Page 1 / 11
2 Perturbations in the leading coefficient of DAEs E(t)ẋ = A(t)x, E, A C(R 0 ; R n n ) Institute of Mathematics, Ilmenau University of Technology Page 2 / 11
3 Perturbations in the leading coefficient of DAEs E(t)ẋ = A(t)x, E, A C(R 0 ; R n n ) [März 1991]: (E, A) is index-1 Q C 1 : Q(t) 2 = Q(t) im Q(t) = ker E(t), and D C 0 : Eẋ = Ax d dt (P x) = ( P + P D)P x, Qx = QDP x P = I Q Institute of Mathematics, Ilmenau University of Technology Page 2 / 11
4 Perturbations in the leading coefficient of DAEs E(t)ẋ = A(t)x, E, A C(R 0 ; R n n ) [März 1991]: (E, A) is index-1 Q C 1 : Q(t) 2 = Q(t) im Q(t) = ker E(t), and D C 0 : Eẋ = Ax d dt (P x) = ( P + P D)P x, Qx = QDP x P = I Q ẋ 1 = x 2 0 = x 1 ẋ1 = D 11 x 1 x 2 = D 21 x 1 Institute of Mathematics, Ilmenau University of Technology Page 2 / 11
5 Eẋ = Ax d dt (P x) = ( P + P D)P x, Qx = QDP x Institute of Mathematics, Ilmenau University of Technology Page 3 / 11
6 Eẋ = Ax d dt (P x) = ( P + P D)P x, Qx = QDP x Eẋ = Ax d dt x = P x + Qx = (I + QD)P x, (P x) = ( P + P D)P x Institute of Mathematics, Ilmenau University of Technology Page 3 / 11
7 Eẋ = Ax d dt (P x) = ( P + P D)P x, Qx = QDP x Eẋ = Ax d dt x = P x + Qx = (I + QD)P x, (P x) = ( P + P D)P x ẏ = ( P + P D)y, y(t 0 ) = P (t 0 )x(t 0 ) uniqueness = y(t) = P (t)x(t) Institute of Mathematics, Ilmenau University of Technology Page 3 / 11
8 crucial: E = EP Eẋ = Ax EP ẋ = Ax d dt (P x) = ( P + P D)P x, Qx = QDP x Institute of Mathematics, Ilmenau University of Technology Page 4 / 11
9 crucial: E = EP Eẋ = (A + A )x EP ẋ = (A + A )x d dt (P x) = ( P + P D)P x + P G A x, Qx = QDP x + QG A x G 1 Institute of Mathematics, Ilmenau University of Technology Page 4 / 11
10 crucial: E = EP i.g. (E + E )ẋ = Ax (E + E )P ẋ = Ax Institute of Mathematics, Ilmenau University of Technology Page 4 / 11
11 crucial: E = EP i.g. (E + E )ẋ = Ax (E + E )P ẋ = Ax (A): E C(R 0 ; R n n ) s.t. (E + E, A) is index-1 and ker E(t) = ker(e(t) + E (t)) Institute of Mathematics, Ilmenau University of Technology Page 4 / 11
12 crucial: E = EP i.g. (E + E )ẋ = Ax (E + E )P ẋ = Ax (A): E C(R 0 ; R n n ) s.t. (E + E, A) is index-1 and ker E(t) = ker(e(t) + E (t)) (E + E )ẋ = Ax d dt (P x) = ( P + P D)P x + P G P x, Qx = QDP x + QG P x Institute of Mathematics, Ilmenau University of Technology Page 4 / 11
13 [ ] ) [ ] ( ) In1 0 (ẋ1 A11 (t) A = 12 (t) x1 0 0 A 21 (t) A 22 (t) ẋ 2 x 2 Institute of Mathematics, Ilmenau University of Technology Page 5 / 11
14 [ ] ) [ ] ( ) In1 0 (ẋ1 A11 (t) A = 12 (t) x1 (t) 0 A 21 (t) A 22 (t) ẋ 2 0 = (A 21 A 11 )(t)x 1 + (A 22 A 12 )(t)x 2 x 2 Institute of Mathematics, Ilmenau University of Technology Page 5 / 11
15 [ ] ) [ ] ( ) In1 0 (ẋ1 A11 (t) A = 12 (t) x1 (t) 0 A 21 (t) A 22 (t) ẋ 2 0 = (A 21 A 11 )(t)x 1 + (A 22 A 12 )(t)x 2 [ ] ) 1 0 (ẋ1 = 0 0 ẋ 2 [ ] ( ) x1 x 2 x 2 x 1 (t) = e t x 0 1 x 2 (t) = 0 exp. stable Institute of Mathematics, Ilmenau University of Technology Page 5 / 11
16 [ ] ) [ ] ( ) In1 0 (ẋ1 A11 (t) A = 12 (t) x1 (t) 0 A 21 (t) A 22 (t) ẋ 2 0 = (A 21 A 11 )(t)x 1 + (A 22 A 12 )(t)x 2 [ ] ) 1 0 (ẋ1 = 0 ε ẋ 2 [ ] ( ) x1 x 2 x 2 x 1 (t) = e t x 0 1 x 2 (t) = e t/ε x 0 2 not exp. stable Institute of Mathematics, Ilmenau University of Technology Page 5 / 11
17 Bohl exponent and perturbation operator E(t) d dt Φ(t, t 0) = A(t)Φ(t, t 0 ), P (t 0 )(Φ(t 0, t 0 ) I) = 0. (E, A) exp. stab. µ, M > 0 t t 0 : Φ(t, t 0 ) Me µ(t t 0) Perturbation theory Institute of Mathematics, Ilmenau University of Technology Page 6 / 11
18 Bohl exponent and perturbation operator E(t) d dt Φ(t, t 0) = A(t)Φ(t, t 0 ), P (t 0 )(Φ(t 0, t 0 ) I) = 0. (E, A) exp. stab. µ, M > 0 t t 0 : Φ(t, t 0 ) Me µ(t t 0) k B (E, A) = inf ρ R N ρ > 0 t s : Φ(t, s) N ρ e ρ(t s) } Perturbation theory Institute of Mathematics, Ilmenau University of Technology Page 6 / 11
19 Bohl exponent and perturbation operator E(t) d dt Φ(t, t 0) = A(t)Φ(t, t 0 ), P (t 0 )(Φ(t 0, t 0 ) I) = 0. (E, A) exp. stab. µ, M > 0 t t 0 : Φ(t, t 0 ) Me µ(t t 0) k B (E, A) = inf ρ R N ρ > 0 t s : Φ(t, s) N ρ e ρ(t s) } Rem.: k B (E, A) < 0 (E, A) exp. stable Perturbation theory Institute of Mathematics, Ilmenau University of Technology Page 6 / 11
20 Bohl exponent and perturbation operator E(t) d dt Φ(t, t 0) = A(t)Φ(t, t 0 ), P (t 0 )(Φ(t 0, t 0 ) I) = 0. (E, A) exp. stab. µ, M > 0 t t 0 : Φ(t, t 0 ) Me µ(t t 0) k B (E, A) = inf ρ R N ρ > 0 t s : Φ(t, s) N ρ e ρ(t s) } Rem.: k B (E, A) < 0 (E, A) exp. stable L t0 : L 2 ([t 0, ); R n ) L 2 ([t 0, ); R n ), f( ) x( ), x solves E(t)ẋ = A(t)x + f(t), P (t 0 )x(t 0 ) = 0 Perturbation theory Institute of Mathematics, Ilmenau University of Technology Page 6 / 11
21 Bohl exponent and perturbation operator E(t) d dt Φ(t, t 0) = A(t)Φ(t, t 0 ), P (t 0 )(Φ(t 0, t 0 ) I) = 0. (E, A) exp. stab. µ, M > 0 t t 0 : Φ(t, t 0 ) Me µ(t t 0) k B (E, A) = inf ρ R N ρ > 0 t s : Φ(t, s) N ρ e ρ(t s) } Rem.: k B (E, A) < 0 (E, A) exp. stable L t0 : L 2 ([t 0, ); R n ) L 2 ([t 0, ); R n ), f( ) x( ), x solves E(t)ẋ = A(t)x + f(t), P (t 0 )x(t 0 ) = 0 Lemma [Du et al. 2006]: (E, A) exp. stable and (BC) hold = L t0 is linear bd. operator and t 0 L t0 is mon. nonincreasing Perturbation theory Institute of Mathematics, Ilmenau University of Technology Page 6 / 11
22 Theorem (Robustness of Bohl exponent) (E, A) index-1, Q bounded, given ε > 0: E satisfies (A), E suff. small = k B (E + E, A) k B (E, A) + ε Perturbation theory Institute of Mathematics, Ilmenau University of Technology Page 7 / 11
23 Theorem (Robustness via perturbation operator) (E, A) index-1 and exp. stable, (BC) hold, E satisfies (A): } κ i = κ i (E, A, Q), i = 1, 2, 3, α := min lim L t t 0 0 1, κ 3, lim E [t0, ) < t 0 = (E + E, A) is exponentially stable α κ 1 + κ 2 α Perturbation theory Institute of Mathematics, Ilmenau University of Technology Page 8 / 11
24 Stability radius P := [ E, A ] B(R 0 ; R n 2n ) inf [ E, A ] P (E + E, A + A ) is index-1, ker E(t) = ker(e(t) + E (t)) S := (E, A) C(R 0 ; R n n ) } 2 (E, A) is exponentially stable, [ r(e, A) := [ E, A ] E, A ] P or (E + E, A + A ) S }, } Stability radius Institute of Mathematics, Ilmenau University of Technology Page 9 / 11
25 Proposition (Properties of the stability radius) r(e, A) = 0 (E, A) S r(α(e, A)) = r(αe, αa) = α r(e, A) for all α 0 V(t) time-varying subspace of R n with constant dimension, K V := [E, A] B(R 0 ; R n 2n ) (E, A) is index-1, ker E(t) = V(t) = K V [E, A] r(e, A) is continuous }, Stability radius Institute of Mathematics, Ilmenau University of Technology Page 10 / 11
26 Theorem (Lower bound for the stability radius) (E, A) index-1 and exp. stable, (BC) holds = } κ i = κ i (E, A, Q), i = 1, 2, 3, α := min lim L t t 0 0 1, κ 3, α r(e, A) κ 1 + κ 2 α Stability radius Institute of Mathematics, Ilmenau University of Technology Page 11 / 11
27 Theorem (Lower bound for the stability radius) (E, A) index-1 and exp. stable, (BC) holds = } κ i = κ i (E, A, Q), i = 1, 2, 3, α := min lim L t t 0 0 1, κ 3, α r(e, A) κ 1 + κ 2 α Cor.: V(t) time-varying subspace of R n with constant dimension, S V := [E, A] B(R 0 ; R n 2n ) (E, A) is index-1 and exp. stable, ker E(t) = V(t) for all t R 0 } = S V is open in K V Stability radius Institute of Mathematics, Ilmenau University of Technology Page 11 / 11
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