Stability analysis of differential algebraic systems

Transcription

1 Stability analysis of differential algebraic systems Volker Mehrmann TU Berlin, Institut für Mathematik with Vu Hoang Linh and Erik Van Vleck DFG Research Center MATHEON Mathematics for key technologies Novi Sad,

2 Introduction Stability analysis for linear differential-algebraic equations DAEs of the form E(t)ẋ = A(t)x + f, with variable coefficients on the half-line I = [0, ). They arise as linearization of nonlinear systems around reference solutions. F(t, x, ẋ) = 0 Stability analysis for DAEs 2 / 58

3 Applications Mechanical multibody systems Electrical circuit simulation Mechatronic systems Chemical reactions Semi-discretized PDEs (Stokes, Navier-Stokes) Automatically generated coupled systems Stability analysis for DAEs 3 / 58

4 Automatic gearboxes Modeling, simulation and software control of automatic gearboxes. Project with Daimler AG (Dissertation: Peter Hamann 2009). Stability analysis for DAEs 4 / 58

5 Technological Application Modeling of multi-physics model: multi-body system, elasticity, hydraulics, friction,.... Development of control methods for coupled system. Real time control of gearbox. Goal: Decrease full consumption, improve smoothness of switching Large control system of nonlinear DAEs. Stability analysis for DAEs 5 / 58

6 Overview 1. Classical theory for ODEs: 2. Extension of theory to DAEs. 3. Numerical computation of spectral intervals for DAEs (Extending work of Dieci/Van Vleck for ODEs). References: V.H. Linh and V.M.. Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations. J. DYNAMICS AND DIFF. EQUATIONS, Vol. 21, , 2009 V.H. Linh and V.M. and E. Van Vleck. QR Methods and Error Analysis for Computing Lyapunov and Sacker-Sell Spectral Intervals for Linear Differential-Algebraic Equations. To appear in ADV. COMP. MATH., 2010 Stability analysis for DAEs 6 / 58

7 Spectral theory for ODEs Classical spectral theory for ODEs ẋ = f (t, x), t I, x(0) = x 0, with x C 1 (I, R n ). By shifting the solution, we may assume that x(t) 0. A constant coefficient system ẋ = Ax with A R n,n is asymptotically stable if all eigenvalues of A have negative real part. Stability analysis for DAEs 7 / 58

8 Asympt. stab. of var. coefficient ODEs Even if for all t R, the matrix A(t) has all eigenvalues in the left half plane, the system ẋ = A(t)x may be unstable. Example For all t R [ cos A(t) = 2 (3t) 5 sin 2 ] (3t) 6 cos(3t) sin(3t) cos(3t)sin(3t) + 3 sin 2 (3t) 5 cos 2 (3t) has a double [ ] eigenvalue 2 but the solution of ẋ = A(t)x, c1 x(0) = is 0 x(t) = [ c1 e t cos(3t) c 1 e t cos(3t) ]. Stability analysis for DAEs 8 / 58

9 Lyapunov exponents For the linear ODE ẋ = A(t)x with bounded coefficient function A(t) and nontrivial solution x we define the upper and lower Lyapunov exponents, λ u (x) = lim sup t 1 t ln x(t), λl (x) = lim inf t 1 t ln x(t). Since A is bounded, the Lyapunov exponents are finite. Theorem (Lyapunov 1907) If the greatest bound of upper Lyapunov exponents for all solutions ẋ = A(t)x is negative, then the system is asymptotically stable. Stability analysis for DAEs 9 / 58

10 Lyapunov exp. of fundamental sol n For the fundamental solution of Ẋ = A(t)X, the Lyapunov exponents for the i-th column of X are λ u (Xe i ), and λ l (Xe i ), i = 1, 2,..., n, (1) where e i denotes the i-th unit vector. W.l.o.g. we assume that the columns of X are ordered such that the upper Lyapunov exponents satisfy λ u (Xe 1 ) λ u (Xe 2 )... λ u (Xe n ). Stability analysis for DAEs 10 / 58

11 Lyapunov spectrum Definition Lyapunov 1907, Perron 1930, Daleckii/Krein 1974 The Lyapunov spectrum Σ L of ẋ = A(t)x is the union of Lyapunov spectral intervals Σ L := n [λ l i, λu i ]. i=1 If each of the Lyapunov spectral intervals shrinks to a point, i.e., if λ l i = λ u i i = 1, 2,..., n, then the system is called Lyapunov-regular. If a system is Lyapunov-regular, then we simply write λ i for the Lyapunov exponents. Stability analysis for DAEs 11 / 58

12 Kinematic similarity Definition A change of variables z = T 1 x with an invertible matrix function T C 1 (I, R n n ) is called a kinematic similarity transformation if T and T 1 are bounded. If Ṫ is bounded as well, then it is called a Lyapunov transformation. Theorem (Perron 1930) For every linear ODE ẋ = A(t)x, there exists a Lyapunov transformation to upper triangular form, and this transformation can be chosen to be pointwise orthogonal. Stability analysis for DAEs 12 / 58

13 Characterization of Lyapunov regularity Theorem (Lyapunov 1907) Let B = [b i,j ] C(I, R n n ) be bounded and upper-triangular. Then the system Ṙ = B(t)R, is Lyapunov-regular if and only if all the limits 1 t lim b i,i (s)ds, i = 1, 2,..., n, t t 0 exist and these limits coincide with the Lyapunov exponents λ i, i = 1, 2,..., n. Stability analysis for DAEs 13 / 58

14 Definition Stability of Lyapunov exponents Consider a homogeneous linear system ẋ = A(t)x with upper Lyapunov exponents λ u i and a perturbed system ẋ = [A(t) + A(t)]x with upper Lyapunov exponents νi u, both decreasingly ordered. The upper Lyapunov exponents, λ u 1... λu n, are called stable if for any ε > 0 there exists a δ = δ(ɛ) > 0 such that sup t 0 A(t) < δ implies λ u i ν u i < ɛ, i = 1,..., n. A fundamental solution matrix X is called integrally separated if for i = 1, 2,..., n 1, there exist b > 0 and c > 0 such that for all t, s with t s. X(t)e i X(s)e i X(s)e i+1 X(t)e i+1 ceb(t s), Stability analysis for DAEs 14 / 58

15 Are Lyapunov exponents stable? Theorem (see e.g. Dieci/Van Vleck 2006) i) Integral separation is invariant under Lyapunov transformations (or kinematic similarity transformations). ii) An integrally separated system has pairwise distinct upper (and pairwise distinct lower) Lyapunov exponents. iii) Distinct upper Lyapunov exponents are stable if and only if there exists an integrally separated fundamental solution matrix. iv) Integral separation is a generic property. Stability analysis for DAEs 15 / 58

16 Computation of Lyapunov exponents Theorem (Dieci/Van Vleck 2002) A system ẋ = B(t)x with B bounded, continuous, and triangular, has an integrally separated fundamental solution matrix iff the diagonal elements of B are integrally separated. If the diagonal of B is integrally separated, then Σ L = Σ CL, where with λ l i,i := lim inf t 1 t t 0 Σ CL := n [λ l i,i, λu i,i ], i=1 b i,i (s)ds, λ u i,i := lim sup t 1 t t 0 b i,i (s)ds, i = 1, 2,..., n. Stability analysis for DAEs 16 / 58

17 Bohl exponents, Piers Bohl Definition Let x be a nontrivial solution of ẋ = A(t)x. The (upper) Bohl exponent κ u B (x) of this solution is the greatest lower bound of all those numbers ρ for which there exist numbers N ρ such that x(t) N ρ e ρ(t s) x(s) for any t s 0. If such numbers ρ do not exist, then one sets κ u B (x) = +. Similarly, the lower Bohl exponent κ l B (x) is the least lower bound of all those numbers ρ for which there exist numbers N ρ such that x(t) N ρe ρ (t s) x(s), 0 s t. The interval [κ l B (x), κu B (x)] is called the Bohl interval of the solution. Stability analysis for DAEs 17 / 58

18 Bohl exponents Theorem (Daleckii/Krein 1974) Bohl and Lyapunov exponents are related via The Bohl exponents are given by κ u B (x) = lim sup s,t s κ l B (x) λl (x) λ u (x) κ u B (x). ln x(t) ln x(s), κ l ln x(t) ln x(s) B (x) = lim inf t s s,t s t s t+1 If A(t) is integrally bounded, i.e., if sup t 0 t A(s) ds <, then the Bohl exponents are finite. Stability analysis for DAEs 18 / 58

19 Lyapunov and Bohl exponents Bohl exponents characterize the uniform growth rate of solutions, while Lyapunov exponents simply characterize the growth rate of solutions departing from t = 0. If the greatest bound of upper Lyapunov exponents for all solutions ẋ = A(t)x is negative, then the system is asymptotically stable. If the same holds for the greatest upper bound of the upper Bohl exponents then the system is (uniformly) exponentially stable. Bohl exponents are stable without any extra assumption. Stability analysis for DAEs 19 / 58

20 Sacker-Sell spectra Definition The fundamental matrix solution X of Ẋ = A(t)X is said to admit an exponential dichotomy if there exist a projector P : R n n R n n and constants α, β > 0, as well as K, L 1, such that X(t)PX 1 (s) Ke α(t s), t s, X(t)(I P)X 1 (s) Le β(t s), t s. The Sacker-Sell (or exponential-dichotomy) spectrum Σ S for is given by those values λ R such that the shifted system ẋ λ = [A(t) λi]x λ does not have exponential dichotomy. The complement of Σ S is called the resolvent set. Stability analysis for DAEs 20 / 58

21 Properties of Sacker-Sell spectra Theorem (Sacker/Sell 1978) The property that a system possesses an exponential dichotomy as well as the exponential dichotomy spectrum are preserved under kinematic similarity transformations. Σ S is the union of at most n disjoint closed intervals, and it is stable. Furthermore, the Sacker-Sell intervals contain the Lyapunov intervals, i.e. Σ L Σ S. Stability analysis for DAEs 21 / 58

22 Characterization of Sacker-Sell spectra Theorem (Dieci/Van Vleck 2006) Consider ẋ = B(t)x with B bounded, continuous, and upper triangular. The Sacker-Sell spectrum of this system and that of the corresponding diagonal system coincide. ẋ = D(t)x, with D(t) = diag(b 1,1 (t),..., b n,n (t)), t 0, Thus, one can retrieve Σ S of ẋ = A(t)x from the diagonal elements of the triangularized system. Stability analysis for DAEs 22 / 58

23 Stability Theory for DAEs Lyapunov theory for regular constant coeff. DAEs Stykel 2002 Index 1 systems Ascher/Petzold 1993, Systems of tractability index 2, Tischendorf 1994, Hanke/Macana/März 1998, Systems with properly stated leading term, Higueras/März/Tischendorf 2003, März 1998, März/Riazza 2002, Riazza 2002, Riazza/Tischendorf Lyapunov exponents and regularity, Cong/Nam 2003/2004. Exponential dichotomy in bound. val. problems, Lentini/März Exponential stability and Bohl exponents, Du/Linh 2006, General theory for linear DAEs Linh/M Perturbation theory Linh/M./Van Vleck 2009 Stability analysis for DAEs 23 / 58

24 A crash course in DAE Theory We put E(t)ẋ = A(t)x + f (t) and its derivatives up to order µ into a large DAE M k (t)ż k = N k (t)z k + g k (t), k N 0 for z k = (x, ẋ,..., x (k) ). M 2 = E 0 0 A Ė E 0 Ȧ 2Ë A Ė E, N 2 = A 0 0 Ȧ 0 0 Ä 0 0, z 2 = x ẋ ẍ. Stability analysis for DAEs 24 / 58

25 Theorem (Kunkel/M. 1996) Under some constant rank assumptions, for a square regular linear DAE there exist integers µ, a, d such that: 1. rank M µ (t) = (µ + 1)n a on I, and there exists a smooth matrix function Z 2 with Z T 2 M µ(t) = The first block column Â2 of Z 2 N µ(t) has full rank a so that there exists a smooth matrix function T 2 such that Â2T 2 = rank E(t)T 2 = d = n a and there exists a smooth matrix function Z 1 of size (n, d) with rank Ê1 = d, where Ê1 = Z T 1 E. Stability analysis for DAEs 25 / 58

26 Reduced problem The quantity µ is called the strangeness-index. It describes the smoothness requirements for the inhomogeneity. It generalizes the usual differentiation index to general DAEs (and counts slightly differently). We obtain a numerically computable strangeness-free formulation of the equation with the same solution. Ê 1 (t)ẋ = Â1(t)x + ˆf 1 (t), d differential equations 0 = Â2(t)x + ˆf 2 (t), a algebraic equations where Â1 = Z T 1 A, ˆf 1 = Z T 1 f, and ˆf 2 = Z T 2 g µ. The reduced system is strangeness-free. This is a Remodelling! For the theory we may assume that this has been done. Stability analysis for DAEs 26 / 58

27 Definition Fundamental solution matrices A matrix function X C 1 (I, R n k ), d k n, is called fundamental solution matrix of E(t)Ẋ = A(t)X if each of its columns is a solution to E(t)ẋ = A(t)x and rank X(t) = d, for all t 0. A fundamental solution matrix is said to be maximal if k = n and minimal if k = d, respectively. A maximal fundamental matrix solution, denoted by X(t, s), is called normalized if it satisfies the projected initial condition E(0)(X(0, 0) I) = 0.. Every fundamental solution matrix has exactly d linearly independent columns and a minimal fundamental matrix solution can be easily made maximal by adding n d zero columns. Stability analysis for DAEs 27 / 58

28 Lyapunov exponents for DAEs Definition For a fundamental solution matrix X of a strangeness-free DAE system E(t)ẋ = A(t)x, and for d k n, we introduce λ u i = lim sup t 1 t ln X(t)e i and λ l 1 i = lim inf t t ln X(t)e i, i = 1, 2,..., k. The λ u i, i = 1, 2,..., n, are called (upper) Lyapunov exponents and the intervals [λ l i, λu i ], i = 1, 2,..., d, are called Lyapunov spectral intervals. The DAE system is said to be Lyapunov-regular if λ l i = λ u i, i = 1, 2,..., d. Stability analysis for DAEs 28 / 58

29 Kinematic equivalence Definition Suppose that W C(I, R n n ) and T C 1 (I, R n n ) are pointwise nonsingular matrix functions such that T and T 1 are bounded. Then the transformed DAE system Ẽ(t) x = Ã(t) x, with Ẽ = WET, Ã = WAT WEṪ and x = T x is called globally kinematically equivalent to E(t)ẋ = A(t)x. If, furthermore, also W and W 1 are bounded then we call this a strong global kinematical equivalence transformation. Stability analysis for DAEs 29 / 58

30 Transformation to triangular form Lemma Consider a strangeness-free DAE E(t)ẋ = A(t)x with continuous coefficients and a minimal fundamental solution matrix X. Then there exist pointwise orthogonal matrix functions U C(I, R n n ) and V C 1 (I, R n n ) such [ that ] in EẊ = AX the change of variables R1 X = VR, with R = and R 0 1 C 1 (I, R d d ) and the multiplication from the left with U T leads to the system E d Ṙ 1 = A d R 1, where E d := U T 1 EV 1 is pointwise nonsingular and A d := U T 1 AV 1 U T 1 E V 1. Here U 1, V 1 are the matrix functions consisting of the first d columns of U, V, respectively. Stability analysis for DAEs 30 / 58

31 Theorem Transformation to block form Consider a reduced strangeness-free DAE system. If the coefficient matrices are sufficiently smooth, then there exists an orthogonal matrix function ˆQ C 1 (I, R n n ) such that with ˆx = ˆQ T x, the submatrix E 1 is compressed, i.e., the transformed system has the form [ Ê ] ˆx = [ Â11 Â 12 Â 21 Â 22 ] ˆx, t 0, Furthermore, this system is still strangeness-free and thus Ê11 and Â22 are pointwise nonsingular. The associated underlying (implicit) ODE is, where Âs := Â11 Â12Â 1 22 Â21. Ê 11 ˆx1 = Âsˆx 1, Stability analysis for DAEs 31 / 58

32 Relation between Lyapunov exponents Theorem Let λ u ( 1 22 Â21) be the upper Lyapunov exponent of the matrix function  1 22 Â21. If the boundedness condition λ u ( 1 22 Â21) 0 holds, then the upper Lyapunov exponents of the transformed DAE and its underlying ODE coincide if they are both ordered decreasingly. Stability analysis for DAEs 32 / 58

33 Perturbed DAE systems Consider perturbed DAEs (E(t) + E(t))ẋ = (A(t) + A(t))x, t 0, with special perturbations ( E, A), E, A C(I, R n n ) that are sufficiently smooth and small enough such that by appropriate orthogonal transformation we obtain [ Ê11 + Ê ] [ Â11 + ˆx = Â11 Â 21 + Â21 Â 12 + Â12 Â 22 + Â22 ] ˆx, t 0. If this is the case then we say that the perturbations are admissible. Stability analysis for DAEs 33 / 58

34 Stability of Lyapunov exponents Definition The upper Lyapunov exponents λ u 1... λu d are said to be stable if for any ɛ > 0, there exists δ > 0 such that the conditions sup t Ẽ(t) < δ, sup t Ã(t) < δ on admissable perturbations imply that the perturbed DAE system is strangeness-free and λ u i γ u i < ɛ, i = 1, 2,..., d, where the γi u are the ordered upper Lyapunov exponents of the perturbed system. A DAE system and an admissably perturbed system are called asymptotically equivalent if they are strangeness-free and lim E(t) = lim A(t) = 0. t t Stability analysis for DAEs 34 / 58

35 Asymptotic equivalence Theorem Suppose that the DAE and an admissibly perturbed DAE are asymptotically equivalent. If the Lyapunov exponents are stable then λ u i = γ u i, for all i = 1, 2,..., d, where again the γ u i are the ordered upper Lyapunov exponents of the perturbed system. Stability analysis for DAEs 35 / 58

36 Integral separation Definition A minimal fundamental solution matrix X for a strangeness-free DAE is called integrally separated if for i = 1, 2,..., d 1 there exist b > 0 and c > 0 such that for all t, s with t s 0. X(t)e i X(s)e i X(s)e i+1 X(t)e i+1 ceb(t s), Stability analysis for DAEs 36 / 58

37 Distinct Lyapunov exponents Proposition Consider a strangeness-free DAE. 1. If the DAE is integrally separated then the same holds for any strongly globally kinematically equivalent system. 2. If the DAE is integrally separated, then it has pairwise distinct upper and pairwise distinct lower Lyapunov exponents. 3. If  1 22 Â21 is bounded, then the DAE is integrally separated if and only if and the underlying ODE is integrally separated. Stability analysis for DAEs 37 / 58

38 Stability of exponents Theorem Consider a strangeness-free DAE and its transformed system, satisfying that  1 22 Â21, Â12 1 22, Ê 11, Ê 1 11 (Â11 Â12 1 22 Â21), are bounded. The DAE has d pairwise distinct upper and pairwise distinct lower Lyapunov exponents and they are stable iff it is integrally separated. If  1 22 Â21, Â12 1 22, Ê11, and Ê 1 11 are bounded, then the system has an integrally separated fundamental solution matrix iff its adjoint has. Stability analysis for DAEs 38 / 58

39 Differences between ODEs and DAEs Example For the DAE system ẋ 1 = x 1 + x 2, 0 = x 1 x 3, ẋ 2 = x 2, 0 = x 2 e t x 4, the underlying ODE is not integrally separated, but the Lyapunov exponents are stable and the minimal fundamental solution X(t) = e t te t 0 e t e t te t 0 e 2t is integrally separated. However, the Lyapunov exponents of the DAE ({1, 2}), are not stable., Stability analysis for DAEs 39 / 58

40 Shifted systems Definition Consider a strangeness-free DAE. For a scalar λ R, the DAE system E(t)ẋ = [A(t) λe(t)]x, t 0, is called a shifted DAE system. The shifted DAE transforms as [ Ê ] [ Â11 λê11 Â ˆx = 12 Â 21 Â 22 ] ˆx, t 0, and clearly, the shifted DAE system inherits the strangeness-free property from the original DAE. Stability analysis for DAEs 40 / 58

41 Definition Exponential Dichotomy A strangeness-free DAE system is said to have an exponential dichotomy if for a maximal fundamental solution ˆX(t), there exists a projection matrix P R d d and constants α, β > 0, and K, L 1 such that [ ] P 0 ˆX(t) ˆX 0 0 (s) Ke α(t s), t s, [ ] ˆX(t) Id P 0 ˆX 0 0 (s) Leβ(t s), t s. Here X (s) is a reflexive generalized inverse. Theorem A strangeness-free DAE system has an exponential dichotomy if and only if in the transformed system  1 22 Â21 is bounded and the underlying ODE has an exponential dichotomy. Stability analysis for DAEs 41 / 58

42 Sacker-Sell spectra Definition The Sacker-Sell (or exponential dichotomy) spectrum of a strangeness-free DAE system is defined by Σ S := {λ R, the shifted DAE does not have an exponential dichotomy}. The complement of Σ S is called the resolvent set. The Sacker-Sell spectrum of a DAE system does not depend on the choice of the orthogonal change of basis that brings it to the transformed system. Stability analysis for DAEs 42 / 58

43 Sacker-Sell spectra Theorem Consider a DAE and suppose that in the transformed DAE Â 1 22 Â21 is bounded. The Sacker-Sell spectrum is exactly the Sacker-Sell spectrum of the underlying ODE. It consists of at most d closed intervals. If the Sacker-Sell spectrum of the DAE system is given by d disjoint closed intervals, then there exists a minimal fundamental solution matrix ˆX with integrally separated columns. In this case it is given exactly by the d (not necessarily disjoint) Bohl intervals associated with the columns of ˆX and Σ L Σ S. Stability analysis for DAEs 43 / 58

44 Stability of Sacker-Sell spectra Theorem Consider a strangeness-free DAE system  1 22 Â21, Â12 1 22, Ê 11, Ê 1 11 (Â11 Â12 1 22 Â21), are bounded and for k d, Σ S = k i=1 [a i, b i ]. Consider an admissible perturbation. Let ε > 0 be sufficiently small such that b i + ε < a i+1 ε for some i, 0 i k. For i = 0 and i = k, set b 0 = and a k+1 =, respectively. Then, there exists δ > 0 such that the inequality max{sup t E(t), sup t A(t) } δ, implies that the interval (b i + ε, a i+1 ε) is contained in the resolvent of the perturbed DAE system. Stability analysis for DAEs 44 / 58

45 Stability of Sacker-Sell spectra Corollary Let the assumptions of the last Theorem hold and let ε > 0 be sufficiently small such that b i 1 + ε < a i ε < a i b i < b i + ε < a i+1 ε, for 0 i k. For i = 0 and i = k, set b 0 = and a k+1 =, respectively. Then, there exists δ > 0 such that the inequality max{sup t Ẽ(t) }, sup Ã(t) δ, implies that the Sacker-Sell interval [a i, b i ] either remains a Sacker-Sell interval under the perturbation or it is possibly split into several new intervals, but the smallest left end-point and the largest right end-point stay in the interval [a i ε, b i + ε]. t Stability analysis for DAEs 45 / 58

46 Numerical methods For the numerical computation we have first have to obtain the strangeness-free form of E(t), A(t). It can be obtained pointwise for every t via the FORTRAN code GELDA Kunkel/M./Rath/Weickert 1997 or the corresponding MATLAB version Kunkel/M./Seidel It comes in the form [ E1 (t) 0 ] [ A1 (t) ẋ = A 2 (t) ] x where A 2 is full row rank. Stability analysis for DAEs 46 / 58

47 Smooth QR factorizations Kunkel/M. 1991, Dieci/Eirola 1999 Suppose that the original DAE has sufficiently smooth coefficients. There exists a pointwise nonsingular, upper triangular matrix function Ã22 C 1 (I, R (n d) (n d) ) and a pointwise orthogonal matrix function Q C 1 (I, R n n ) such that A 2 = [ 0 Ã22 ] ˆQ. To make the factorization unique and to obtain smoothness, we require the diagonal elements of Ã22 to be positive. Alternatively we can derive differential equations for Q (or its Householder factors) and to solve the corresponding initial value problems. Stability analysis for DAEs 47 / 58

48 Transformed DAE The transformation x = Q T x leads to [ ] [ ] [ Ẽ11 Ẽ 12 E1 Ã11 Ã = Q, Ã 22 ] = [ A1 A 2 ] Q [ E1 0 ] Q. To evaluate Q at time t, we may use either a finite difference formula or the smooth QR method derived in Kunkel/M The solution component x 2 associated with the algebraic equations is simply 0, thus we only have to deal with x 1. Stability analysis for DAEs 48 / 58

49 Basic Idea for Computing Exponents Determine for every point t E = [e ij ] = P T Ẽ 11 Q, A = [a ij ] = P T Ã 11 Q P T Ẽ 11 Q = P T Ã 11 Q EQ T Q, such that E, A are upper triangular. Determine strictly lower triangular part of the skew symmetric S(Q) = Q T Q by corresponding part of E 1 P T Ã 11 Q and the remaining part by skew-symmetry. Determine P and E via a smooth QR-factorization Ẽ11Q = PE. Keep orthogonality via orthogonal integrators Hairer/Lubich/Wanner 2002 or projected ODE integrators Dieci/Van Vleck Compute the spectral intervals from E 1 (t)ṙ1 = A 1 (t)r 1, t I, where R 1 is the fundamental solution matrix of the triangularized underlying implicit ODE. Stability analysis for DAEs 49 / 58

50 Continuous QR algorithm Compute a smooth QR factorization of A 2 [ ] [ E1 U1 T Ẽ12 A1 U ż = 1 T Ã Ã 22 with E 1, A 1, Ã22 upper triangular. Apply ODE methods of Dieci/Van Vleck to Compute λ i (t j ) = 1 t j ln[r 1 (t j )] i,i = 1 t j ln E 1 (t)ṙ1 = A 1 (t)r 1, t I, ] z, j [Θ l ] i,i = 1 j ln[θ l ] i,i, i = 1, 2,..., d. t j l=1 Solve optimization problems inf τ t T λ i (t) and sup τ t T λ i (t), i = 1, 2,..., d for a given τ (0, T ). l=1 Stability analysis for DAEs 50 / 58

51 Discrete QR algorithm Take a mesh 0 = t 0 < t 1 <... < t N 1 < t N = T. Compute the fundamental solution X [j] on [t j 1, t j ] by solving EẊ [j] = AX [j], t j 1 t t j, X [j] (t j 1 ) = χ j 1, using the DAE integrator GELDA. Determine QR factorizations [ Q(t j ) T X [j] Zj (t j ) = 0 Compute λ i (t j ) = 1 t j ln[r 1 (t j )] i,i = 1 t j ln ], Z j = Q j Θ j, j = 1, 2,..., N j [Θ l ] i,i = 1 j ln[θ l ] i,i, i = 1, 2,..., d, t j l=1 Solve the optimization problems inf τ t T λ i (t) and sup τ t T λ i (t), i = 1, 2,..., d for a given τ (0, T ). Stability analysis for DAEs 51 / 58 l=1

52 Lyapunov exp. via cont. QR-Euler method Lyapunov regular 2 2 DAE λ 1 = 5, λ 2 = 0. T h λ 1 λ 2 CPU(s) Stability analysis for DAEs 52 / 58

53 Lyapunov exp. via disc. QR-Euler method Lyapunov regular 2 2 DAE λ 1 = 5, λ 2 = 0. T h λ 1 λ 2 CPU(s) Stability analysis for DAEs 53 / 58

54 Lyapunov exp. via cont. QR-Euler method Lyapunov non regular 2 2 DAE with Lyapunov intervals [ 1, 1], [ 6, 4]. T t 0 h [λ l 1, λu 1 ] [λl 2, λu 2 ] CPU(s) [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] Stability analysis for DAEs 54 / 58

55 Sacker-Sell interv. via cont. QR-Euler Sacker-Sell intervals [ 2, 2] and [ 5 2, 5 + 2]. T H h [κ l 1, κu 1 ] [κl 2, κu 2 ] CPU(s) [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] Stability analysis for DAEs 55 / 58

56 Conclusions The classical Theory of Lyapunov/Bohl/Sacker-Sell has been extended to linear DAEs with variable coefficients. Boundedness conditions and strangeness-free formulations are the key tools. In principle we can compute spectral intervals. Numerical methods for computing the Sacker-Sell spectra of DAEs extending work of Dieci/Van Vleck. They are expensive. Perturbation theory of Dieci/Van Vleck for ODEs has been extended and the methods have been modified to deal only with partial exponents (e.g. the largest Sacker-Sell interval) Linh/M./Van Vleck Dec SVD based methods for more accurate spectra. Stability analysis for DAEs 56 / 58

57 Future work Extension to analysis of nonlinear systems Production code for classes special classes of DAEs Extension to operator DAEs/PDEs Stability analysis for DAEs 57 / 58

58 Thank you very much for your attention. Stability analysis for DAEs 58 / 58