The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro The Computable Differential Equation Lecture ... - Bruce E. Shapiro
172 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS we conclude that v = (I − ND + · · · + N k−1 D k−1 )h (9.60) = h − Nh ′ + N 2 h ′′ − · · · + N k−1 h ( k − 1) (9.61) In other words, equation 9.55 is not a differential equation at all, despite the presence of the derivative, but is an algebraic equation for v. This is sometimes referred to as a hidden constraint of the DAE. Two key points emerge from the preceding discussion. First, we see that DAE’s may have hidden constraints, namely, part of the solution may be fully determined independently of any initial conditions. Second, two many initial conditions can over-determine the system. If too many initial conditions are provided, that is, more than are necessary to determine u, then there may not be any solution at all. Theorem 9.3. Let Ay ′ + By = f be a solvable linear constant coefficient DAE with index k ≥ 1. Then (A + λB) −1 has a pole of order k at λ = 0, and (A + λB) −1 A has pole of order k − 1 at λ = 0. Proof. We will give the proof for the first statement. The second is left as an exercise. Hence I = Q −1 Q (9.62) = Q −1 (A + λB) −1 (A + λB)Q (9.63) = Q −1 (A + λB) −1 P −1 P (A + λB)Q (9.64) = Q −1 (A + λB) −1 P −1 (P AQ + λP BQ) (9.65) Q −1 (A + λB) −1 P −1 = (P AQ + λP BQ) −1 (9.66) (A + λB) −1 = Q(P AQ + λP BQ) −1 P (9.67) ( ) −1 I + λC 0 = Q P (9.68) 0 N + λI ( ) (I + λC) −1 0 = Q 0 (N + λI) −1 P (9.69) Since (this can be verified by multiplying out the product, using the fact that N k = 0) I = 1 [I − 1 λ λ N + 1 ] λ 2 N 2 + · · · + (−1)k−1 λ k−1 N k−1 [N + λI] (9.70) we know that (N + λI) −1 = 1 λ ∑k−1 i=0 −1 i λ i N i (9.71) Hence A + λB has elements that are proportional to 1/λ k . This gives a pole of oder k when λ → 0. Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 173 9.3 General Linear Differential Algebraic Equations In the general linear system the matrices A and B may be functions of t, A(t)y ′ (t) + B(t)y(t) = f(t) (9.72) Then theorem 9.1 is still valid, but theorem 9.2 is not. In general, the index of the pencil and the index of the DAE are not the same! Definition 9.7. Let A(t)y ′ + B(t)y = f be a linear time-varying DAE with pencil λA(t) + B(t). We say the DAE is a regular DAE if the pencil is regular for all values of t. Definition 9.8. Let A(t)y ′ + B(t)y = f be a linear time-varying DAE with pencil λA(t) + B(t).Then the local index of the pencil is the index of the pencil at a given time t. Theorem 9.4. Let A(t)y ′ + B(t)y = f be a solvable linear time-varying DAE with pencil λA(t) + B(t).Then the following are equivalent: 1. A(t)y ′ + B(t)y = f is an ordinary differential equation (system). 2. The index of the DAE A(t)y ′ + B(t)y = f is zero. 3. The local index of the pencil λA(t) + B(t) is zero for all t. 4. The local index of the pencil is zero for some t = t 0 . Theorem 9.5. Let A(t)y ′ + B(t)y = f be a solvable linear time-varying DAE with pencil λA(t) + B(t). Then the index of the pencil is one if and only if the index of the DAE is one. Definition 9.9. Let A(t)u ′ (t)+B(t)u(t) = f be a solvable linear time-varying DAE. An analytically equivalent transformation of the DAE is given by 1. Pre-multiply the DAE by a non-singular matrix P (t), P Au ′ + P Bu = P f (9.73) 2. Make the change of variables u = Q(t)y, for some non-singular matrix Q(t) P A(Qy ′ + Q ′ y) + P BQy = P f (9.74) P AQy ′ + (P AQ ′ + P BQ)y = P f (9.75) Theorem 9.6. An analytically equivalent transformation preserves the following properties of a solvable linear DAE: 1. The local index of the pencil is zero 2. The local index of the pencil is one c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 173<br />
9.3 General Linear <strong>Differential</strong> Algebraic <strong>Equation</strong>s<br />
In the general linear system the matrices A and B may be functions of t,<br />
A(t)y ′ (t) + B(t)y(t) = f(t) (9.72)<br />
<strong>The</strong>n theorem 9.1 is still valid, but theorem 9.2 is not. In general, the index of the<br />
pencil and the index of the DAE are not the same!<br />
Definition 9.7. Let A(t)y ′ + B(t)y = f be a linear time-varying DAE with pencil<br />
λA(t) + B(t). We say the DAE is a regular DAE if the pencil is regular for all<br />
values of t.<br />
Definition 9.8. Let A(t)y ′ + B(t)y = f be a linear time-varying DAE with pencil<br />
λA(t) + B(t).<strong>The</strong>n the local index of the pencil is the index of the pencil at a<br />
given time t.<br />
<strong>The</strong>orem 9.4. Let A(t)y ′ + B(t)y = f be a solvable linear time-varying DAE with<br />
pencil λA(t) + B(t).<strong>The</strong>n the following are equivalent:<br />
1. A(t)y ′ + B(t)y = f is an ordinary differential equation (system).<br />
2. <strong>The</strong> index of the DAE A(t)y ′ + B(t)y = f is zero.<br />
3. <strong>The</strong> local index of the pencil λA(t) + B(t) is zero for all t.<br />
4. <strong>The</strong> local index of the pencil is zero for some t = t 0 .<br />
<strong>The</strong>orem 9.5. Let A(t)y ′ + B(t)y = f be a solvable linear time-varying DAE with<br />
pencil λA(t) + B(t). <strong>The</strong>n the index of the pencil is one if and only if the index of<br />
the DAE is one.<br />
Definition 9.9. Let A(t)u ′ (t)+B(t)u(t) = f be a solvable linear time-varying DAE.<br />
An analytically equivalent transformation of the DAE is given by<br />
1. Pre-multiply the DAE by a non-singular matrix P (t),<br />
P Au ′ + P Bu = P f (9.73)<br />
2. Make the change of variables u = Q(t)y, for some non-singular matrix Q(t)<br />
P A(Qy ′ + Q ′ y) + P BQy = P f (9.74)<br />
P AQy ′ + (P AQ ′ + P BQ)y = P f (9.75)<br />
<strong>The</strong>orem 9.6. An analytically equivalent transformation preserves the following<br />
properties of a solvable linear DAE:<br />
1. <strong>The</strong> local index of the pencil is zero<br />
2. <strong>The</strong> local index of the pencil is one<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007<br />
Math 582B, Spring 2007<br />
California State University Northridge
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