A Geometric Index Reduction Method for DAE Systems Gabriela Jeronimo - - PowerPoint PPT Presentation

A Geometric Index Reduction Method for DAE Systems

Gabriela Jeronimo(1)

Joint work with L. D’Alfonso(1), F. Ollivier(2),

• A. Sedoglavic(3) and P. Solern´
• (1)

(1) Universidad de Buenos Aires, Argentina (2) ´ Ecole Polytechnique, France (3) Universit´ e de Lille I, France

DART IV – October 2010

Gabriela Jeronimo Index reduction for DAE systems

DAE systems

Consider a Differential Algebraic Equation (DAE) system (Σ) =      f1(X, . . . , X (e)) = . . . fn(X, . . . , X (e)) = X (k) := {x(k)

1 , . . . , x(k) n } for every k ∈ Z≥0,

F := f1, . . . , fn polynomials with coefficients in C or C(t).

Gabriela Jeronimo Index reduction for DAE systems

DAE systems

Consider a Differential Algebraic Equation (DAE) system (Σ) =      f1(X, . . . , X (e)) = . . . fn(X, . . . , X (e)) = X (k) := {x(k)

1 , . . . , x(k) n } for every k ∈ Z≥0,

F := f1, . . . , fn polynomials with coefficients in C or C(t). If det

• ∂F

∂X (e)

• = 0, (Σ) is equivalent to an ODE system:

( Σ) =

• X (e) = G(X, . . . , X (e−1)).

Gabriela Jeronimo Index reduction for DAE systems

DAE systems

Consider a Differential Algebraic Equation (DAE) system (Σ) =      f1(X, . . . , X (e)) = . . . fn(X, . . . , X (e)) = X (k) := {x(k)

1 , . . . , x(k) n } for every k ∈ Z≥0,

F := f1, . . . , fn polynomials with coefficients in C or C(t). If det

• ∂F

∂X (e)

• = 0, (Σ) is equivalent to an ODE system:

( Σ) =

• X (e) = G(X, . . . , X (e−1)).

What can be done when det

• ∂F

∂X (e)

• = 0?

Gabriela Jeronimo Index reduction for DAE systems

Semi-explicit systems (Σ0) :=          ˙ x1 = g1(X) . . . X = (x1, . . . , xn) ˙ xn−1 = gn−1(X) g(X) =

Gabriela Jeronimo Index reduction for DAE systems

Semi-explicit systems (Σ0) :=          ˙ x1 = g1(X) . . . X = (x1, . . . , xn) ˙ xn−1 = gn−1(X) g(X) = From the last equation: ∂g ∂t (X) +

n−1

• i=1

∂g ∂xi (X) gi(X) + ∂g ∂xn (X) ˙ xn = 0.

Gabriela Jeronimo Index reduction for DAE systems

Semi-explicit systems (Σ0) :=          ˙ x1 = g1(X) . . . X = (x1, . . . , xn) ˙ xn−1 = gn−1(X) g(X) = From the last equation: ∂g ∂t (X) +

n−1

• i=1

∂g ∂xi (X) gi(X) + ∂g ∂xn (X) ˙ xn = 0. If ∂g ∂xn = 0, we can solve ˙ xn = gn(X).

Gabriela Jeronimo Index reduction for DAE systems

Semi-explicit systems (Σ0) :=          ˙ x1 = g1(X) . . . X = (x1, . . . , xn) ˙ xn−1 = gn−1(X) g(X) = From the last equation: ∂g ∂t (X) +

n−1

• i=1

∂g ∂xi (X) gi(X) + ∂g ∂xn (X) ˙ xn = 0. If ∂g ∂xn = 0, we can solve ˙ xn = gn(X). The first n − 1 equations plus this one form an ODE system.

Gabriela Jeronimo Index reduction for DAE systems

Differentiation index of DAE systems

The differentiation index of a DAE system is an integer σ ∈ Z≥0 which measures the implicitness of the given system.

Gabriela Jeronimo Index reduction for DAE systems

Differentiation index of DAE systems

The differentiation index of a DAE system is an integer σ ∈ Z≥0 which measures the implicitness of the given system. σ = minimum number of differentiations of the system required in

• rder to obtain an ODE.

Gabriela Jeronimo Index reduction for DAE systems

Differentiation index of DAE systems

The differentiation index of a DAE system is an integer σ ∈ Z≥0 which measures the implicitness of the given system. σ = minimum number of differentiations of the system required in

• rder to obtain an ODE.

Examples. For a DAE system F(X, ˙ X, . . . , X (e)) = 0 such that det

• ∂F

∂X (e)

• = 0, we have σ = 0.

Gabriela Jeronimo Index reduction for DAE systems

Differentiation index of DAE systems

The differentiation index of a DAE system is an integer σ ∈ Z≥0 which measures the implicitness of the given system. σ = minimum number of differentiations of the system required in

• rder to obtain an ODE.

Examples. For a DAE system F(X, ˙ X, . . . , X (e)) = 0 such that det

• ∂F

∂X (e)

• = 0, we have σ = 0.

For a semi-explicit DAE system, σ = 1.

Gabriela Jeronimo Index reduction for DAE systems

Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ0) with low index (preferably 0 or 1, and semi-explicit).

Gabriela Jeronimo Index reduction for DAE systems

Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ0) with low index (preferably 0 or 1, and semi-explicit). Motivation: low-index DAE systems are easier to solve than high-index DAE systems.

Gabriela Jeronimo Index reduction for DAE systems

Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ0) with low index (preferably 0 or 1, and semi-explicit). Motivation: low-index DAE systems are easier to solve than high-index DAE systems. Previous work: [Gear 1988, 1989], [Brenan-Campbell-Petzold 1996], [Kunkel-Mehrmann 2006].

Gabriela Jeronimo Index reduction for DAE systems

Index reduction problem Given a high-index DAE system (Σ), obtain an equivalent DAE system (Σ0) with low index (preferably 0 or 1, and semi-explicit). Motivation: low-index DAE systems are easier to solve than high-index DAE systems. Previous work: [Gear 1988, 1989], [Brenan-Campbell-Petzold 1996], [Kunkel-Mehrmann 2006]. Tools: Computation of successive derivatives of the equations, rewriting techniques relying on the Implicit Function Theorem, etc.

Gabriela Jeronimo Index reduction for DAE systems

Our main results

(Σ) =      f1(X, X (1), . . . , X (e)) = 0 . . . fn(X, X (1), . . . , X (e)) = 0 with certain assumptions.

Gabriela Jeronimo Index reduction for DAE systems

Our main results

(Σ) =      f1(X, X (1), . . . , X (e)) = 0 . . . fn(X, X (1), . . . , X (e)) = 0 with certain assumptions. (Σ) is generically equivalent to a first order semi-explicit DAE system with differentiation index 1

(Σ0) :=          ˙ u1 = g1(U, v) . . . ˙ ur = gr(U, v) q(U, v) = U = (u1, . . . , ur)

Gabriela Jeronimo Index reduction for DAE systems

Our main results

(Σ) =      f1(X, X (1), . . . , X (e)) = 0 . . . fn(X, X (1), . . . , X (e)) = 0 with certain assumptions. (Σ) is generically equivalent to a first order semi-explicit DAE system with differentiation index 1

(Σ0) :=          ˙ u1 = g1(U, v) . . . ˙ ur = gr(U, v) q(U, v) = U = (u1, . . . , ur)

a probabilistic algorithm to compute the differentiation index

• f (Σ) and the associated system (Σ0).

Gabriela Jeronimo Index reduction for DAE systems

Remarks

(Σ) and (Σ0) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ0) and conversely.

Gabriela Jeronimo Index reduction for DAE systems

Remarks

(Σ) and (Σ0) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ0) and conversely. Our algorithms rely on:

Gabriela Jeronimo Index reduction for DAE systems

Remarks

(Σ) and (Σ0) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ0) and conversely. Our algorithms rely on:

an alternative characterization of the differentiation index ([DAlJeSo08]),

Gabriela Jeronimo Index reduction for DAE systems

Remarks

(Σ) and (Σ0) are equivalent in the sense that almost all analytic solution of (Σ) can be obtained from an analytic solution of (Σ0) and conversely. Our algorithms rely on:

an alternative characterization of the differentiation index ([DAlJeSo08]), the polynomial time Kronecker algorithm for the computation

• f geometric resolutions of algebraic polynomial systems

([GiLeSa01], [Schost03]).

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

K{X} = K[x(j)

i

: 1 ≤ i ≤ n, j ∈ N0] with the derivation δ induced by δ(x(j)

i

) = x(j+1)

i

.

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

K{X} = K[x(j)

i

: 1 ≤ i ≤ n, j ∈ N0] with the derivation δ induced by δ(x(j)

i

) = x(j+1)

i

. [F] = [f1, . . . , fn] ⊂ K{X} differential ideal associated with (Σ).

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

K{X} = K[x(j)

i

: 1 ≤ i ≤ n, j ∈ N0] with the derivation δ induced by δ(x(j)

i

) = x(j+1)

i

. [F] = [f1, . . . , fn] ⊂ K{X} differential ideal associated with (Σ). pr(ℓ)F = (F, F (1), . . . , F (ℓ)) ⊂ K[X, X (1), . . . , X (e+ℓ)] ∀ ℓ ∈ Z≥0.

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

K{X} = K[x(j)

i

: 1 ≤ i ≤ n, j ∈ N0] with the derivation δ induced by δ(x(j)

i

) = x(j+1)

i

. [F] = [f1, . . . , fn] ⊂ K{X} differential ideal associated with (Σ). pr(ℓ)F = (F, F (1), . . . , F (ℓ)) ⊂ K[X, X (1), . . . , X (e+ℓ)] ∀ ℓ ∈ Z≥0. Assumptions:

1 pr(ℓ−1)F + (f (ℓ)

1 , . . . , f (ℓ) j

) is prime ∀ ℓ ∈ Z≥0, ∀ 1 ≤ j ≤ n.

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

K{X} = K[x(j)

i

: 1 ≤ i ≤ n, j ∈ N0] with the derivation δ induced by δ(x(j)

i

) = x(j+1)

i

. [F] = [f1, . . . , fn] ⊂ K{X} differential ideal associated with (Σ). pr(ℓ)F = (F, F (1), . . . , F (ℓ)) ⊂ K[X, X (1), . . . , X (e+ℓ)] ∀ ℓ ∈ Z≥0. Assumptions:

1 pr(ℓ−1)F + (f (ℓ)

1 , . . . , f (ℓ) j

) is prime ∀ ℓ ∈ Z≥0, ∀ 1 ≤ j ≤ n. ⇒ pr(ℓ)F is prime ∀ ℓ ∈ Z≥0 and [F] ⊂ K{X} is prime.

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

K{X} = K[x(j)

i

: 1 ≤ i ≤ n, j ∈ N0] with the derivation δ induced by δ(x(j)

i

) = x(j+1)

i

. [F] = [f1, . . . , fn] ⊂ K{X} differential ideal associated with (Σ). pr(ℓ)F = (F, F (1), . . . , F (ℓ)) ⊂ K[X, X (1), . . . , X (e+ℓ)] ∀ ℓ ∈ Z≥0. Assumptions:

1 pr(ℓ−1)F + (f (ℓ)

1 , . . . , f (ℓ) j

) is prime ∀ ℓ ∈ Z≥0, ∀ 1 ≤ j ≤ n. ⇒ pr(ℓ)F is prime ∀ ℓ ∈ Z≥0 and [F] ⊂ K{X} is prime.

2 (Σ) is quasi-regular = the Jacobian matrix

∂(F,...,F (ℓ)) ∂(X,...,X (e+ℓ)) has

maximal rank modulo [F] ∀ ℓ ∈ Z≥0 [Johnson 1978].

Gabriela Jeronimo Index reduction for DAE systems

Assumptions on the system

K{X} = K[x(j)

i

: 1 ≤ i ≤ n, j ∈ N0] with the derivation δ induced by δ(x(j)

i

) = x(j+1)

i

. [F] = [f1, . . . , fn] ⊂ K{X} differential ideal associated with (Σ). pr(ℓ)F = (F, F (1), . . . , F (ℓ)) ⊂ K[X, X (1), . . . , X (e+ℓ)] ∀ ℓ ∈ Z≥0. Assumptions:

1 pr(ℓ−1)F + (f (ℓ)

1 , . . . , f (ℓ) j

) is prime ∀ ℓ ∈ Z≥0, ∀ 1 ≤ j ≤ n. ⇒ pr(ℓ)F is prime ∀ ℓ ∈ Z≥0 and [F] ⊂ K{X} is prime.

2 (Σ) is quasi-regular = the Jacobian matrix

∂(F,...,F (ℓ)) ∂(X,...,X (e+ℓ)) has

maximal rank modulo [F] ∀ ℓ ∈ Z≥0 [Johnson 1978]. ⇒ pr(ℓ)F is a complete intersection ideal ∀ℓ ∈ Z≥0.

Gabriela Jeronimo Index reduction for DAE systems

The differentiation index of the system

Gabriela Jeronimo Index reduction for DAE systems

The differentiation index of the system

The following ideal chain in K[X [e]] = K[X, . . . , X (e)] eventually becomes stationary: (0) ⊂ pr(0)F ∩ K[X [e]] ⊂ pr(1)F ∩ K[X [e]] ⊂ . . . · · · ⊂ pr(ℓ)F ∩ K[X [e]] ⊂ · · · ⊂ [F] ∩ K[X [e]].

Gabriela Jeronimo Index reduction for DAE systems

The differentiation index of the system

The following ideal chain in K[X [e]] = K[X, . . . , X (e)] eventually becomes stationary: (0) ⊂ pr(0)F ∩ K[X [e]] ⊂ pr(1)F ∩ K[X [e]] ⊂ . . . · · · ⊂ pr(ℓ)F ∩ K[X [e]] ⊂ · · · ⊂ [F] ∩ K[X [e]]. The differentiation index of the system (Σ) is σ = min

• ℓ ∈ Z≥0 | pr(ℓ)F ∩ K[X [e]] = [F] ∩ K[X [e]]
• = min
• ℓ ∈ Z≥0 | pr(ℓ)F ∩ K[X [e]] = pr(ℓ+1)F ∩ K[X [e]]
• Gabriela Jeronimo

Index reduction for DAE systems

The differentiation index of the system

The following ideal chain in K[X [e]] = K[X, . . . , X (e)] eventually becomes stationary: (0) ⊂ pr(0)F ∩ K[X [e]] ⊂ pr(1)F ∩ K[X [e]] ⊂ . . . · · · ⊂ pr(ℓ)F ∩ K[X [e]] ⊂ · · · ⊂ [F] ∩ K[X [e]]. The differentiation index of the system (Σ) is σ = min

• ℓ ∈ Z≥0 | pr(ℓ)F ∩ K[X [e]] = [F] ∩ K[X [e]]
• = min
• ℓ ∈ Z≥0 | pr(ℓ)F ∩ K[X [e]] = pr(ℓ+1)F ∩ K[X [e]]
• The second identity leads to a recursive algorithm to compute σ.

Gabriela Jeronimo Index reduction for DAE systems

Parametric geometric resolutions

Gabriela Jeronimo Index reduction for DAE systems

Parametric geometric resolutions

Let V ⊂ Cn be an irreducible algebraic variety defined over K and K(V ) = Frac(K[X]/I(V )), I(V ) = {f ∈ K[X] | f (ξ) = 0 ∀ξ ∈ V }.

Gabriela Jeronimo Index reduction for DAE systems

Parametric geometric resolutions

Let V ⊂ Cn be an irreducible algebraic variety defined over K and K(V ) = Frac(K[X]/I(V )), I(V ) = {f ∈ K[X] | f (ξ) = 0 ∀ξ ∈ V }. U ⊂ X a transcendence basis of K(V )/K,

Gabriela Jeronimo Index reduction for DAE systems

Parametric geometric resolutions

Let V ⊂ Cn be an irreducible algebraic variety defined over K and K(V ) = Frac(K[X]/I(V )), I(V ) = {f ∈ K[X] | f (ξ) = 0 ∀ξ ∈ V }. U ⊂ X a transcendence basis of K(V )/K, v a primitive element of K(V )/K(U) (linear form in X \ U) ⇒ K(V ) = K(U)[v],

Gabriela Jeronimo Index reduction for DAE systems

Parametric geometric resolutions

Let V ⊂ Cn be an irreducible algebraic variety defined over K and K(V ) = Frac(K[X]/I(V )), I(V ) = {f ∈ K[X] | f (ξ) = 0 ∀ξ ∈ V }. U ⊂ X a transcendence basis of K(V )/K, v a primitive element of K(V )/K(U) (linear form in X \ U) ⇒ K(V ) = K(U)[v], q ∈ K[U, Y ] minimal polynomial of v over K(U),

Gabriela Jeronimo Index reduction for DAE systems

Parametric geometric resolutions

Let V ⊂ Cn be an irreducible algebraic variety defined over K and K(V ) = Frac(K[X]/I(V )), I(V ) = {f ∈ K[X] | f (ξ) = 0 ∀ξ ∈ V }. U ⊂ X a transcendence basis of K(V )/K, v a primitive element of K(V )/K(U) (linear form in X \ U) ⇒ K(V ) = K(U)[v], q ∈ K[U, Y ] minimal polynomial of v over K(U), wj ∈ K(U)[Y ] such that, for every j, xj = wj(U, v) in K(V ).

Gabriela Jeronimo Index reduction for DAE systems

Parametric geometric resolutions

Let V ⊂ Cn be an irreducible algebraic variety defined over K and K(V ) = Frac(K[X]/I(V )), I(V ) = {f ∈ K[X] | f (ξ) = 0 ∀ξ ∈ V }. U ⊂ X a transcendence basis of K(V )/K, v a primitive element of K(V )/K(U) (linear form in X \ U) ⇒ K(V ) = K(U)[v], q ∈ K[U, Y ] minimal polynomial of v over K(U), wj ∈ K(U)[Y ] such that, for every j, xj = wj(U, v) in K(V ). The 4-tuple (U, v, q, (w1, . . . , wn)) is a parametric geometric resolution of V .

Gabriela Jeronimo Index reduction for DAE systems

(U, v, q, (w1, . . . , wn)) a parametric geometric resolution of V .

Gabriela Jeronimo Index reduction for DAE systems

(U, v, q, (w1, . . . , wn)) a parametric geometric resolution of V . If H = {(U, v) ∈ Cr+1 | q(U, v) = 0}, then φ : H V , (U, v) → (w1(U, v), . . . , wn(U, v)) is a parametrization of a dense open set of V by the points of a dense open set of H

Gabriela Jeronimo Index reduction for DAE systems

(U, v, q, (w1, . . . , wn)) a parametric geometric resolution of V . If H = {(U, v) ∈ Cr+1 | q(U, v) = 0}, then φ : H V , (U, v) → (w1(U, v), . . . , wn(U, v)) is a parametrization of a dense open set of V by the points of a dense open set of H A parametric geometric resolution of V can be computed from a finite set of polynomials defining V in polynomial time: Kronecker algorithm.

Gabriela Jeronimo Index reduction for DAE systems

An algebraic variety associated with (Σ)

V ([F] ∩ K[X [e]]) = V (pr(σ)F ∩ K[X [e]]) is an irreducible variety of dimension r = ord([F]).

Gabriela Jeronimo Index reduction for DAE systems

An algebraic variety associated with (Σ)

V ([F] ∩ K[X [e]]) = V (pr(σ)F ∩ K[X [e]]) is an irreducible variety of dimension r = ord([F]). U = (u1, . . . , ur) ⊂ X, . . . , X (e−1) transcendence basis of Frac(K[X [e]]/pr(σ)F ∩ K[X [e]])/K,

Gabriela Jeronimo Index reduction for DAE systems

An algebraic variety associated with (Σ)

V ([F] ∩ K[X [e]]) = V (pr(σ)F ∩ K[X [e]]) is an irreducible variety of dimension r = ord([F]). U = (u1, . . . , ur) ⊂ X, . . . , X (e−1) transcendence basis of Frac(K[X [e]]/pr(σ)F ∩ K[X [e]])/K, Z ⊂ X (e+1), . . . , X (e+σ) such that {U, Z} is a transcendence basis of Frac(K[X, . . . , X (e+σ)]/pr(σ)F)/K.

Gabriela Jeronimo Index reduction for DAE systems

An algebraic variety associated with (Σ)

V ([F] ∩ K[X [e]]) = V (pr(σ)F ∩ K[X [e]]) is an irreducible variety of dimension r = ord([F]). U = (u1, . . . , ur) ⊂ X, . . . , X (e−1) transcendence basis of Frac(K[X [e]]/pr(σ)F ∩ K[X [e]])/K, Z ⊂ X (e+1), . . . , X (e+σ) such that {U, Z} is a transcendence basis of Frac(K[X, . . . , X (e+σ)]/pr(σ)F)/K. For a generic specialization Z = z0, we have [F] ∩ K[X (e)] = (F|z0, . . . , F (σ)|z0) ∩ K[X (e)].

Gabriela Jeronimo Index reduction for DAE systems

An algebraic variety associated with (Σ)

V ([F] ∩ K[X [e]]) = V (pr(σ)F ∩ K[X [e]]) is an irreducible variety of dimension r = ord([F]). U = (u1, . . . , ur) ⊂ X, . . . , X (e−1) transcendence basis of Frac(K[X [e]]/pr(σ)F ∩ K[X [e]])/K, Z ⊂ X (e+1), . . . , X (e+σ) such that {U, Z} is a transcendence basis of Frac(K[X, . . . , X (e+σ)]/pr(σ)F)/K. For a generic specialization Z = z0, we have [F] ∩ K[X (e)] = (F|z0, . . . , F (σ)|z0) ∩ K[X (e)]. V = {ξ ∈ Cn(σ+1)+r | F|z0(ξ) = 0, . . . , F (σ)|z0(ξ) = 0}

Gabriela Jeronimo Index reduction for DAE systems

An algebraic variety associated with (Σ)

V ([F] ∩ K[X [e]]) = V (pr(σ)F ∩ K[X [e]]) is an irreducible variety of dimension r = ord([F]). U = (u1, . . . , ur) ⊂ X, . . . , X (e−1) transcendence basis of Frac(K[X [e]]/pr(σ)F ∩ K[X [e]])/K, Z ⊂ X (e+1), . . . , X (e+σ) such that {U, Z} is a transcendence basis of Frac(K[X, . . . , X (e+σ)]/pr(σ)F)/K. For a generic specialization Z = z0, we have [F] ∩ K[X (e)] = (F|z0, . . . , F (σ)|z0) ∩ K[X (e)]. V = {ξ ∈ Cn(σ+1)+r | F|z0(ξ) = 0, . . . , F (σ)|z0(ξ) = 0} V is equidimensional of dimension r, U is a transcendence basis of K(Vi) for each irreducible component Vi of V.

Gabriela Jeronimo Index reduction for DAE systems

The associated system (Σ0)

(U, v, q, (wj(U, v))1≤j≤n(σ+1)+r) a parametric geometric resolution of V = V (F|z0, . . . , F (σ)|z0) ⊂ Cn(σ+1)+r.

Gabriela Jeronimo Index reduction for DAE systems

The associated system (Σ0)

(U, v, q, (wj(U, v))1≤j≤n(σ+1)+r) a parametric geometric resolution of V = V (F|z0, . . . , F (σ)|z0) ⊂ Cn(σ+1)+r. H = {(U, v) ∈ Cr+1 | q(U, v) = 0}.

Gabriela Jeronimo Index reduction for DAE systems

The associated system (Σ0)

(U, v, q, (wj(U, v))1≤j≤n(σ+1)+r) a parametric geometric resolution of V = V (F|z0, . . . , F (σ)|z0) ⊂ Cn(σ+1)+r. H = {(U, v) ∈ Cr+1 | q(U, v) = 0}. Φ : H V, (U, v) → (wj(U, v))1≤j≤n(σ+1)+r

• isom. between dense open subsets UH ⊂ H and UV ⊂ V.

Gabriela Jeronimo Index reduction for DAE systems

The associated system (Σ0)

(U, v, q, (wj(U, v))1≤j≤n(σ+1)+r) a parametric geometric resolution of V = V (F|z0, . . . , F (σ)|z0) ⊂ Cn(σ+1)+r. H = {(U, v) ∈ Cr+1 | q(U, v) = 0}. Φ : H V, (U, v) → (wj(U, v))1≤j≤n(σ+1)+r

• isom. between dense open subsets UH ⊂ H and UV ⊂ V.

Recover solutions of (Σ) from curves in H by means of Φ.

Gabriela Jeronimo Index reduction for DAE systems

The associated system (Σ0)

(U, v, q, (wj(U, v))1≤j≤n(σ+1)+r) a parametric geometric resolution of V = V (F|z0, . . . , F (σ)|z0) ⊂ Cn(σ+1)+r. H = {(U, v) ∈ Cr+1 | q(U, v) = 0}. Φ : H V, (U, v) → (wj(U, v))1≤j≤n(σ+1)+r

• isom. between dense open subsets UH ⊂ H and UV ⊂ V.

Recover solutions of (Σ) from curves in H by means of Φ. If γ : (−ε, ε) → UH, and π : V → Cn(e+1) projection, π(Φ(γ(t))) = (ϕ0(t), . . . , ϕe(t)) ∈ π(V) ∀ t ∈ (−ε, ε) and each point satisfies the polynomial equations of (Σ).

Gabriela Jeronimo Index reduction for DAE systems

The associated system (Σ0)

(U, v, q, (wj(U, v))1≤j≤n(σ+1)+r) a parametric geometric resolution of V = V (F|z0, . . . , F (σ)|z0) ⊂ Cn(σ+1)+r. H = {(U, v) ∈ Cr+1 | q(U, v) = 0}. Φ : H V, (U, v) → (wj(U, v))1≤j≤n(σ+1)+r

• isom. between dense open subsets UH ⊂ H and UV ⊂ V.

Recover solutions of (Σ) from curves in H by means of Φ. If γ : (−ε, ε) → UH, and π : V → Cn(e+1) projection, π(Φ(γ(t))) = (ϕ0(t), . . . , ϕe(t)) ∈ π(V) ∀ t ∈ (−ε, ε) and each point satisfies the polynomial equations of (Σ). ϕ0 is a solution to (Σ) provided that ϕi+1(t) = ˙ ϕi(t) ∀ 0 ≤ i ≤ e − 1.

Gabriela Jeronimo Index reduction for DAE systems

We add differential equations on the curves γ so that the required conditions hold.

Gabriela Jeronimo Index reduction for DAE systems

We add differential equations on the curves γ so that the required conditions hold. For each uk ∈ U:

Gabriela Jeronimo Index reduction for DAE systems

We add differential equations on the curves γ so that the required conditions hold. For each uk ∈ U:

1 If ˙

uk ∈ U, that is, ∃ 1 ≤ ik ≤ r such that ˙ uk = uik, add this equation.

Gabriela Jeronimo Index reduction for DAE systems

We add differential equations on the curves γ so that the required conditions hold. For each uk ∈ U:

1 If ˙

uk ∈ U, that is, ∃ 1 ≤ ik ≤ r such that ˙ uk = uik, add this equation.

2 If ˙

uk / ∈ U, since ˙ uk ∈ (X, . . . , X (e+σ)) \ {U, Z}, it corresponds to some coordinate of V; then ˙ uk = wjk(U, v) via Φ for some 1 ≤ jk ≤ n(1 + σ) + r.

Gabriela Jeronimo Index reduction for DAE systems

We add differential equations on the curves γ so that the required conditions hold. For each uk ∈ U:

1 If ˙

uk ∈ U, that is, ∃ 1 ≤ ik ≤ r such that ˙ uk = uik, add this equation.

2 If ˙

uk / ∈ U, since ˙ uk ∈ (X, . . . , X (e+σ)) \ {U, Z}, it corresponds to some coordinate of V; then ˙ uk = wjk(U, v) via Φ for some 1 ≤ jk ≤ n(1 + σ) + r. Add the equation ˙ uk = wjk(U, v).

Gabriela Jeronimo Index reduction for DAE systems

We add differential equations on the curves γ so that the required conditions hold. For each uk ∈ U:

1 If ˙

uk ∈ U, that is, ∃ 1 ≤ ik ≤ r such that ˙ uk = uik, add this equation.

2 If ˙

uk / ∈ U, since ˙ uk ∈ (X, . . . , X (e+σ)) \ {U, Z}, it corresponds to some coordinate of V; then ˙ uk = wjk(U, v) via Φ for some 1 ≤ jk ≤ n(1 + σ) + r. Add the equation ˙ uk = wjk(U, v). (Σ0) =    ˙ uk = uik ∀ k such that ˙ uk ∈ U ˙ uk = wjk(U, v) ∀ k such that ˙ uk / ∈ U q(U, v) =

Gabriela Jeronimo Index reduction for DAE systems

Theorem Given a DAE system (Σ) as before, we can compute algorithmically within a bounded complexity a rational map Φ and a first order semi-explicit DAE system (Σ0) equivalent to (Σ) in the following sense: There is a dense open subset U of V ([F] ∩ K[X [e]]) such that for every ξ ∈ U, there exists a unique solution ϕ of (Σ) with initial condition ξ, which can be obtained as ϕ = πX(Φ(γ)) for a solution γ of (Σ0) (πX denotes the projection to the first n coordinates).

Gabriela Jeronimo Index reduction for DAE systems

Theorem Given a DAE system (Σ) as before, we can compute algorithmically within a bounded complexity a rational map Φ and a first order semi-explicit DAE system (Σ0) equivalent to (Σ) in the following sense: There is a dense open subset U of V ([F] ∩ K[X [e]]) such that for every ξ ∈ U, there exists a unique solution ϕ of (Σ) with initial condition ξ, which can be obtained as ϕ = πX(Φ(γ)) for a solution γ of (Σ0) (πX denotes the projection to the first n coordinates). Key facts used in the proof: Parametric geometric resolution of V, which gives the map Φ that enables to lift solutions of (Σ0) to solutions of (Σ). A theorem of existence and uniqueness of solutions for DAE systems.

Gabriela Jeronimo Index reduction for DAE systems

Example: the pendulum

(Σ) =    x(2) − λ · x = y(2) − λ · y + g = x2 + y2 = 1

Gabriela Jeronimo Index reduction for DAE systems

Example: the pendulum

(Σ) =    x(2) − λ · x = y(2) − λ · y + g = x2 + y2 = 1 σ = 4 U = {x, ˙ x}, Z = {λ(3), λ(4), λ(5), λ(6)}, v = y

Gabriela Jeronimo Index reduction for DAE systems

Example: the pendulum

(Σ) =    x(2) − λ · x = y(2) − λ · y + g = x2 + y2 = 1 σ = 4 U = {x, ˙ x}, Z = {λ(3), λ(4), λ(5), λ(6)}, v = y (Σ0) =      ˙ u1 = u2 ˙ u2 = gu1v − u1u2

2

1−u2

1

v2 + u2

1 − 1

= Φ(U, v) =

• u1, v, gv −

u2

2

1 − u2

1

• Gabriela Jeronimo

Index reduction for DAE systems