Differential Algebraic Equation

In a differential-algebraic equation (also differential-algebraic equation , algebro-differential equation or descriptor system ), ordinary differential equations and algebraic (ie here: derivative-free) constraints are coupled and understood as an equation or system of equations. In some cases this structure is already laid out in the form of the system of equations, e.g. B. in

${\displaystyle {\begin{aligned}{\dot {x}}(t)&=f(t,x(t),y(t))\\0&=g(t,x(t),y(t))\end{aligned}}}$

This form regularly results from problems arising from the mechanics of bodies under constrained conditions; the pendulum is often chosen as an instructive example .

The most general form of a differential-algebraic equation is an implicit differential equation in the form

${\displaystyle F({\dot {x}}(t),x(t),t)=0}$, ${\displaystyle F\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{m}}$

for a vector-valued function with . An equation in this implicit form is (locally) solvable for if the partial derivative is regular. This follows from the classical theorem about implicit functions . In this particular case, the implicit equation can be rewritten in the form ${\displaystyle x\colon I\to \mathbb {R} ^{n}}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle {\dot {x}}}$ ${\displaystyle F_{\dot {x}}}$

${\displaystyle {\dot {x}}=g(x,t)}$

and thus again has an explicit ordinary differential equation.

A true differential-algebraic equation exists when the partial derivative is singular. Then the implicit differential equation splits locally into an inherent differential equation and an algebraic constraint. This practically corresponds to a differential equation considered on a manifold . The practical problem with the implicit differential equation, however, is that this manifold is initially not explicitly known. ${\displaystyle F_{\dot {x}}}$

In contrast to ordinary differential equations, the solution of which is determined by integration, parts of the solution of a differential-algebraic equation result from differentiation. This places further demands on the system function . If this only has to be continuously or continuously differentiable for ordinary differential equations in order to guarantee solvability, higher derivatives are now also required for the solution. The exact order of the required derivatives depends on the chosen approach and is generally referred to as the index of the differential-algebraic equation. ${\displaystyle F}$

The derivation of components of the equation system to be included in the solution process results in an overdetermined system. One consequence of this is that the solutions must also satisfy a number of explicit or implicit algebraic constraints. This applies in particular to initial values ​​of initial value problems . The search for consistent initial values, e.g. B. in the vicinity of predetermined inconsistent initial values, is a nontrivial first problem in the practical solution of differential-algebraic equations.

Types of differential algebraic equations

Semi-explicit differential-algebraic equation

A special case for a differential-algebraic equation is a system in the form

${\displaystyle {\dot {x}}_{1}=f_{1}(x_{1},x_{2},t)\quad ,\quad 0=f_{2}(x_{1},x_{2},t)}$.

By differentiating the second differential equation and inserting the first one obtains a further condition for a solution

${\displaystyle 0=\partial _{1}f_{2}(x_{1},x_{2},t)f_{1}(x_{1},x_{2},t)+\partial _{2}f_{2}(x_{1},x_{2},t){\dot {x}}_{2}+\partial _{t}f_{2}(x_{1},x_{2},t)}$.

If the factor vor is different from zero, an explicit system of ordinary differential equations results. However, initial values ​​for this system must also satisfy the undifferentiated second equation, so that only one parameter can be freely selected. ${\displaystyle {\dot {x}}_{2}}$

Linear differential-algebraic equation

Differential-algebraic equations very often appear in the form

${\displaystyle E{\dot {x}}+Cx=q\,,\qquad x(t)\in \mathbb {R} ^{m}\,,\quad t\in I\subset \mathbb {R} }$

with continuous matrix coefficients

${\displaystyle E(t)\in L(\mathbb {R} ^{m},\mathbb {R} ^{k})\,,\quad C(t)\in L(\mathbb {R} ^{m},\mathbb {R} ^{k})\,,\quad t\in I}$

and a function

${\displaystyle q\colon I\to \mathbb {R} ^{k}}$.

A true differential-algebraic equation is here when the matrix function on a non-trivial core has. A particularly simple case occurs when the matrices are square with constant entries. ${\displaystyle E}$ ${\displaystyle I}$

Linear differential-algebraic equation with a properly formulated main term

Another notation for linear differential-algebraic equations is the form

${\displaystyle A(Bx)'+Cx=q\,,\qquad x(t)\in \mathbb {R} ^{m}\,,\quad t\in I\subset \mathbb {R} }$

with (at least) continuous matrix coefficients

${\displaystyle A(t)\in L(\mathbb {R} ^{n},\mathbb {R} ^{k})\,,\quad B(t)\in L(\mathbb {R} ^{m},\mathbb {R} ^{n})\,,\quad C(t)\in L(\mathbb {R} ^{m},\mathbb {R} ^{k})\,,\quad t\in I}$

and a function

${\displaystyle q\colon I\to \mathbb {R} ^{k}}$.

This notation takes into account the fact that in a differential-algebraic equation only part of the variable vector is differentiated. In fact, only the component is differentiated here and not the entire variable vector . Functions from space are the classic solutions to this equation ${\displaystyle x}$ ${\displaystyle Bx}$ ${\displaystyle x}$

${\displaystyle C_{B}^{1}(I,\mathbb {R} ^{m}):=\left\{x\in C(I,\mathbb {R} ^{m})\mid Bx\in C^{1}(I,\mathbb {R} ^{n})\right\}}$

considered, i.e. the space of continuous functions for which the component is continuously differentiable. ${\displaystyle x}$ ${\displaystyle Bx}$

The two matrix functions and form the main term of the equation and this is properly formulated if two properties are met: ${\displaystyle A}$ ${\displaystyle B}$

1. It applies

   ${\displaystyle \ker A(t)\oplus \operatorname {im} B(t)=\mathbb {R} ^{n}\quad ,\quad t\in I}$.

2. There is a continuously differentiable projector function

   ${\displaystyle R(t)\in L(\mathbb {R} ^{n},\mathbb {R} ^{n})\quad ,\quad R^{2}(t)=R(t)\quad ,\quad t\in I}$

with the property

${\displaystyle \ker R(t)=\ker A(t)\quad ,\quad \operatorname {im} R(t)=\operatorname {im} B(t)\quad ,\quad t\in I}$.

The first condition here ensures that “nothing is lost” between the two matrix functions and . In the core of the matrix nothing can disappear from the picture of the matrix . The projector function realizes exactly the decomposition of the room given by the matrix functions and and is helpful for the analysis of the equation. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \mathbb {R} ^{n}}$

A simple special case for a properly formulated main term is given by matrix functions and with the property ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \ker A(t)=\{0\}\quad ,\quad \operatorname {im} B(t)=\mathbb {R} ^{n}\quad ,\quad t\in I}$.

The identity matrix can then be selected for the projector function . ${\displaystyle R}$

Index terms for DAEs

Differentiation index

The solution of an algebro differential equation system can often be represented by (special) solution curves of an ordinary differential equation system, although it is singular. The differentiation index of the algebro differential equation system plays a key role here . ${\displaystyle f_{\dot {x}}}$

Numerical methods for solving algebro differential equation systems can usually only integrate systems whose differentiation index does not exceed a certain maximum value. For example, the differentiation index of the system in the implicit Euler method must not be greater than one.

The differentiation index of a system of algebro differential equations

${\displaystyle f({\dot {x}},x,t)=0}$

is the number of time derivatives that are necessary to obtain from the resulting system of equations ${\displaystyle N\geq 0}$

${\displaystyle {\begin{aligned}f({\dot {x}},x,t)&=0\\{\frac {d}{dt}}\left(f({\dot {x}},x,t)\right)&=0\\&\vdots \\{\frac {d^{N}}{dt^{N}}}\left(f({\dot {x}},x,t)\right)&=0\end{aligned}}}$

through algebraic transformations an ordinary differential equation system

${\displaystyle {\dot {x}}=g(x,t)}$

to be able to extract.

Examples

A system of algebro differential equations with a regular matrix , which can therefore be rearranged algebraically , has the differentiation index zero. ${\displaystyle f_{\dot {x}}}$ ${\displaystyle {\dot {x}}}$

A purely algebraic equation

${\displaystyle F(x,t)=0}$

with a regular Jacobi matrix , which is also interpreted as an algebro differential equation , has differentiation index one: After differentiating once, the equation is obtained ${\displaystyle F_{x}(x,t)}$ ${\displaystyle f({\dot {x}},x,t)=F(x,t)}$

${\displaystyle 0={\frac {d}{dt}}f({\dot {x}},x,t)=F_{t}(x,t)+F_{x}(x,t){\dot {x}}}$,

which is resolvable by: ${\displaystyle {\dot {x}}}$

${\displaystyle {\dot {x}}=-F_{x}(x,t)^{-1}F_{t}(x,t)}$.

This fact is sometimes used in the construction of homotopy methods .

The Euler-Lagrange equations for the mathematical pendulum (with gravitational acceleration and pendulum length normalized to one) are as follows

${\displaystyle {\begin{aligned}{\ddot {x}}_{1}&=2x_{1}\lambda \\{\ddot {x}}_{2}&=2x_{2}\lambda -1\\0&=x_{1}^{2}+x_{2}^{2}-1\end{aligned}}}$

This algebro differential equation system has the differentiation index three: Provides a double time derivative of the constraint (third equation) according to time

${\displaystyle 2\left(x_{1}{\ddot {x}}_{1}+{\dot {x}}_{1}^{2}+x_{2}{\ddot {x}}_{2}+{\dot {x}}_{2}^{2}\right)=0}$.

With the help of the two differential equations in the Euler-Lagrange equations, the second time derivatives and what ${\displaystyle {\ddot {x}}_{1}}$ ${\displaystyle {\ddot {x}}_{2}}$

${\displaystyle 2\left(2x_{1}^{2}\lambda +{\dot {x}}_{1}^{2}+2x_{2}^{2}\lambda -x_{2}+{\dot {x}}_{2}^{2}\right)=0}$

supplies. With you get the equation ${\displaystyle x_{1}^{2}+x_{2}^{2}=1}$

${\displaystyle \lambda =-{\frac {1}{2}}\left({\dot {x}}_{1}^{2}+{\dot {x}}_{2}^{2}\right)+{\frac {1}{2}}x_{2}}$.

By time derivative of this equation (that is the third time derivative) one arrives at the missing differential equation for ${\displaystyle \lambda }$

${\displaystyle {\begin{aligned}{\dot {\lambda }}&=-{\dot {x}}_{1}{\ddot {x}}_{1}-{\dot {x}}_{2}{\ddot {x}}_{2}+{\frac {1}{2}}{\dot {x}}_{2}\\&=-2\lambda ({\dot {x}}_{1}x_{1}+{\dot {x}}_{2}x_{2})+{\frac {3}{2}}{\dot {x}}_{2}\\&={\frac {3}{2}}{\dot {x}}_{2}\end{aligned}},}$

again using the differential equations of the Euler-Lagrange equations were used to and to replace, and has also taken into consideration that applies. ${\displaystyle {\ddot {x}}_{1}}$ ${\displaystyle {\ddot {x}}_{2}}$ ${\displaystyle 2({\dot {x}}_{1}x_{1}+{\dot {x}}_{2}x_{2})={\frac {d}{dt}}\left(x_{1}^{2}+x_{2}^{2}\right)={\frac {d}{dt}}(1)=0}$

Geometric index

A mathematically clearly defined and geometrically well interpretable term is the geometric index of an algebro differential equation system. The basic idea is to use the iterative method shown below to determine the maximum constrained manifold on which the algebro differential equation describes a vector field (as a vector field on a manifold). The geometric index of the algebro differential equation system is then the minimum number of iteration steps that is required for this method.

The geometric index is equal to the differentiation index.

An autonomous algebro differential equation is given

${\displaystyle f({\dot {x}}(t),x(t))=0}$

with a sufficiently often differentiable function . ${\displaystyle f\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$

Within the framework of the algorithm, the is interpreted as a manifold with the tangential bundle . The pairs are also referred to as tangent vectors des . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle M_{0}:=\mathbb {R} ^{n}}$ ${\displaystyle TM_{0}=\mathbb {R} ^{n}\times \mathbb {R} ^{n}}$ ${\displaystyle (x,y)\in TM_{0}}$ ${\displaystyle \mathbb {R} ^{n}}$

The function defines the set that assigns to each point all velocity vectors admissible for solutions of the algebro-DGL system in this point. ${\displaystyle f}$ ${\displaystyle N:=\left\{(x,v)\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid f(v,x)=0\right\}}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle v}$

It is possible that no pair at all , exactly one such pair or several such pairs exist for a point . ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle (x,v)}$ ${\displaystyle N}$

The points through which solutions can possibly go are captured in the set

${\displaystyle \left.M_{1}:=\operatorname {pr} _{1}N,\right.}$

(with the projection onto the first component, that is ). At this point, is to be assumed that a differentiable submanifold of representing. ${\displaystyle \operatorname {pr} _{1}}$ ${\displaystyle \operatorname {pr} _{1}N=\{x\mid (x,y)\in N\}}$ ${\displaystyle M_{1}}$ ${\displaystyle \mathbb {R} ^{n}}$

Every tangential vector to a solution of the algebro differential equation must also be in the tangent bundle ${\displaystyle (x(t),{\dot {x}}(t))}$ ${\displaystyle x\colon I\to \mathbb {R} ^{n}}$

${\displaystyle TM_{1}:=\left\{({\bar {x}}(0),{\dot {\bar {x}}}(0))\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid {\bar {x}}\in C^{1}((-\epsilon ,\epsilon ),M_{1}){\text{ mit einem }}\epsilon >0\right\}}$

of lie (here means that there is a curve defined on an interval , continuously differentiable once, which lies completely in ). ${\displaystyle M_{1}}$ ${\displaystyle {\bar {x}}\in C^{1}((-\epsilon ,\epsilon ),M_{1})}$ ${\displaystyle {\bar {x}}}$ ${\displaystyle (-\epsilon ,\epsilon )}$ ${\displaystyle M_{1}}$

The tangential vectors in solutions of the algebro differential equation must also be in the set and thus the solutions themselves in the set . ${\displaystyle N\cap TM_{1}}$ ${\displaystyle M_{2}:=\mathrm {pr} _{1}(N\cap TM_{1})}$

This process can be continued (under certain conditions) and the constrained manifold becomes the constrained manifold ${\displaystyle M_{k}}$

${\displaystyle M_{k+1}:=\mathrm {pr} _{1}(N\cap TM_{k})}$

form. It is possible that as a any point in exactly one tangent vector is assigned. Then describes a vector field on the manifold . ${\displaystyle k\in \{0,1,\dotsc \}}$ ${\displaystyle x\in M_{k+1}}$ ${\displaystyle N\cap TM_{k}}$ ${\displaystyle (x,v)}$ ${\displaystyle N\cap TM_{k}}$ ${\displaystyle M_{k+1}}$

The geometric index of the algebro differential equation is precisely the minimal number for which a vector field describes on the manifold . ${\displaystyle k\in \{0,1,\dotsc \}}$ ${\displaystyle N\cap TM_{k}}$ ${\displaystyle M_{k+1}}$

example

The by the equation

${\displaystyle f({\dot {x}},x):={\begin{pmatrix}x_{3}\\{\dot {x}}_{1}-\cos(x_{2})\\{\dot {x}}_{3}-\sin(x_{2})\end{pmatrix}}}$

The defined function and the associated algebro differential equation serve as an accompanying example in the following text.

In the example there are no pairs for all points that are not in the plane defined by . So in this example there are no solutions to the algebro differential equation outside of this level. ${\displaystyle x\in \mathbb {R} ^{3}}$ ${\displaystyle x_{3}=0}$ ${\displaystyle (x,v)\in N}$

It arises and and with it ${\displaystyle M_{1}=\{x\in \mathbb {R} ^{3}\mid x_{3}=0\}}$ ${\displaystyle TM_{1}=\{(x,v)\in \mathbb {R} ^{3}\times \mathbb {R} ^{3}\mid x_{3}=v_{3}=0\}}$

${\displaystyle N\cap TM_{1}=\left\{(x,v)\in \mathbb {R} ^{3}\times \mathbb {R} ^{3}\mid x_{3}=0,\,v_{1}=\cos(x_{2}),\,v_{3}=\sin(x_{2}),\,v_{3}=0\right\}.}$

As you can see, the given tangential vector (des ) for values with due does not lie in the tangent space , so it cannot correspond to a solution of the algebro differential equation system. This results in ${\displaystyle N}$ ${\displaystyle (x,v)}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle x_{2}\neq k\pi }$ ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle v_{3}=\sin(x_{2})\neq 0}$ ${\displaystyle TM_{1}}$

${\displaystyle M_{2}=\operatorname {pr} _{1}(N\cap TM_{1})=\left\{x\in \mathbb {R} ^{3}\mid k\in \mathbb {Z} ,\,x_{2}=k\pi ,x_{3}=0\right\}}$

We obtain

${\displaystyle TM_{2}=\{(x,v)\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid k\in \mathbb {Z} ,\,x_{2}=k\pi ,\,x_{3}=0,\,v_{2}=v_{3}=0\}}$

and the crowd

${\displaystyle N\cap TM_{2}=\{(x,v)\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid k\in \mathbb {Z} ,\,x_{2}=k\pi ,\,x_{3}=0,\,v_{2}=v_{3}=0,\,v_{1}=(-1)^{k}\}}$

assigns exactly one tangential vector to each point from the set (which is exactly the same here ). This is not yet the case for the set, since the component is not yet restricted for tangential vectors from this set . ${\displaystyle x}$ ${\displaystyle M_{3}=\operatorname {pr} _{1}(N\cap TM_{2})}$ ${\displaystyle M_{2}}$ ${\displaystyle N\cap TM_{1}}$ ${\displaystyle v_{2}}$

The geometric index of the algebro differential equation system in this example is therefore equal to two.

If there is a manifold, then this can be expressed with the help of a function in the form ${\displaystyle M_{1}}$ ${\displaystyle g_{1}\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m_{1}}}$

${\displaystyle M_{1}=\{x\in \mathbb {R} ^{n}\mid g_{1}(x)=0\}}$

being represented. The constraining equations in this illustration are called the constraints of the algebro differential equation. ${\displaystyle g_{1}(x)=0}$

In the example: . ${\displaystyle g_{1}(x)=x_{3}}$

In addition to the diversity of a function by means of the manifold are discarded: . The equations with are also referred to as hidden constraints of the algebro differential equation (English: hidden constraints ). ${\displaystyle k=2,3,\dotsc }$ ${\displaystyle M_{k}}$ ${\displaystyle g_{k}\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m_{k}}}$ ${\displaystyle M_{k-1}}$ ${\displaystyle M_{k}=\{x\in M_{k-1}\mid g_{k}(x)=0\}}$ ${\displaystyle g_{k}(x)=0}$ ${\displaystyle k=2,3,\dotsc }$

In the example: . ${\displaystyle g_{2}(x)=\sin(x_{2})}$

Remarks

• The fact that only autonomous algebro differential equations are considered in this section simplifies the geometric interpretation and is not really a restriction, since every time-dependent algebro differential equation can be rewritten into an autonomous algebro differential equation by introducing an additional variable and an additional differential equation . ${\displaystyle f({\dot {x}},x,t)=0}$ ${\displaystyle x_{n+1}:=t}$ ${\displaystyle {\dot {x}}_{n+1}=1}$
• This section has assumed that a submanifold is. If this is not the case, the geometric index is not explained for the algebro differential equation in question. ${\displaystyle M_{k+1}:=\operatorname {pr} _{1}(N\cap TM_{k})}$ ${\displaystyle \mathbb {R} ^{n}}$
• There are also algebro differential equations in which the geometric index is infinite.

Consistent initial values

Again an algebro differential equation is given

${\displaystyle f({\dot {x}},x,t)=0}$

with sufficiently often differentiable. ${\displaystyle f\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n}}$

A point is called the consistent initial value at the time if there is a solution of the algebro differential equation defined in an open interval with for which holds. ${\displaystyle x_{0}\in \mathbb {R} ^{n}}$ ${\displaystyle t_{0}\in \mathbb {R} }$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle t_{0}\in I}$ ${\displaystyle x}$ ${\displaystyle x(t_{0})=x_{0}}$

When calculating, it should be noted that of consistent initial values, in addition to the constraints, the hidden constraints must also be met (see section Geometric Index ).

literature

• Ernst Hairer and Gerhard Wanner: Solving Ordinary Differential Equations II , Stiff and Differential-Algebraic Problems . Second Revised Edition, Springer-Verlag, Berlin, 1996, ISBN 978-3-642-05220-0 (print), ISBN 978-3-642-05221-7 (online), doi : 10.1007 / 978-3-642- 05221-7 .
• Uri M. Ascher and Linda R. Petzold: Computer Methods for Ordinary Differential equations and Differential-Algebraic equations . SIAM, Philadelphia, 1998, ISBN 0-89871-412-5 .
• Peter Kunkel and Volker Mehrmann: Differential-Algebraic Equations . EMS Textbooks in Mathematics, EMS Publishing House, Zurich, 2006, ISBN 3-03719-017-5 , doi : 10.4171 / 017 .
• René Lamour, Roswitha März and Caren Tischendorf. Differential-Algebraic Equations: A Projector Based Analysis . Differential-Algebraic Equations Forum, Springer Berlin Heidelberg, 2013, ISBN 978-3-642-27554-8 (print), ISBN 978-3-642-27555-5 (online), doi : 10.1007 / 978-3-642- 27555-5 .

Individual evidence

1. ^ G. Reissig: Contributions to the theory and applications of implicit differential equations . Dissertation, Dresden University Press, 1998.