2 Basic Definitions and Notions

In a very general form, a DAE can be written as

\[\begin{equation} F(t, x(t), \dot x(t)) = 0 \tag{2.1} \end{equation}\]

with \(F\colon \mathcal I \times D_x \times D_{\dot x} \to \mathbb R^m\) and with a time interval \(\mathcal I=[t_0,t_e) \subset \mathbb R\) and state spaces \(D_x\), \(D_{\dot x} \subset \mathbb R^{n}\) and the task to find a function

\[\begin{equation*} x \colon \mathcal I \to \mathbb R^{n} \end{equation*}\]

with time derivative \(\dot x \colon \mathcal I \to \mathbb R^{n}\) such that (2.1) is fulfilled for all \(t\in \mathcal I\).

A dynamical process that evolves in time needs an initial state. Thus, one can expect a unique solution to the DAEs only if an initial value is prescribed

\[\begin{equation} x(t_0) = x_0 \in \mathbb R^{n}. \tag{2.2} \end{equation}\]

The form of \(F(t, x(t), \dot x(t))\) is a very formal way to write down a system of differential and algebraic equations. X: Write down the equations of the previous examples in this form – i.e. define suitable functions \(F\), \(x\), and \(\dot x\).

2.1 Solution Concept

In order to talk of solutions, we need to define what we understand as a solution.

Definition 2.1 \(\quad\)

  1. A function \(x \in \mathcal C^1(\mathcal I, \mathbb R^{n})\) is called a solution to the DAE (2.1), if \(F(t, x(t), \dot x(t)) = 0\) holds for all \(t\in \mathcal I\).

  2. A function \(x \in \mathcal C^1(\mathcal I, \mathbb R^{n})\) is called a solution to the initial value problem (2.1) and (2.2), if, furthermore, \(x(t_0)= x_0\) holds.

  3. An initial condition (2.2) is called consistent for the DAE (2.1), if there exists at least one solution as defined in 2.

Some remarks

  • The requirement that \(x \in \mathcal C^1\) could be relaxed. Compare Example 1.3, where certain components of the solution where smoother than others.
  • Consistency of initial values is a major issue in the treatment of DAEs. See the pendulum…

2.2 Initial Conditions and Consistency

We consider again the equations of motions of the pendulum (Example 1.1)

\[\begin{align*} \dot x(t) &= u(t) \\ \dot y(t) &= v(t) \\ m \dot u(t) &= - 2 \lambda(t) (x(t) - c_x) \\ m \dot v(t) &= - 2 \lambda(t) (y(t) - c_y) - mgy(t) \end{align*}\]

with the constraint

\[\begin{equation} 0=(x(t) - c_x)^2 + (y(t) - c_y)^2 - l^2. \tag{2.3} \end{equation}\]

To use this model to predict the time evolution of the system, a starting point needs to be known, say for \(t=0\). This means initial positions and initial velocities:

\[ \begin{bmatrix} x(0) \\ y(0) \end{bmatrix} = \begin{bmatrix} x_0 \\ y_0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} u(0) \\ v(0) \end{bmatrix} = \begin{bmatrix} u_0 \\ v_0 \end{bmatrix}. \]

The constraint (2.3) needs to be fulfilled at all times and also at \(t=0\), which gives the constraint for the initial positions:

\[\begin{equation*} (x_0 - c_x)^2 + (y_0 - c_y)^2 - l^2=0. \end{equation*}\]

Moreover, if a constraint \(h(x(t), y(t))=0\) holds for all \(t\), then, necessarily, \(\frac{d}{dt}h=0\). For the pendulum this means that

\[\begin{equation} 2(x(t) - c_x)u(t) + 2(y(t) - c_y)v(t) = 0 \tag{2.4} \end{equation}\]

must hold for all \(t\) and in particular at \(t=0\) which gives constraints on the initial velocities \(u_0\) and \(v_0\):

\[\begin{equation*} 2(x_0 - c_x)u_0 + 2(y_0 - c_y)v_0 = 0. \end{equation*}\]

Some remarks on consistency, constraints, and derivations:

  • The so-called consistency conditions on \((x_0, y_0, u_0, v_0)\) have the physical interpretation that the initial positions lie on the prescribed circle and that the velocities are tangent to this circle.
  • One can show that the variable \(\lambda\) is completely defined in terms of \(x\) and \(y\) and their derivatives. Thus, in the formulation (1.1), both in the analysis and in the numerical treatment, there is no need for an initial value for \(\lambda\). However, as we will see, DAEs can be reformulated as ODEs through differentiation and substitutions. In such an ODE formulation, a necessary initial condition for \(\lambda\) will have to fulfill similar consistency conditions as \((x_0, y_0, u_0, v_0)\).

Condition (2.4) is an example for a hidden-constraint – an algebraic constraint to the system that is not explicit in the original formulation. In theory, condition (2.3) can be replaced by (2.4). Moreover, through differentiation and elimination of constraints, a DAE can be brought into the form of an ODE: in the case of the circuit of Example 1.2 one only needs to replace the constraints by their derivatives:

\[\begin{equation} \begin{split} C(\dot x_3 - \dot x_2) &= - \frac{x_1 - x_2}{R} \\ \dot x_1 - \dot x_3 &= \dot U \\ \dot x_3 &= 0. \end{split} \tag{2.5} \end{equation}\]

Note that (2.5) can be written as \(B\dot x = Ax + f\) with an invertible matrix \(B\) and, thus, is an ODE.

For an ODE there is no constraint on the initial values. However, a solution to (2.5) only solves the original DAE (1.2), if the initial values are consistent with the DAE. In this case, this means \(x_3(t_0)=0\) and \(x_1(t_0) - x_3(t_0) = U(t_0)\).

2.3 Additional Remarks

  • It just took a single derivation to turn the circuit model into an ODE (2.5). For the pendulum this wouldn’t be that easy.

  • The extend of how much algebraic and differential parts are intertwined is measured by indices which is the classifier for DAEs.

  • There are many indices. We will learn about some of the concepts. But first we will introduce some more theory.

A low index means that differential and algebraic parts are relatively well separated. (The circuit example is of index 1). A high index means that the structure is more involved. (The pendulum is of index 3).