# 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 $$x_1(t_0) - x_3(t_0) = U(t_0)$$.