*2.1. The ODE Initial-Value Problem (or Cauchy Problem [1])*

Let us consider the following Ordinary Differential System:

$$\frac{d\mathbf{x}(t)}{dt} = f(\mathbf{x}(t), \mathbf{u}(t)) \tag{1}$$

where *x*(*t*) = *x*(0) at t = 0 is the initial value.

The Picard-Lindelöf theorem [50] guarantees the existence of a unique solution to (1). The principle of this theorem consists in reformulating the problem as an equivalent integral equation:

$$\mathbf{x}(t) = I\_t^1 \left[ \frac{d\mathbf{x}(t)}{dt} \right] + \mathbf{x}(0) = \int\_0^t f(\mathbf{x}(\tau), \boldsymbol{\mu}(\tau)) d\tau + \mathbf{x}(0) \tag{2}$$

and to construct a sequence of functions

$$\phi\_{k+1}(t) = \int\_0^t f(\phi\_k(\tau), u(\tau))d\tau + x(0) \text{ with } \phi\_1(t) = x(0), \tag{3}$$

which converges to the solution of (1) and thus to the solution of the initial-value problem. Such a construction is called Picard's method [51] or the method of successive approximations. In the linear and multidimensional case, Equation (1) can be expressed as:

$$\frac{d\underline{x}(t)}{dt} = A\underline{x}(t) + Bu(t) \text{ x}(t) = x(0) \text{ at } t = 0,\tag{4}$$

where *x*(*t*) -*RN*, and *A* and *B* are matrices of appropriate dimensions.

It is well known that its solution, based on the exponential matrix function or transition matrix

$$\Phi(t) = e^{At} \text{ with } \Phi(t) \in \mathbb{R}^{N \times N}$$

is given by [52]:

$$\mathbf{x}(t) = \Phi(t)\mathbf{x}(0) + \int\_0^t \Phi(t-\tau)B u(\tau)d\tau. \tag{5}$$
