## 3 Target Functions  
In this section, we introduce various target functions, abbreviated as STF--System Target Function, useful for achieving the goal. The main targets to pursue are as follows:  
1. Knowledge of the system's state and the situations active on it;
2. Completeness and, therefore, the reliability of the information acquired through georeferencing information and the temporal localization of events;
3. The ability to associate and contextualize events within classifiable scenarios, allowing for the identification of the following: 1. Decision strategies (DS); 2. Management strategies (MS); 3. Operational strategies (OS). The mathematical formulation of the problem can be presented in two different types and with different levels of abstraction. A very advanced and extremely realistic modelling should consider memory effects, which, from a modelling point of view, imply the presence of first-order (or even higher) derivative terms for the variables at play, i.e., \(X_{i}\). Conversely,a simpler approach is typical of control theory, which often stops at formulation in terms of linear combinations. In other words, we are assuming that we do not know the exact f and are analyzing it at different levels of approximation (non-linear in the first case, linear in the second). Regarding the level of abstraction, as we will see, in a first approximation, we will deal with an \(f:X\subset\mathbb{R}^{10}\to Y\subset\mathbb{R}^{7}\), but from a more refined analysis, it is understood that such a function lives in a weighted Sobolev space \(\mathrm{H}^{2,2}\mathrm{G}\), where G is an appropriate weight. In fact, concerning the components of f, if we consider the first one, which represents \"knowledge of the state of the system and the situation active on it\", it is important to know both how the rate (\(f_{1}^{\prime}\)) varies and how this variation (\(f_{1}^{\prime\prime}\)) varies. Since we want these functions not to diverge from the norm, they must be square-summable. Furthermore, needing weighted spaces explains why it is necessary to consider \(\mathrm{H}^{2,2}\mathrm{G}\). It is also evident that for complete use of the model, historical information will also be necessary to create a system that is both adaptive and contextualized to support decision-making. Going into implementation details, we define the first target function STF_1 = \(f_{1}\) in (1) and (2) as follows:  
\[f_{1}:A\times\ldots\times A\subset\mathbb{R}^{7}\to B\times C\subset \mathbb{R}^{2} \tag{1}\]  
where \(A=\{0,1,2,3,4,5\}\subset\mathbb{N}\) e \(B=[0,12]\subset\mathbb{N}\), \(C=[7,35]\subset\mathbb{N}\). From a computational point of view, we have  
\[f_{1}=(E,S)=\bigg{(}\frac{1}{7}\sum_{i=1}^{7}\alpha_{i}X_{i},\frac{1}{7}\sum_ {i=1}^{7}\alpha_{i}\bigg{|}X_{i}-\widetilde{X}\bigg{|}\bigg{)} \tag{2}\]  
where E represents the internal energy of the system represented by the considered state, S is its Entropy, \(\widetilde{X}\) is the median, and the coefficients \(\alpha_{i}\) are suitable normalization coefficients, defined as follows:  
\[\alpha_{i}=\frac{\beta_{i}}{5} \tag{3}\]  
with \(\beta_{i}\) as the activation parameter (or relative weight), and as such, \(\beta_{i}\in[0,1]\subset\mathbb{R}\). In the absence of historical information, we assume \(\beta_{i}=1\), and it is also possible to choose not to normalize (i.e., multiply by \(1/7\)), as we will do later for simplicity, without a loss of generality. Formally, it is common to say that \(f_{i}\in BV(0,1)\), where BV is the space of functions with bounded values in the interval \([0,1]\subset\mathbb{R}\). The choice of the aforementioned interval comes from the fact that, this way, the target function f1 will return a value between zero and one, which can be directly associated, through a fuzzification process, with a probability of incidence of the CSFs on a specific systemic alteration. Therefore, the STF function f1 responds to the first objective, i.e., knowing the state of the system and the situations active on it.