## 3 Mathematical Modeling
### Suboptimal Control in Nonlinear Dynamics  
Control laws are intrinsically connected to model constructions for physical systems, essentially relying on up-to-date knowledge about the system under study. However, accurate models for real physical systems may be difficult to realize, or even impossible, as the models will be impregnated by uncertainties, mostly coming from unmodeled dynamics, input disturbances, hardware precision and so on, which will engender deviations when comparing the predicted/modeled to real system behaviors. As stated by [87], \"No mathematical system can precisely model a real physical system; there is always uncertainty\". An effective way to correct these deviations is by means of feedback laws (assuming the system is observable), in which the states or outputs depicting actual system behavior are compared to those predicted by the model and, in the case of any discrepancy beyond a pre-assigned limit, correction actions are executed to bring the system back to the prescribed behavior, either via regulation (station-keeping around an equilibrium) or tracking (following a reference trajectory), in due time and considering energy availability. To implement such approaches, consider a dynamical system with a state-space nonlinear model  
\[\dot{\mathbf{x}}(t) =\mathbf{f}(t,\,\mathbf{x}(t),\,\mathbf{u}(t)),\,\,\,\,\mathbf{t }\geq t_{0}\, \tag{8}\] \[\mathbf{x}(t_{0}) =\mathbf{x}_{0}\,,\]  
in which \(\mathbf{x}\in\Omega\subset\mathbb{R}^{n}\) are the allowed or reachable states, \(\mathbf{u}\in\mathcal{U}\subset\mathbb{R}^{m}\) are unconstrained admissible or executable controls, and the dynamics \(\mathbf{f}:\mathbb{R}\times\Omega\times\mathcal{U}\rightarrow\mathbb{R}^{n}\) is assumed sufficiently differentiable. An unconstrained optimization problem for nonlinear regulation (station-keeping) may then be stated based on the determination of a stabilizing feedback control law \(\mathbf{u}\) that \"regulates the system states \(\mathbf{x}\) around a prescribed equilibrium \(\mathbf{x}_{\epsilon}\)\" while minimizing a mission-designed cost function, or performance index (PI), usually conditioning the control effort \(\mathbf{u}\in\mathcal{U}\), the states \(\mathbf{x}\in\Omega\) and the execution time interval \(I=[t_{0},\,t_{f}]\subset\mathbb{R}\), which may be translated to [88; 89]  
\[PI=\mathbf{\Phi}_{j}(\mathbf{x}_{j})+\int\limits_{t_{0}}^{t_{f}}\Psi(\,t,\, \mathbf{x}(t),\,\mathbf{u}(t)\,)dt\, \tag{9}\]  
Figure 5: Planar _ZVCs_ for 2001\(SN\)263_Alpha. Equilibrium points are located only some tens of meters above the body’s 2.8 km \(\times\) 2.7 km equatorial line (black circle). (St)table equilibrium: \((x_{\epsilon},y_{\epsilon})=(0,\pm 1450.75\,\mathrm{m})\); \(J=-0.8424\); Green curves surrounding the stable equilibria: \(J=-0.8465\); (U)nstable equilibrium: \((x_{u},y_{u})=(\pm 1480.39\,\mathrm{m},0)\); \(J=-0.853\). Blue curve: \(J=-1.2\). for some sufficiently differentiable function \(\Psi\colon\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{+}=\{z \in\mathbb{R}:z>0\}\) and a symmetric positive semi-definite weighting matrix \(\mathbf{\Phi}\colon\mathbb{R}^{n}\to\mathbb{R}^{+}_{o}=\{z\in\mathbb{R}:z\geq 0\}\) which accounts for under-performance costs/penalties due to eventual errors in achieving \(\mathbf{x}_{\text{\tiny$\prime$}}\) and may also translate eventual bounds imposed to the final states.