## 4 Impulse PIN-Type Control  
Epidemic diseases are a complex phenomenon in which numerous factors have effects, and due to their dynamic nature in terms of propagation, control theory methods are an area of science that provide us with interesting and useful alternatives for the distribution of vaccines or treatments. In these types of systems [15], constant or impulsive controllers can be developed to control the spread of infectious diseases. For the case of constant control, vaccine application occurs in a constant period of time \(0\leq p\leq 1\), where 95% of the entire population needs to be vaccinated in order to fully eradicate the disease [16], resulting in a high-cost solution. On the other hand, impulse control, as its name implies, applies a periodical vaccination to a portion p of the population for a very short period of time [15; 16; 17; 18; 19], this is named as impulsive dynamical control [13], modeled as follows:  
\[\dot{x}(t)=f_{c}(x(t)),\,\,\,x(0)=x_{0},\,\,\,t\
eq t_{k} \tag{8}\]  
\[\Delta_{x}(t)=f_{d}(x(t)),\,\,\,t=t_{k} \tag{9}\]  
In an impulse control scheme, control inputs are introduced at established time instances, where \(t\geq 0\), \(x(t)\in D\subseteq\Re^{n},\,\,D\) is an open set with \(0\in D\), \(\Delta_{x}(t)\triangleq x(t^{+})-x(t)\), with \(x(t^{+})\triangleq x(t)+f_{d}(x(t))=[\lim_{t\to 0}x(t+\varepsilon)],\,\,\,f_{c}:D \rightarrow\Re^{n}\) is continuous, \(f_{d}:D\rightarrow\Re^{n}\) is continuous and \(T=\{t_{k}\}\) is the reset set with \(t_{k}\in\Re\), \(t_{k}<t_{k+1}\), \(k=1,2,\cdots\). The dynamics of \(f_{c}(\cdot)\) are assumed to ensure that the solution of (8) is continuous with respect to both \(t\) and \(x_{0}\) in time interval \(t_{k}\). Equation (8) is referred to as the continuous-time dynamic while (9) is the impulsive control law. The function \(s(t,\tau,x_{0})\) represents the solution of \(x(t)\) subject to (8) and (9) for \(t\geq\tau\) with an initial value defined as \(x=x_{0}\). Finally, the point \(x_{\varepsilon}\in D\) is the equilibrium of (8) and (9) if and only if \(s(t,\tau,x_{\varepsilon})=x_{\varepsilon}\) for all \(\tau\geq 0\) and \(t\geq\tau\). It should be noted that \(x_{\varepsilon}\in D\) is an equilibrium of (8) and (9) if and only if \(f_{c}(x_{\varepsilon})=0\) and \(f_{d}(x_{\varepsilon})=0\). \[\dot{x}(t)=f_{c}(x(t))+G_{c}(x(t))u_{c}(t),\,\,\,x(0)=x_{0}\,\,\,\,t\
eq t_{k} \tag{10}\]  
\[\Delta x(t)=f_{d}(x(t))+G_{d}(x(t))u_{d}(t),\,\,t=t_{k} \tag{11}\]  
where \(u_{c}(\cdot)\) and \(u_{d}(\cdot)\) are restricted to a class of admissible hybrid controls consisting of measurable functions such that \((u_{c}(t),u_{d}(t_{k}))\in U_{c}\times U_{d}\) for all \(t\geq 0\) and \(t_{k}\in\Re\), \(t-k<t_{k}+1\), \(k=1,2,\ldots\), \(L_{1c}:\Re^{n}\rightarrow\Re\) with \(L_{1c}(x)\geq 0\), \(L_{1d}:D\rightarrow\Re\) in addition to \(L_{1d}(x)\geq 0\), \(x\in\Re^{n}\) and \(R_{2d}:\,\,D\to p^{m_{d}}\). \[J(x_{0},u_{c}(\cdot),u_{d}(\cdot)) =\int_{0}^{\infty}\Bigl{[}L_{1c}(x(t))+{u_{c}}^{T}(t)R_{2c}(x(t))u_{ c}(t)\Bigr{]}dt \tag{12}\] \[+\sum_{k\in Z_{[0,\infty)}}\Bigl{[}L_{1d}(x(t_{k}))+{u_{d}}^{T}(t_ {k})R_{2d}(x(t_{k}))u_{d}(t_{k})\Bigr{]}\]  
Then, the solution \(x(t)\equiv 0\) of the closed-loop system (10) and (11) is globally asymptotically stable with the optimal hybrid feedback control law. \[u_{c}(t)=\phi_{c}(x)=-\frac{1}{2}R_{2c}^{-1}(x)G_{c}^{T}(x){V^{\prime}}^{T}(x), \ t\
eq t_{k} \tag{13}\]  
\[u_{d}(t)=\phi_{d}(x)=-\frac{1}{2}\Bigl{(}(R_{2d}(x)+P_{2}(x))^{-1}P_{12}^{T}(x )\Bigr{)},\ t=t_{k} \tag{14}\]  
and (12), with  
\[L_{1c}(x) =\phi_{c}^{T}(x)R_{2c}(x)\phi_{c}(x)-Vt(x)f_{c}(x) \tag{15}\] \[L_{1d}(x) =\phi_{d}^{T}(x)(R_{2d}(x)+P_{2}(x))\phi_{d}(x)-V(x+f_{d}(x))+V(x)\]  
for  
\[\Delta_{x}(t)=f_{d}(x(t)),\,\,\,t=t_{k} \tag{9}\]0 \[\Delta_{x}(t)=f_{d}(x(t)),\,\,\,t=t_{k} \tag{9}\]1  
where \(\phi_{c}:D\to U_{c}\), \(\phi_{d}:D\to U_{d}\) and \((u_{c}(t),u_{d}(t_{k}))=(\phi_{c}(x(t)),\phi_{d}(x(t_{k})))\) satisfying (10) and (11) for \(x(t)\), \(t\geq 0\).