# New Aspects of Solution Feasibility in a Context of Personalized Therapy Optimization
## 2 Problem Formulation
Then, a real-valued performance index can be considered  
\[\min_{u(t)}\quad F(y(t),z(t),u(t),p,t) \tag{9}\]  
with a system of dynamical constraints  
\[dy/dt=f(y(t),z(t),u(t),p,t) \tag{10}\]  
and continuous algebraic relations  
\[0=g(y(t),z(t),u(t),p,t), \tag{11}\]  
with  
\[F:\mathcal{R}^{ny}\times\mathcal{R}^{n_{z}}\times\mathcal{R}^{n_{u}}\times \mathcal{R}^{n_{p}}\times\mathcal{R}\rightarrow\mathcal{R} \tag{12}\]  
\[f:\mathcal{R}^{ny}\times\mathcal{R}^{n_{z}}\times\mathcal{R}^{n_{u}}\times \mathcal{R}^{n_{p}}\times\mathcal{R}\rightarrow\mathcal{R}^{n_{y}} \tag{13}\]  
\[g:\mathcal{R}^{ny}\times\mathcal{R}^{n_{z}}\times\mathcal{R}^{n_{u}}\times \mathcal{R}^{n_{p}}\times\mathcal{R}\rightarrow\mathcal{R}^{n_{z}} \tag{14}\]  
where \(\mathcal{R}\) denotes a set of real numbers. Moreover,  
\[y(t)=\left[ \begin{array}{c}y_{1}(t)\\ \vdots\\ y_{ny}(t)\end{array} \right]\in\mathcal{R}^{ny} \tag{15}\]  
is a vector of differential state trajectories;  
\[z(t)=\left[ \begin{array}{c}z_{1}(t)\\ \vdots\\ z_{n_{z}}(t)\end{array} \right]\in\mathcal{R}^{n_{z}} \tag{16}\]  
is a vector of algebraic state trajectories;  
\[u(t)=\left[ \begin{array}{c}u_{1}(t)\\ \vdots\\ u_{n_{u}}(t)\end{array} \right]\in\mathcal{R}^{n_{u}} \tag{17}\]  
is a vector of control functions;  
\[p=\left[ \begin{array}{c}p_{1}(t)\\ \vdots\\ p_{n_{p}}(t)\end{array} \right]\in\mathcal{R}^{n_{p}} \tag{18}\]  
is a vector of global constant parameters; \(t\in\mathcal{R}\) is the independent variable; and  
\[t\in[t_{0}\quad t_{f}]. \tag{19}\]  
A vector of initial conditions for state trajectories \(y(t)\) at the initial point \(t=t_{0}\) is given  
\[y(t_{0})=y_{0}. \tag{20}\]The constraints on the control function take a form of lower and upper bounds  
\[u_{L}\leq u(t)\leq u_{U}. \tag{21}\]  
The pointwise constraints related to the final state trajectories \(y(t)\), in particular  
\[y_{i}(t_{f})=y_{i,t_{f}}\qquad\text{for}\quad i=1,\ldots,n_{y} \tag{22}\]  
can be reflected in the performance index in the compact form  
\[\min_{u(t)}\sum_{i=1}^{n_{y}}\rho_{i}\Big{(}y_{i}(t_{f})-y_{i,t_{f}}\Big{)}^{2} \tag{23}\]  
where \(\rho_{i}\in\mathcal{R},i=1,\ldots,n_{y}\) are problem-related adjustable penalty parameters. The presented infinitely dimensional task can be transformed into a finite-dimensional one with, e.g., a control parametrization method. This approach is based on dividing the range of the independent variable into an assumed number \(N\) sub-intervals such that  
\[t_{0}=t_{0}^{1}<t_{f}^{1}=t_{0}^{2}<\cdots=t_{0}^{N}<t_{f}^{N}=t_{f}. \tag{24}\]  
The parametrization of the control function \(u(t)\) is carried out separately in each sub-interval. In particular, a step function is often used, which causes the control function to take a constant value in each subinterval  
\[u(t^{i})=u^{i}=const\qquad\text{for}\quad t^{i}\in[t_{0}^{i}\quad t_{f}^{i}] \quad\text{and}\quad i=1,\ldots,N. \tag{25}\]  
The values of the state variables at the end of a subinterval are treated as starting values for the next subinterval. Therefore, Equations (9) and (11) can take the form of a finite-dimensional optimization task with an objective function  
\[\min_{\mathbf{U}\in\mathcal{R}^{n_{u}\times N}}\widetilde{F}(\mathbf{U}) \tag{26}\]  
with a matrix of decision variables, which combine relation (25) with the vector representation (17)  
\[\mathbf{U}=\left[ \begin{array}{ccc}u_{1}^{1}&\cdots&u_{1}^{N}\\ \vdots&\ddots&\vdots\\ u_{n_{u}}^{1}&\cdots&u_{n_{u}}^{N}\end{array} \right]\in\mathcal{R}^{n_{u} \times N} \tag{27}\]  
as well as the constraints vector  
\[c(\mathbf{U})=0 \tag{28}\]  
with  
\[c:\mathcal{R}^{n_{u}\times N}\rightarrow\mathcal{R}^{n_{c}} \tag{29}\]  
The vector of decision variables \(\mathbf{U}^{*}\), which is considered to be a solution to the given optimization problem, should meet all of the imposed constraints (28). The key problem at this point is how to understand the fulfillment of the constraints and what kind of constraints should be taken into account.