Optimal control problems with singular modes for a model from microbiology (Q5938978)

A bilinear control system, which simulates the process of microbial colony growth, with the limited control of the form \[ dx/dt = a-(a+x)v, \quad t\in[0,T],\quad x(0) =x_0, \] \[ J=\int_0^T -\frac{x}{(1+x)}v dt\to\min_{v(\cdot)}, \quad |v(t)|\leq 1, \] where \(x\) and \(v\) are a one-dimensional phase and a controlling variable, respectively, and the left end of the trajectory \(x(0)= x_0\) is fixed and the right end \(x(T)\) is free, is considered. The parameters \(a\) and \(T\) are assumed to be given. The control function \(v\) is searched in a class of piece-wise continuous functions. On the basis of the Pontryagin maximum principle the optimal control containing sites of the special control is constructed. The analysis and substantiation of the obtained solutions are carried out.