Problems of control of distributions in dynamical systems
Some problems of optimal control of distributions are considered. One of the most characteristic problem of this type may be formulated as follows: The dynamic of the system is given by the equation \[ \dot x(t)= f(x(t), u(t))\quad\text{with }x(t)\in \mathbb{R}^n\quad\text{and }u(t)\in U\subset \mathbb{R}^n.\tag{1} \] The performance index is of the form \[ \int_{\mathbb{R}^n} \phi(x)\zeta(T, x) dx\to \min,\tag{2} \] where the function \(\zeta= \zeta(t, x)\), \(t\in [0,T]\), \(x\in\mathbb{R}^n\) satisfies the continuity equation \[ {\partial\over\partial t} \zeta(t,x)= \text{div}_x[\zeta(t, x) f(x,u)]= 0\tag{3} \] and \[ \int_{\mathbb{R}^n} \zeta(t, x) dx\equiv 1,\quad \zeta(x,t)\geq 0. \] The function \(\zeta(0,y)= \zeta_0(y)\) is interpreted as the probability density that the initial value for the dynamical system (1) takes the value \(x(0)= y\) at the moment \(t= 0\). For systems of type (1)--(3) some necessary optimality conditions in the form of the Pontryagin maximum principle and some analogs of the Feldbaum theorem on the number of switchings are proved.
scientific article; zbMATH DE number 1290433