On solution problem of optimal control by variational methodsстатья
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Аннотация:We consider the optimal control problem of the system $$\gathered \dot x= f(x,u),\quad u\in V,\quad \dim u=m;\\ x\in\bbfR^n,\quad x(t_0)= x_0,\quad x(t_1)= x_1.\endgathered\tag 1$$ The optimality of control is determined by the functional $$J= \int^{t_2}_{t_1} f_0(x(t), u(t),t) dt.\tag 2$$ A control is optimal if it transfers the system from the point $x_0$ to a point $x_1$ and minimizes the functional $J$. This paper reduces the optimal control problem to a variational problem of finding a constrained extremum of the functional, which can be solved by the Lagrange multiplier method. The main difficulty connected with the fact that the controls $u(t)$ are restricted to a domain is overcome by introducing new controls that are not constrained