Application of the averaging method to the problems of optimal control for ordinary differential equations on the semiaxis (Q2417275)

The control system is of the form \[ x'(t) = X\Big (\frac{t}{\varepsilon}, x(t), u(t) \Big) \qquad x(0) = x_0, \tag{1} \] so that, if $X(x, u)$ is oscillatory in $x$ the right side of (1) oscillates more amd more rapidly as $\varepsilon \to 0$. The control problem is that of minimizing the performance criterion \[ J(u) = \int_0^\infty e^{-jt}L(t, x_\varepsilon(t), u(t)) dt\tag{2} \] among the trajectories of the system. Assuming that the average \[ X_0(x, u) = \lim_{s \to \infty}\frac{1}{s}\int_0^s X(t, x, u) dt \] exists, one can set up the averaged control problem \[ x'(t) = X_0(x(t), u(t)) \qquad x(0) = x_0 \tag{3} \] with the same performance criterion (2). The objective of this paper is that of studying convergence of optimal values of the performance criterion, optimal controls and optimal trajectories of (1) to the optimal value, optimal control and optimal trajectory of (3). The results imply that for small $\varepsilon$ the solution of the optimal control problem for (3) is approximately optimal for (1). \par There are special results for the case where the control appears linearly \[ X\Big (\frac{t}{\varepsilon}, x, u \Big) = f\Big(\frac{t}{\varepsilon}, x\Big) + f_1(x)u \] and the integrand of the performance index is $L(t, x, u) = A(t, x) + B(t, u)$.