Asymptotically optimal control in a linear control system with fast and slow variables (Q1683301)

The system is \[ \begin{alignedat}{2} x'(t) &= \varepsilon (f(t, x(t), y(t)) + A(x(t))u(t)) , \quad &x(0) &= x_0 \\ y'(t) &= g(t, x(t), y(t)) , \quad &y(0) &= y_0 \end{alignedat} \qquad 0 \leq t \leq T \tag{1} \] where \(\varepsilon\) is a small parameter, \(T = L/\varepsilon,\) the components of \(x(t) \in {\mathbb R}^n\) are the \textit{slow variables} and the components of \(y(t) \in {\mathbb R}^m\) are the \textit{fast variables}. Controls take values in a convex compact set \(U \subset {\mathbb R^r}\) and are \textit{admissible} if the solution exists in \(0\leq t \leq T.\) The control problem is \[ \text{minimize }J(u) = \varphi(x(T)) \tag{2} \] over admissible controls. The \textit{degenerate} system is obtained setting \(\varepsilon = 0\) in (1), which makes \(x\) constant, \[ y'(t) = g(t, x, y(t)) \, , \qquad y(0) = y_0 \, .\, \quad x = x(0) \tag{3} \] and the \textit{averaged} control problem is \begin{align*} z'(t) &= \varepsilon (f_0(z(t)) + A(z(t))v(t)) , \qquad z(0) = x_0 \tag{4} \\ \text{minimize }J(v) &= \varphi(z(T)) \tag{5} \end{align*} where \(f_0(x)\) is the average \[ f_0(x) = \lim_{T \to \infty} \int_{t_0}^{t_0 + T} f(s, x, h(s, x, y_0))ds \] with \(h(t, x, y_0)\) the solution of (3). The main result establishes that, under suitable conditions the solution of (1)--(2) approximates the solution of (4)--(5) when \(\varepsilon \to 0\).