Internal layer solution of singularly perturbed optimal control problem with integral boundary condition (Q1641832)

The paper deals with the optimal control problem for linear singularly perturbed first-order ordinary differential equations with an integral boundary condition of the form \[ J[u]=\int_0^T f(y,u,t)dt\rightarrow \min_u, \] \[ \mu\frac{dy}{dt}=a(t)y+b(t)u, \] \[ y(0,\mu)=y^0,\;y(T,\mu)=\int_0^T g(s)y(s)ds, \] where the state variable \(y\in\mathbb{R},\) control input \(u\in\mathbb{R}\) and \(\mu\) is a small positive parameter; the functions \(f(y,u,t),\) \(a(t),\) \(b(t)\) and \(g(t)\) are sufficiently smooth, moreover, \(b\) is positive and \(f\) is convex as function of the variable \(u\) for every \(t\in[0,T]\) and \(|y|< A,\) where \(A\) is some constant. By using the \(k+\sigma\) Exchange Lemma the authors proved the existence of optimal solution with an internal layer, and consequently, the boundary function method and asymptotic expansion techniques are applied to construct the formal asymptotic solution and to determine the location of the internal transition point \(t^*\).