The existence of optimal controls for problems defined on time scales (Q493053)

In this well written paper, the existence of optimal controls is discussed for optimal control problems on time scales. The theory starts with a very nice survey on integration theory on time scales \(\mathbb{T}\), for example \(\mathbb{T} = \mathbb{R}, \, \mathbb{T} = \mathbb{Z}\), or \(\mathbb{T}= 1 -q^{\mathbb{N}\cup \{0\}}\) with \(q \in (0,1)\). In particular, the notion of the \(\Delta\)-derivative \(x^{\Delta}\), of the Lebesgue-\(\Delta\)-measure and of the Lebesgue-\(\Delta\)-integral \(\int_{[r,s)\cap \mathbb{T}} f(t) \Delta t\) are introduced. Main result is a theorem on the existence of an optimal control for the problem of minimizing \[ J[x,u] = g(t_1,x(t_1),t_2,x(t_2)) + \int_{[t_1,t_2)\cap \mathbb{T}} f_0(t,x(t),u(t))\, \Delta t \] subject to the differential equation on the time scale \(\mathbb{T}\) \[ x^{ \Delta}(t) = f(t,x(t),u(t)) \quad \Delta \text{-a.e. on } (t_1,t_2)\cap {\mathbb{T}}, \] under appropriate constraints on \((t_1,x(t_1),t_2,x(t_2))\) and pointwise constraints on \(x\) and \(u\).