Necessary conditions of optimality for measure driven differential inclusions

Minimize h(x(0), x(1)) subject to dx(t) ∈ F (t, x(t))dt + G(t, x(t))dμ(t) ∀t∈[0, 1] (x(0), x(1)) ∈ C ⊂ IR × IR l(t, x(t)) ≤ 0 ∀t∈[0, 1] dμ ∈ K where h : IR × IR → IR, and l : [0, 1]× IR → IR are given functions, F : [0, 1]× IR → P(IRn), and G : [0, 1] × IR → P(IRn×q) are given set-valued maps, K ⊂ C∗([0, 1];K) is the set of control measures supported on [0, 1] with range in a given set K ⊂ IR, and C ⊂ IR × IR is closed. This control paradigm can be regarded as an idealization of systems with fast and slow dynamics. This is pertinent to important classes of systems with multi-phase missions or reconfigurable dynamics for which the switching between different “productive” activities represented by slow dynamics are modeled by fast dynamics. Note that, if we consider F (t, x) := {f(t, x, u) : u ∈ Ω} and G(t, x) := {G(t, x, u) : u ∈ Ω}, where the measurable function u plays the role of the conventional control taking values in a given compact set Ω ⊂ IR, then it is not difficult to see that this paradigm encompasses impulsive optimal control problems where dynamics are specified by controlled differential equations. Moreover, the dependence of the singular dynamics on the “conventional” control constitutes an interesting challenge with practical implications. This issue is partially addressed in [12]. This optimal control problem has also been considered in [9], but the stated optimality conditions are of different character. The relationship between hybrid systems whose evolution is defined by the interaction of time-driven and event-driven discrete dynamics and impulsive systems has addressed in [4, 5]. The importance of the former is due to the emergence of the so-called networked systems. To better understand the extent to which the measure driven differential paradigm enables the composition of dynamic control systems, property at the crux of hybrid automata, a popular model for hybrid systems, just consider x = col(y, z), a certain index set A, and Z = {zα : α ∈ A}, and note that the impulsive system    ẏ = f(y, z, u), u ∈ Ω dz = g(y, z)dμ