General Control Systems: Necessary Conditions

Abstract

We begin with an autonomous control problem and a point target, i. e., we consider the equation

$${\rm \dot x}\,{\rm = }\,{\rm f(x}\,{\rm ,}\,{\rm u)}\,{\rm ,}$$

(A)

any Control region Ω ⊂R^m, the admissible class U = U_M, the point target

$${\rm G(t)}\, \equiv \,{\rm x}_{\rm 1} \,{\rm for}\,{\rm - }\,\infty \,{\rm < }\,{\rm t}\,{\rm < }\,\infty \,{\rm ,}$$

an initial point x_o∈Rⁿ, and the cost functional

$${\rm C(u(} \cdot {\rm ))}\,{\rm = }\,\int_{{\rm t}_{\rm o} }^{{\rm t}_{\rm 1} } {{\rm f}^{\rm o} \,({\rm x(t)}\,{\rm ,}\,{\rm u(t))}\,{\rm dt}\,{\rm .}}$$

(8.1)