Linear Time Optimal Systems

Abstract

In this section we return to the linear system

$${\rm \dot x}\, = \,{\rm A(t)x}\,{\rm + }\,{\rm B(t)u}\,{\rm + }\,{\rm c(t),} $$

(L)

where A(·), B(·), and c(·) are continuous on [0, ∞), the general target

$${\rm G}\,{\rm :}\,[0,\infty ) \to {\rm P}$$

which is continuous (in the Hausdorff metric), the cost functional

$${\rm C}\, = \,{\rm C(u(} \cdot {\rm )}\,{\rm )}\, = \,{\rm t}_{\rm 1} \, - \,{\rm t}_{\rm O} ,$$

the control region

$$\Omega _{\rm C} \, = \,\left\{ {{\rm u}\,{\rm :}\,\left| {{\rm u}^{\rm i} } \right|\, \le \,1\,{\rm for}\,{\rm all}\,{\rm 1}\, \le {\rm i} \le \,{\rm m}} \right\},$$

and the admissible control class U = U_M. We shall prove an existence theorem (Theorem 6.2) and a necessary condition (Theorem 6.5) for optimality, neither of which is as general as those which will be presented in §7 and §8. However, in linear time optimal case, we are able to see some of the geometric aspects of control which are not at all obious in the general case.