Nonlinear optimal controls

✍ Scribed by A.A. Goldstein; J.S. Meditch

• Publisher: Elsevier Science
• Year: 1970
• Tongue: English
• Weight: 296 KB
• Volume: 4
• Category: Article
• ISSN: 0022-0000
• DOI: 10.1016/s0022-0000(70)80010-1

✦ Synopsis

In this paper we state a necessary condition for optimality for unconstrained controls. For certain systems a constructive proof is given showing the existence of controls satisfying this condition. I. STATIONARY CONTROLS For the system ~(t) = f(x(t), u(t), t), (t) on the fixed time interval [0, T] with fixed boundary conditions x(O) and x(T) E E~, assume that there exists a continuous control u, u(t) ~ E r , such that T fo g(u(t)) dt is minimized. Assume further that f has partial Jacobians with respect to x and u, denoted respectively by Jx(x(t), u(t), t) and J~(x(t), u(t), t), and let Xu(t ) denote the fundamental matrix of the system

with u fixed, where x(t) is the corresponding solution of (1), and xo(t ) the solution of (2).

We denote by U the Hilbert space of r-tuples of square-integrable functions u on [0, T], with norm I1 u ]i 2 = f0rl u(t)l 2 dt, where I u(t)] * = [u(t), u(t)]. By L we denote the Hilbert space of n-tuples of such functions normed as above.