Some remarks on the existence of optimal controls for quasilinear systems

Abstract

Let a quasilinear control system having the state space$\bar X \subseteq R^n$ be governed by the vector differential equation

$$\dot x = G(u(t))x,$$

wherex(0) =x ₀ andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR ^m.LetG:U ⇑R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G ⁻¹ can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional

$$I(u) = \int_0^T {L(u(t))x(t)dt,} $$

whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.