On the existence of optimal controls for nonlinear systems

Abstract

Let a nonlinear control system having the state space$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\bar X} \subseteq R^n$ be governed by the vector differential equation

$$\dot x\left( t \right) = f\left( {t, x\left( t \right), u\left( t \right)} \right),$$

wherex(0)=x ₀ and

is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR ^m. Letg:U→R ^m be a bounded measurable function such thatg(U) is compact and convex, and letF be a function from$\left[ {0, T} \right] \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\bar X}$ intoR ^n×m. If, among other conditions, fori=1, ...,n,

$$f^i \left( {t, x, u_1 } \right) - f^i \left( {t, x, u_2 } \right) \leqslant F^i \left( {t, x} \right)\left( {g\left( {u_1 } \right) - g\left( {u_2 } \right)} \right),$$

whereF ⁱ is theith row ofF, then the main result of the paper establishes the existence of a control

which minimizes the cost functional

$$I\left( u \right) = \int {_0^T } c\left( {t, x\left( t \right), u\left( t \right)} \right)dt,$$

wherec(t,x,u) is convex inu for each (t,x). An example is worked out in detail.