ISSN:
1573-2878
Keywords:
Optimal control
;
nonlinear systems
;
existence theorems
;
convexity
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01767452
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