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| On Applications of Vector Measure to the Optimal Control Theory for Distributed Parameter Systems |
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Citation： |
Li Xunjing.On Applications of Vector Measure to the Optimal Control Theory for Distributed Parameter Systems[J].Chinese Annals of Mathematics B,1982,3(5):655~662 |
| Page view： 844
Net amount： 906 |
Authors： |
Li Xunjing; |
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| Abstract： |
Let X and Z be two reflexive Banach spaces, U\in Z and b(\cdot,\cdot):[t_0,T]*U\rightarrow X continuous. Suppose $x(t)\equiv x(t,u(\cdot))$ is a function from [t_0, T] into X , satisfying the distrbnted parameter system
$dx(t)\dt=A(t)x(t)+b(t,u(t)),t_0+\int_t_0^T {+r(t,u(t))dt}$.
We have proved the following theorem.
Theorem. Suppose u^*(\cdot) is the optimal control function, $x^*(t)=x(t,u^*(\cdot))$ and
$\psi (t)=-U'(T,t)Q_1x^*(T)-\int_t^T{U'(\sigma,t)Q(\sigma)x^*(\sigma)d\sigma}$,
then the maximum principle
$<\psi(t),b(t,u^*(t))>-1/2r(t,u^*(t))=\mathop {\max }\limits_{u \in U} {\psi (t),b(t,u)>-1/2r(t,u)}$ (16)
holds for almost all t on [t_0, T ]. |
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