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| Maximum Principle of Semi-Linear Distributed System (I) |
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Citation： |
Yao Yunlong.Maximum Principle of Semi-Linear Distributed System (I)[J].Chinese Annals of Mathematics B,1982,3(5):679~690 |
| Page view： 787
Net amount： 885 |
Authors： |
Yao Yunlong; |
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| Abstract： |
In this paper, an optimal control problem of non-linear Volterra systems $x(\cdot)=h(t)+\int_0^t G(t,s)f(s,x(s),u(s))ds$ on Banach space X with a general cost functional $Q(u(\cdot)) = \int_0^T J(s,x(s,u(\cdot)),u(s))ds$ is discussed, where $G(t,s)\in \varphi(X)$ is strongly continuous in (t, s), h(\cdot)\in C([0,T],G),f(s,x,u):[0,T]*X*U \rightarrow X and J (s, x, u) : [0, T] *X*U \rightarrow R. The control region U is an arbitrary set in a Banach space. Under some other assumptions of f and J, we have proved the following Theorem. The optimal control u^*(\cdot) of the above problem satisfies
max $H(t,u)=H(t,u^*(t))$ for a.e.t\in [0,T],
Where $H(t,u)=-J(t,x^*(t),u)+(\phi(t),f(t,x^*(t),u))$,
$\phi(t)=\int_t^T J_x(s,x^*(s),u^*(s))U(s,t)ds$
and $x^*(t)=x(t,u^*(\cdot)),U(s,t)\in \phi(X)$ is the solution of
$U(s,t)=G(s,t)+\int _t^s G(s,w)f_x(w,x^*(w),u^*(w))U(w,t)dw$.
We have applied the results to semi-linear distributed systems. |
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