【摘要】 <正> 有限维线性系统的可控性讨论已有比较完善的结果。但对无限维线性系统可控性的讨论仍在发展中。本文讨论Hilbert空间上线性系统的可控性。对此,Balakrish-nan,Ruth Curtain,Anthony Pritchard已有专著论述了系统可控性的判别条件,所讨论的方程为更多还原

【Abstract】 In this note, the followiog systemsx=Ax+Bu x(0)=x0(1)x=Ax+Bu+Cv x(0)=x0(2)arc discussed, where x denotes element of a Hubert space H, A is a closed linear operator, u, v denote the elements of another Hilbert space Hc which are controll elements, u(t),v(t)∈L2(0,t1;H), B and C are bounded operators, B is called control operator and C is called compensator operator. Ln this note the following problem is discussed; if the system (1) is not controllable, under what conditions is the system (2) controllable? The problem is solved in terms of mild solution of (2) by the follywing equation,(3)where T(t) is the semigroup generated by A. The following theorem holds.Theorem: Let A be a closed operator, T(s) is the semigroup generated by A, x(t)∈C(0,t,H), H is a Hilbert space, u(s) ,v(s) belong to L2(0,t13 H), Hc in another Hilbert space, B, C are linear bounded operators. The necessary and sufficient condition for system (2) to the approximately controllable is thatis dense in the Hilbert space H, where R(T(t)B), R(T(t)C) reprent the ranges of T(t)B, T(t)C respectively.In the note, the notions of effective compensator and optimal compensator are discussed and two criteria are established.更多还原