An Application of Maximal Dissipative Sets in Control Theory

摘要

Consider the linear control system in Hilbert space given by dx/dt = Ax + Bu. Here A is the infinitesimal generator of a C sub o semigroup of bounded linear operators T(t), t > or = O, on a real Hilbert space E. The author assumes that T(t) is such that norm//T(t)//(sub L(E,E)) or = 1 and omega > O. B is a bounded linear operator from a real Hilbert space H to E and N(B) is properly contained in H. The author attempts to synthesize a feedback control u(t) = f(x(t)) whose active part is bounded, preserves the property of exponential asymptotic stability possessed by the uncontrolled system u = O, and is suboptimal in some sense. The synthesis is formally obtained but leads to a nonlinear singular evolution equation for the state variable X(t). The theory of maximal dissipative sets is then applied to show that the state evolution equation possesses a unique solution when the synthesis is modified in an appropriate multivalued way at the singularities. (Author)