摘要： 

In the linear version of our results, we consider the control system $$\dot x = Ax+Bu,$$ where $A$ generates a strongly continuous semigroup $\tline$ on the Hilbert space $X$ and the control operator $B$ maps into the dual of $\Dscr(A^*)$, but it is not necessarily admissible for $\tline$. We prove that if the pair $(A,B)$ is both forward and backward optimizable (our definition of this concept is slightly more general than the one in the literature), then the system is exactly controllable. This is a generalization of a well-known result called Russell's principle. Moreover the usual stabilization by state feedback $u=Fx$, where $F$ is an admissible observation operator for the closed-loop semigroup, can be replaced with a more complicated periodic (but still linear) controller. The period $\tau$ of the controller has to be chosen large enough to satisfy an estimate. This controller can improve the exponential decay rate of the system to any desired value, including $-\infty$ (dead-beat control). The corresponding control signal $u$, generated by alternately solving two exponentially stable homogeneous evolution equations on each interval of length $\tau$, back and forth in time, will still be in $L^2$. The better the decay rate that we want to achieve, the more iterations the controller needs to perform, but (unless we want to achieve $-\infty$) the number of iterations needed on each period is finite.

We have recently realized that the results can be generalized to nonlinear systems of the type $$\dot x = Ax + B_1 N(x) + Bu,$$ where $B_1$ is an admissible (possibly unbounded) control operator for the semigroup generated by $A$, and $N$ is a nonlinear Lipschitz function, whose Lipschitz bound satisfies a certain estimate. The stabilizability and backward stabilizability assumptions are made on the nonlinear system.