Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time $T>0$ by taking into account the orbit of the initial value under the semigroup for $t\in [0,T]$, measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the Gauss-Weierstrass semigroup and the Ornstein-Uhlenbeck semigroup on the space of bounded continuous functions. Moreover, we generalise the duality between cost-uniform approximate null-controllability and final state observability estimates to the setting of locally convex spaces for the case of bounded and continuous control functions, which seems to be new even for the Banach spaces case

On the continuity of the state constrained minimal time function

We obtain results on the propagation of the (Lipschitz) continuity of the minimal time function associated with a finite dimensional autonomous differential inclusion with state constraints and a closed target. To this end, we first obtain new regularity results of the solution map with respect to initial data

Controllability of linear systems in Banach spaces with bounded operators.

本文研究 Banach空间线性系统 x( t) =Ax( t) +Bu( t)的可控性 ,其中 A,B为有界算子 ,得到其可控的充分必要条件 .这是文献 [3]结果的推广 .The controllability of linear systems in Banach spaces (t)=Ax(t)+Bx(t) was studied, where A and B are bounded operators. A necessary and sufficient condition for exact controllability was obtained and the results of reference [3] were extended

Controllability of linear systems in Banach spaces with bounded operators(Banach空间有界算子对的可控性)

本文研究Banach空间线性系统的可控性，其中A，B为有界算子，得到其可控的充分必要条件.这是文献［3]结果的推广