Weighted Admissibility and Wellposedness of linear systems in Banach spaces

Perturbations of Admissibility, Exact Controllability, Exact Observability and Regularity

This paper is concerned with the notions of admissibility, exact controllability, exact observability and regularity of linear systems in the Banach space setting. It is proved that admissible controllability, exact controllability, admissible observation, exact observability and regularity are invariant under some regular perturbations of the generators, such results are generalizations of some previous references. Moreover, the related boundary linear systems and some illustrative examples are presented.

Wellposedness of Mean Field Games with Common Noise Under a Weak Monotonicity Condition

In this paper, we consider Mean Field Games in the presence of common noise relaxing the usual independence assumption of individual random noise. We assume a simple linear model with terminal cost satisfying a convexity and a weak monotonicity property. Our main result is showing existence and uniqueness of a Mean Field Game solution using the Stochastic Maximum Principle. The uniqueness is a result of a monotonicity property similar to that of Lasry and Lions. We use the Banach Fixed Point Theorem to establish an existence over small time duration and show that it can be extended over an arb…

控制算子的可接受性与输入到状态稳定性鲁棒性的研究

Admissibility of control operators for positive semigroups and robustness of input-to-state stability

本文研究了分布式参数系统的well-posedness和稳定性，重点探讨线性正控制系统的特性，推导了正算子控制算子的残差条件，并研究了在无界扰动下的ISS鲁棒性。

On the admissibility of observation operators for evolution families

This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix $τ> 0$ and let $\left(A(t)\right)_{t \in [0,τ]} \subset \mathcal{L}(D,X)$, where $D$ and $X$ are two Banach spaces such that $D$ is continuously and densely embedded into $X$. We assume that the operator $A(t)$ has maximal regularity for all $t \in [0,τ]$ and that $ A(\cdot) : [0,τ] \to \mathcal{L}(D,X) $ satisfies a regularity condition (viz. relative $p$-Dini for some $p \in (1,\infty)$). At first sight, we show that there exists an evolution family on $X$ associated to the problem $$ \do…

Feedback stabilization of linear and bilinear unbounded systems in Banach space

We consider linear control systems of the form $\dot{y}(t)=Ay(t)-μB C y(t)$ where $μ$ is a positive real parameter, $A$ is the state operator and generates a linear $C_0-$semigroup of contractions $S(t) $ on a Banach space $X$, $B$ and $C$ are respectively the operators of control and observability, which are defined in appropriate spaces in which they are unbounded in some sense. We aim to show the exponential stability of the above system under sufficient conditions which are expressed in term of admissibility and observability properties. The uniform exponential stabilization using bilinear…

Native Banach spaces for splines and variational inverse problems

We propose a systematic construction of native Banach spaces for general spline-admissible operators ${\rm L}$. In short, the native space for ${\rm L}$ and the (dual) norm $\|\cdot\|_{\mathcal{X}'}$ is the largest space of functions $f: \mathbb{R}^d \to \mathbb{R}$ such that $\|{\rm L} f\|_{\mathcal{X}'}<\infty$, subject to the constraint that the growth-restricted null space of ${\rm L}$be finite-dimensional. This space, denoted by $\mathcal{X}'_{\rm L}$, is specified as the dual of the pre-native space $\mathcal{X}_{\rm L}$, which is itself obtained through a suitable completion process. Th…