Let the function $\Lambda : [0,T] \times \mathbb{R^n} \times \mathbb{R^n} \to \mathbb R$ be continuously differentiable. Assume the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ is finite value for all $x \in C^{2}[0 ,2]$ on a neighborhood of $x_0$.

My question: Is the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ lipschitz on a neighborhood of $x_0$ on the space $C^2 [0,T]$ equipped with the norm $W^{1,1}$ ?

P.S: $AC[0, T]$ stands for the space of all absolutely continuous function $x: [0,T] \to \mathbb R^n$ equipped with $W^{1,1}$ norm which is $$ \| x \| := \int_{0}^{T} \|x(t)\| \; dt + \int_{0}^{T} \|x' (t)\| \; dt$$ clearly $C^2 [0 ,T] \subset AC [0 ,T]$.

Let the function $\Lambda : [0,T] \times \mathbb{R^n} \times \mathbb{R^n} \to \mathbb R$ be continuously differentiable. Then the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ is finite value for all $x \in C^{2}[0 ,2]$ .

My question: Is the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ lipschitz on a neighborhood of $x_0$ on the space $C^2 [0,T]$ equipped with the norm $W^{1,1}$ ?

P.S: $AC[0, T]$ stands for the space of all absolutely continuous function $x: [0,T] \to \mathbb R^n$ equipped with $W^{1,1}$ norm which is $$ \| x \| := \int_{0}^{T} \|x(t)\| \; dt + \int_{0}^{T} \|x' (t)\| \; dt$$ clearly $C^2 [0 ,T] \subset AC [0 ,T]$.