CPS Dependability Framework Based on Inhomogeneous Stochastic Hybrid Systems

Abstract

In this paper, we propose a time-inhomogeneous stochastic hybrid system, as the appropriate modelling framework for the performance of complex engineering systems. One important hypothesis for this hybrid system is the ergodicity that ensures the existence of some sort of invariant measures. The invariant measures constitute an important tool for defining performance measures for dependability. First, we define an appropriate model of time-inhomogeneous stochastic hybrid systems. Then, we adapt the concept of invariant measure for the nonhomogeneous case. Under the assumption of geometric (or exponential) ergodicity of the underlying hybrid process, we prove the existence and uniqueness of this kind of invariant measure. The paper ends with some sort of sensitivity analysis of this invariant measure under appropriate perturbations of the infinitesimal generators associated to the stochastic process.

A Appendix

1.1 A.1 Metrics and Norms on Measure Spaces

In this section, Wasserstein metrics and Kantorovich-Rubinstein type norms on the space of finite signed Borel measures are briefly presented. We suppose that $\mathbf {S}$ is a Polish space (i.e. complete, separable, metric space), such that any finite signed Borel measure is a Radon measure (i.e. $\mathbf {S}$ is a Radon space). Recall that a measure $\mu$ defined on the Borel $\sigma$-algebra $\mathcal {B}(\mathbf {S})$ is called Radon measure if it is locally finite (i.e. for any point $x\in \mathbf {S}$ there is a neighborhood that has finite measure) and the following (inner regularity) property is satisfied:

$$\begin{aligned} \mu (B)=\sup \{\mu (K)\vert K\subset B\text {, }K\text { compact}\}\text {.} \end{aligned}$$

Let $\mathcal {M}(\mathbf {S})$ be the Banach space of finite signed Borel measures with the norm $||\mu ||:=|\mu |(\mathbf {S})$, which is called the variation norm of the measure. Let $\mathcal {M}^{+}(\mathbf {S})$ denote the subset of $\mathcal {M}(\mathbf {S})$ consisting in all nonnegative Borel measures. By $\mathcal {M}_{1}(\mathbf {S})$ we denote the set of $\mathcal {M}^{+}(\mathbf {S})$ that are probability measures (i.e. $\mu ( \mathbf {S})=1$ for $\mu \in \mathcal {M}_{1}(\mathbf {S})$).

If d is a metric on $\mathbf {S,}$ the space of Lipschitz functions is defined as follows

$$\begin{aligned} Lip(\mathbf {S}):=\{f:\mathbf {S}\rightarrow {\mathbb {R}}|\exists L\ge 0\text { s.t. }|f(x)-f(y)|\le Ld(x,y),\forall x,y\in \mathbf {S}\}. \end{aligned}$$

For $f\in Lip(\mathbf {S})$, the quantity

$$\begin{aligned} |f|_{L}:={\mathop {\sup }\limits _{\begin{array}{c} x,y\in \mathbf {S} \\ x\ne y \end{array}}}\frac{|f(x)-f(y)|}{ d(x,y)} \end{aligned}$$

is called Lipschitz constant and this is a seminorm on $Lip(\mathbf {S })$ ($|f|_{L}=0$ if and only if f is constant). The quotient space with respect to the constant functions is denoted by $Lip_{0}(\mathbf {S})$ and it is a Banach space. We will need also the set of Lipschitz contraction, which is

$$\begin{aligned} Lip_{1}(\mathbf {S}):=\{f:\mathbf {S}\rightarrow {\mathbb {R}}{{\vert }}|f|_{L}\le 1\}\text {.} \end{aligned}$$

As usual, $\mathbf {B}(\mathbf {S})$ is the space of bounded measurable functions $f:\mathbf {S}\rightarrow {\mathbb {R}}$ and $C(\mathbf {S})$ is the space of all bounded continuous functions. Both spaces are equipped with the supremum norm: $||f||_{\infty }:=\sup _{x\in \mathbf {S}}|f(x)|$.

For a function $f:\mathbf {S}\rightarrow {\mathbb {R}}$ which is integrable with respect to $\mu \in \mathcal {M}(\mathbf {S})$ we write

$$\begin{aligned} {<}f,\mu {>}\,:=\,\int _{\mathbf {S}}f(x)d\mu (x)\text {.} \end{aligned}$$

The space of signed measures $\mathcal {M}(\mathbf {S})$ is complete (Banach space) with respect to the Radon norm: $||\mu ||_{\mathcal {R }}:=\sup \{|{<}f,\mu {>}|$: $||f||_{\infty }\le 1\}$.

We denote by $\mathcal {M}^{d}(\mathbf {S})$ the space of signed measures $\nu$ such that

$$\begin{aligned} \int _{\mathbf {S}}d(x,y)|\nu |(dx)<\infty \end{aligned}$$

(11)

for some (or, equivalently for all) $y\in \mathbf {S}$. Similarly, one can define the space $\mathcal {M}_{1}^{d}(\mathbf {S})$. The condition (11) ensures that the Lipschitz functions are integrable with respect to the measures from $\mathcal {M}^{d}(\mathbf {S})$.

In the following, we will introduce some well-known metrics.

On the space $\mathcal {M}^{d}(\mathbf {S})$, the Fortet-Mourier metric is given by the formula

$$\begin{aligned} ||\mu _{1}-\mu _{2}||_{\mathcal {F}}:= {\mathop {\sup }\limits _{\begin{array}{c} ||f||_{\infty }\le 1 \\ |f|_{L}\le 1 \end{array}}}|{<}f,\mu _{1}-\mu _{2}{>}| \end{aligned}$$

The space of probability measures $\mathcal {M}_{1}^{d}(\mathbf {S})$ with respect to the Fortet-Mourier metric is a complete metric space [3]. Since the state space $\mathbf {S}$ is supposed to be a Polish space, then the weak convergence in $\mathcal {M}_{1}(\mathbf {S})$ is generated by this metric [27].

In the space $\mathcal {M}_{1}^{d}(\mathbf {S})$, we also can introduce the Kantorovich-Wasserstein (called also Hutchinson) metric by the formula

$$\begin{aligned} ||\mu _{1}-\mu _{2}||_{\mathcal {KW}}:=\sup \{|{<}f,\mu _{1}-\mu _{2}{>}|\text {: } |f|_{L}\le 1\}\text {.} \end{aligned}$$

The norms associated to the above metrics can be defined in a standard way, as follows:

• the Kantorovich-Rubinstein norm (or dual Lipschitz norm); which is defined on the closed subspace $\mathcal {M}_{0}(\mathbf {S})$ of zero-charge measures (i.e. measures with total mass equal to zero)

  $$\begin{aligned} ||\mu ||_{\mathcal {K}}:=||\mu ^{+}-\mu ^{-}||_{\mathcal {KW}} \end{aligned}$$

• the Fortet-Mourier (Wasserstein) norm

  $$\begin{aligned} ||\mu ||_{\mathcal {F}}:=\sup \{|\int _{\mathbf {S}}fd\mu |\text {: }|f|_{L}\le 1\text {; }||f||_{\infty }\le 1\}\text {.} \end{aligned}$$

We can employ the equivalent norm

$$\begin{aligned} ||\mu ||_{BL}^{*}:=\sup \{|\int _{\mathbf {S}}fd\mu |\text {: }f\in BL(\mathbf {S })\text {, }||f||_{BL}\le 1\}\text {,} \end{aligned}$$

where $BL(\mathbf {S})$ is the space of all bounded Lipschitz functions on $\mathbf {S}$ with the norm: $||f||_{BL}:=||f||_{\infty }+|f|_{L}$.

Theorem 3

(Kantorovich-Rubinstein Maximum Principle). [27] For any $\mu _{1}$, $\mu _{2}\in \mathcal {M}_{1}^{d}( \mathbf {S})$, $\mu _{1}\ne \mu _{2}$, there exists $f\in Lip_{1}(\mathbf {S})$ such that

$$\begin{aligned} {<}f,\mu _{1}-\mu _{2}{>}=\,||\mu _{1}-\mu _{2}||_{\mathcal {KW}}\text {.} \end{aligned}$$

Moreover, every such a function f satisfies the condition: $|f(x)-f(y)|=d(x,y)$, $\forall x,y\in \mathbf {S}$, $x\ne y$.

1.2 A.2 Proofs

Proof of Theorem 1

Proof

$\Longrightarrow :$ Suppose that the evolution system $(P_{s,t}^{q})$ is exponentially Lipschitz with positive exponent $K>0$. For $\mu _{1}$, $\mu _{2}\in \mathcal {M}_{1}^{d}(\mathbf {S})$, we have, using the Kantorovich-Rubinstein Maximum Principle, the following computations:

$$\begin{aligned}&||(\mu _{1}-\mu _{2})P_{st}^{q}||_{\mathcal {KW}}=\mathop {\sup }\nolimits _{f\in Lip_{1}(\mathbf {S})}|{<}f,(\mu _{1}-\mu _{2})P_{st}^{q}{>}|\\&\quad \qquad =\mathop {\sup }\nolimits _{f\in Lip_{1}(\mathbf {S})}|{<}P_{st}^{q}f,\mu _{1}-\mu _{2}{>}|\le e^{(-K)(t-s)}||(\mu _{1}-\mu _{2})||_{\mathcal {KW}}. \end{aligned}$$

$\Longleftarrow$: Suppose that the evolution system $(P_{s,t}^{q})$ is uniformly geometrically ergodic. It is known that the dual of $\mathcal {M} _{0}(\mathbf {S})$ is the quotient of space of Lipschitz functions by constant functions $Lip_{0}(\mathbf {S})$ with norm $|\cdot |_{L}$. Then the dual space of $Lip_{0}(\mathbf {S})$ is the completion of $\mathcal {M}_{0}( \mathbf {S})$ with respect to the Kantorovich-Rubinstein norm.Then the semi-norm $|P_{s,t}^{q}f|_{L}$, for any $f\in Lip(\mathbf {S})$ can be calculated as the norm of a linear functional on $\overline{\mathcal {M}_{0}( \mathbf {S})}$, i.e.

$$\begin{aligned}&|P_{s,t}^{q}f|_{L}=\mathop {\sup }\nolimits _{||\mu ||_{\mathcal {K}}\le 1}||\mu P_{s,t}^{q}f||_{ \mathcal {K}}\\&\quad \qquad =\mathop {\sup }\nolimits _{||\mu _{1}-\mu _{2}||_{\mathcal {KW}}\le 1}|{<}f,(\mu _{1}-\mu _{2})P_{s,t}^{q}{>}|\le |f|_{L}\cdot Ce^{(-\varDelta )(t-s)}. \end{aligned}$$

Proof of Proposition 1

Proof

Fix $q\in Q$ and define the semigroup of operators associated the evolution system $P_{s,t}^{q}$ via

$$\begin{aligned} ({\widehat{\mu }}T_{t}^{q})(s):=\mu _{s-t}P_{s-t,s}^{q}\text {, }s\in {\mathbb {R}} \text {, }t\ge 0\text {.} \end{aligned}$$

Let ${\widehat{\mu }}:=(\mu _{l})_{l}\$be an $\mathcal {M}_{1}^{d}(\mathbf {S})$-valued sequence. Then

$$\begin{aligned}&||{\widehat{\mu }}T_{t+s}^{q}-{\widehat{\mu }}T_{t}^{q}||_{\infty }\\&\qquad =\sup _{l}||({\widehat{\mu }}T_{t+s}^{q})(l)-({\widehat{\mu }}T_{t}^{q})(l)||_{ \mathcal {KW}} \\&\qquad =\sup _{l}||\mu _{l-(t+s)}P_{l-(t+s),l}^{q}-\mu _{l-t}P_{l-t,l}^{q}||_{ \mathcal {KW}} \\&\qquad \qquad {\mathop {=}\limits ^{(1)}}\sup _{l}||[\mu _{l-(t+s)}P_{l-(t+s),l-t}^{q}-\mu _{l-t}]P_{l-t,l}^{q}||_{\mathcal {KW}} \\&\qquad \le \sup _{l}||P_{l-t,l}^{q}||\cdot ||\mu _{l-(t+s)}P_{l-(t+s),l-t}^{q}-\mu _{l-t}||_{\mathcal {KW}} \\&\qquad \le C^{\prime }e^{(-\varDelta t)} \end{aligned}$$

with $C^{\prime }=C\cdot K$, where K is a bound for $||\mu _{l-(t+s)}P_{l-(t+s),l-t}^{q}-\mu _{l-t}||_{\mathcal {KW}}$ (this bound exists due to the fact that $(P_{s,t}^{q})$ is uniformly exponentially Lipschitz).

Therefore, the sequence $({\widehat{\mu }}T_{t}^{q})$ is a Cauchy sequence in the complete space $L^{\infty }({\mathbb {R}},\mathcal {M}_{1}^{d}(\mathbf {S}))$, thus it converges to a sequence of probability measures $\widehat{\pi } ^{q}=(\pi _{t}^{q})$, which is trivially the unique evolution system of invariant measures.