Stability Analysis of Planar Probabilistic Piecewise Constant Derivative Systems^††thanks: This work was partially supported by NSF CAREER Grant No. 1552668 and NSF Grant No. 2008957.

Let $Dist(S)$ denote the set of all probability distributions on the set $S$. Let us define Discrete-time Markov Chain (DTMC) on the set of states $S$.

We use $\text{P}(s_{1},s_{2})$ to denote $\text{P}(s_{1})(s_{2})$ and $\text{P}^{n}(s_{1},s_{2})$ to denote the probability of going from $s_{1}$ to $s_{2}$ in $n$-steps.

A path of a DTMC $\mathcal{M}$ is a sequence of states $\sigma=s_{1},s_{2},\dots$ such that for all $i<|\sigma|$, $\text{P}(s_{i},s_{i+1})>0$, where $|\sigma|$ is the length of the sequence. A path of length $2$ is called an edge and the set of all edges is denoted as $\mathcal{E}$. The $i^{\textit{th}}$ state of the path $\sigma$ is denoted by $\sigma_{i}$ and the last state of $\sigma$ is denoted as $\sigma_{end}$. $\sigma[i:j]$ denotes the subsequence $\sigma_{i},\sigma_{i+1},\dots,\sigma_{j}$. We say $s_{2}$ is reachable from $s_{1}$ (denoted $s_{1}\leadsto s_{2}$) if there is a path $\sigma$ on $\mathcal{M}$ such that $\sigma_{1}=s_{1}$ and $\sigma_{end}=s_{2}$. The set of all finite paths of a DTMC $\mathcal{M}$ is denoted as $\textit{Paths}_{\textit{fin}}(\mathcal{M})$ and the set of all infinite paths is denoted as $\textit{Paths}(\mathcal{M})$.

The probability of a finite path $\sigma$, denoted $\text{P}(\sigma)$, is the product of the probabilities of each of its edges, $\text{P}(\sigma)\coloneqq\prod_{i<|\sigma|}\text{P}(\sigma_{i},\sigma_{i+1})$. The probability of $\sigma$ with respect to a distribution $\rho$, denoted $P_{\rho}(\sigma)$ is the product of $\text{P}(\sigma)$ and the probability of $\sigma_{1}$ under $\rho$, i.e., $P_{\rho}(\sigma)\coloneqq\rho(\sigma_{1})\cdot\text{P}(\sigma)$. We can associate a probability measure Pr to the set of infinite paths $\textit{Paths}(\mathcal{M})$ of a DTMC $\mathcal{M}$ using probability of the cylinder sets of the finite paths as discussed in [3]. A path property $\mathbb{P}$ is said to be almost surely satisfied if the set of all paths having property $\mathbb{P}$ has probability $1$, i.e., $\textit{Pr}\{\sigma\mid\sigma\text{ has }\mathbb{P}\}=1$.

Next we define some subclasses of DTMC and show that it has some nice convergence properties.

We say a probability distribution is stationary for a DTMC $\mathcal{M}$ if the next step distribution remains unchanged.

For finite, irreducible DTMC, the stationary distribution is unique. The following theorem guarantees existence of limiting distribution for finite, irreducible and aperiodic DTMC and associates it with the stationary distribution of the DTMC (see [27]).

Note that, $\text{P}^{n}(s_{1},s_{2})$ does not depend on $s_{1}$ as $n\rightarrow\infty$.

Let us now define Weighted Discrete-time Markov Chain (WDTMC) that extend DTMC with weighted edges. Basically, a WDTMC can be observed as a Markov Reward Process where rewards are associated to individual transitions rather than nodes.

We also define disjoint union of two WDTMC $\mathcal{M}_{W}^{1}$ and $\mathcal{M}_{W}^{2}$ as a WDTMC $\mathcal{M}_{W}^{1}\sqcup\mathcal{M}_{W}^{2}$ whose states and edges are disjoint unions of states and edges of $\mathcal{M}_{W}^{1}$ and $\mathcal{M}_{W}^{2}$ respectively. With the weight function W defined, it is possible to associate weights to individual paths of $\mathcal{M}_{W}$.

For $\sigma\in\textit{Paths}(\mathcal{M}_{W})$, the quantity $\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\text{W}(\sigma_{i},\sigma_{i+1})$ is denoted by $\text{W}(\sigma[1:\infty])$. It is easy to observe that, $\text{W}(\sigma)=\text{W}(\sigma[1:\infty])$. A simple path is a path without state repetition and a simple cycle is a path where only the starting and the ending states are same. We use the notation $\mathcal{SP}(\mathcal{M}_{W})$ for the set of all simple paths and the notation $\mathcal{SC}(\mathcal{M}_{W})$ for the set of all simple cycles of a WDTMC $\mathcal{M}_{W}$.

Let us define the notions of absolute and probabilistic convergence of WDTMC. A WDTMC is said to be absolutely convergent if the weight of every infinite path diverges to $-\infty$.

Further, a WDTMC is said to be almost surely convergent if the weight of an infinite path diverges to $-\infty$ with probability $1$.

Probabilistic bisimulation [3] on a WDTMC is an equivalence relation on its set of states such that probabilities of corresponding edges agree for two related states.

Note that, $\text{P}(s,T)=\sum_{t\in T}\text{P}(s,t)$ for $s\in S$. Let us now use probabilistic bisimulation to relate infinite paths of a WDTMC.

A set of infinite paths is $\sim$ bisimulation-closed for some probabilistic bisimulation $\sim$, if for any path in the set, all its bisimulation-equivalent paths are also in the set. In other words, $\Pi\subseteq\textit{Paths}(\mathcal{M}_{W})$ is $\sim$ bisimulation-closed if for any $\pi\in\Pi$ and any $\tilde{\pi}\sim\pi$, $\tilde{\pi}\in\Pi$. Let us denote by $\textit{Pr}_{s}(\Pi)$ the set of all paths in $\Pi$ that start from $s\in S$. The following lemma [3] equates the probability of two sets of paths that start from $\sim$ related states and are subset of the same $\sim$ bisimulation-closed set.

We denote the set of all polyhedral subsets of ⁿ by $Poly(n)$. The facets of a polyhedral subset $A$ are the largest polyhedral subsets of the boundary of $A$. We denote the boundary of a polyhedral subset $A$ by $\partial(A)$ and the set of all facets of $A$ by $\mathbb{F}(A)$. We say a polyhedral subset $P$ is positive scaling invariant if for all $x\in P$ and $\alpha>0$, $\alpha x\in P$.

Here we provide a necessary and sufficient condition for analyzing absolute convergence of a WDTMC. We begin with the following proposition (proved in the Appendix) which states that for any finite path $\sigma\in\textit{Paths}_{\textit{fin}}(\mathcal{M}_{W})$, we can get one simple path and a set of simple cycles such that their total weight equals the weight of $\sigma$.

We use Proposition 1 to prove the following main theorem which states that, a WDTMC is absolutely convergent iff there is no edge of infinite weight and no cycle of weight greater or equal to $0$ reachable from the initial point.

In this subsection, we will provide a necessary and sufficient condition for almost sure convergence of a WDTMC. We assume a WDTMC $\mathcal{M}_{W}$ is finite, irreducible and aperiodic and thus has the limiting distribution equal to its stationary distribution $\rho^{*}$ (Theorem 4.1).

Given a WDTMC $\mathcal{M}_{W}$, we begin by defining random variables $\{X^{e}_{j}\mid e\in\mathcal{E};j\in\mathbb{N}\}$ on the set of infinite paths $\textit{Paths}(\mathcal{M}_{W})$, that captures the information of whether an edge $e\in\mathcal{E}$ appears on the $j^{\textit{th}}$ step of an infinite path $\sigma$. More precisely,

Note that for some $e\in\mathcal{E}$ and $\sigma\in\textit{Paths}(\mathcal{M}_{W})$, $\sum_{j=1}^{n}X^{e}_{j}(\sigma)$ gives the number of times $e$ appears on $\sigma[1:n+1]$. Now, the following lemma (proved in Appendix) gives that, for any edge $e\in\mathcal{E}$, the average of $\{X^{e}_{j}\mid j\in\mathbb{N}\}$ almost surely converges to $P_{\rho^{*}}(e)$, which is the probability of $e$ with respect to the stationary distribution $\rho^{*}$.

Next, we define partial average weight upto $n$ for an infinite path $\sigma$ as

and note that,

We now state the main lemma of this subsection which essentially states that, the average weight of an infinite path almost surely converges to a quantity that depends only on the weights and probabilities of the edges.

We say $\sum_{e\in\mathcal{E}}P_{\rho^{*}}(e)\text{W}(e)$ is the effective weight of the WDTMC $\mathcal{M}_{W}$ and denote it as $\text{W}_{\mathcal{E}}$. The main theorem basically states that a WDTMC is almost surely convergent iff its effective weight is strictly less than $0$.

Based on Theorems 5.1 and 5.2 we present two algorithms here for checking absolute and almost sure convergence of a WDTMC. For the first algorithm, assuming the WDTMC is finite, we first check for existence of an infinite weight edge by Breadth First Search (BFS) [16] and then for a cycle with non-negative weight using a variant of the Bellman-Ford algorithm [16]. If neither of them is found then the WDTMC is deemed absolutely convergent by Theorem 5.1. Since BFS takes time linear to the size of its input and Bellman-Ford takes time quadratic to the size of its input, the time complexity of this algorithm is $O(|S|^{2})$, where $S$ is the set of states of $\mathcal{M}_{W}$.

For the second algorithm, assuming the WDTMC is finite, irreducible and aperiodic, existence of an infinite weight edge is checked by Breadth First Search (BFS). If such an edge exists then the WDTMC is deemed not almost surely convergent (by Theorem 5.2). Otherwise, the stationary distribution $\rho^{*}$ of the WDTMC is calculated by solving a set of linear equations mentioned in Definition 4. The value $\sum_{e\in\mathcal{E}}P_{\rho^{*}}(e)\text{W}(e)$ is then calculated (where $\mathcal{E}$ is the set of transitions of the WDTMC) and compared to $0$. The WDTMC is deemed almost surely convergent only if $\sum_{e\in\mathcal{E}}P_{\rho^{*}}(e)\text{W}(e)<0$. Since BFS takes time linear to its input size and solving a set of linear equations takes time at most cubic in the number of variables, the time complexity of this algorithm is $O(|S|^{3})$, where $S$ is the set of states of $\mathcal{M}_{W}$.

We model PPCDs as consisting of a discrete set of modes, each associated with an invariant and probabilistic transitions between modes that are enabled at the boundaries of the invariants.

Next, we discuss the semantics of the PPCD. An execution starts from a location $q_{0}\in Q$ and some continuous state $x_{0}\in\mathcal{X}$ and evolves continuously for some time $T$ according to the dynamics of $q_{0}$ until it reaches a facet $f_{0}$ of the invariant of $q_{0}$. Then a probabilistic discrete transition is taken if there is an edge $(q_{0},f_{0},\rho_{0})$ and the state $q_{0}$ is probabilistically changed to $q_{1}$ with probability $\rho_{0}(q_{1})$. The execution (tree) continues with alternating continuous and discrete transitions.

Formally, for any two continuous states $x_{1},x_{2}\in\mathcal{X}$ and $q\in Q$, we say that there is a continuous transition from $x_{1}$ to $x_{2}$ with respect to $q$ if $x_{1},x_{2}\in\textit{Inv}(q)$, there exists $T\geq 0$ such that $x_{2}=x_{1}+\textit{Flow}(q)\cdot T$, $x_{1}+\textit{Flow}(q)\cdot t\not\in\partial(\textit{Inv}(q_{0}))$ for any $0\leq t<T$ and $x_{2}\in\partial(\textit{Inv}(q_{0}))$. We note that there is a unique continuous transition from any state $(q,x)$ since it requires the state to evolve until it reaches the boundary for the first time, which corresponds to a unique time of evolution $T$. Further, if for all $t\geq 0$, $x_{1}+\textit{Flow}(q)\cdot t\in\textit{Inv}(q)$ then we say $x_{1}$ has an infinite edge with respect to $q$. For two locations $q_{1},q_{2}\in Q$, we say there is a discrete transition from $q_{1}$ to $q_{2}$ with probability $p$ via $\rho\in Dist(Q)$ and $f\in\mathbb{F}(q_{1})$ if $f\subseteq\textit{Inv}(q_{2})$, $(q_{1},f,\rho)\in\textit{Edges}$ and $p=\rho(q_{2})$.

We capture the semantics of a PPCD using a WDTMC, wherein we combine a continuous transition and a discrete transition to represent a probabilistic transition of the DTMC. In addition, to reason about convergence, we also need to capture the relative distance of the states from the equilibrium point, which is captured using edge weights. Let us fix $0$ as the equilibrium point for the rest of the section. The weight on a transition from $(q_{1},x_{1})$ to $(q_{2},x_{2})$ captures the logarithm of the relative distance of $x_{1}$ and $x_{2}$ from $0$, that is, it is $\left(\lvert\!\lvert x_{2}\rvert\!\rvert/\lvert\!\lvert x_{1}\rvert\!\rvert\right)$, where $\lvert\!\lvert x\rvert\!\rvert$ captures the distance of state $x$ from $0$.

Since all executions of the PPCD $\mathcal{H}$ start from location $q_{0}$ and state $x_{0}$, we consider only those paths of the semantics $\mathcal{M}_{\mathcal{H}}$ which start from $(q_{0},x_{0})$ and denote them as $\textit{Paths}(\mathcal{M}_{\mathcal{H}})$. We say a path $\sigma=(q_{0},x_{0}),(q_{1},x_{1}),\dots$ converges to $0$ if norm of the corresponding state sequence $\lvert\!\lvert x_{0}\rvert\!\rvert,\lvert\!\lvert x_{1}\rvert\!\rvert,\dots$ converges to $0$. Stability of a PPCD $\mathcal{H}$ is defined in terms of convergence of paths of its semantics $\mathcal{M}_{\mathcal{H}}$ as follows,

We now characterize stability of a PPCD $\mathcal{H}$ in terms of its semantics $\mathcal{M}_{\mathcal{H}}$. Basically we state that, $\mathcal{H}$ is absolutely (almost surely) stable iff $\mathcal{M}_{\mathcal{H}}$ is absolutely (almost surely) convergent.

In general, semantics of a PPCD has infinite number of states and thus the algorithms developed in section 5.3 cannot be applied to decide absolute (almost sure) convergence of the semantics. However, if the continuous state space of a PPCD $\mathcal{H}$ is ², then we can reduce $\mathcal{M}_{\mathcal{H}}$ to a finite WDTMC that provides an exact characterization of $\mathcal{M}_{\mathcal{H}}$. A PPCD with $\mathcal{X}={}^{2}$ is called a planar PPCD. Since for each location $q$, $\textit{Inv}(q)$ is positively scaled, the facets of $\textit{Inv}(q)$ are rays emanating from origin. Given constant flow for each location $q$, a continuous transition starting at a point of some facet $f_{1}\in\cup_{q\in Q}\mathbb{F}(\textit{Inv}(q))$ ends up at a unique point of a unique facet $f_{2}\in\cup_{q\in Q}\mathbb{F}(\textit{Inv}(q))$. This property is not observed if the continuous state space is of three or higher dimensions (Figure 2). Also, if two continuous transitions start from different points $x_{1},x_{1}^{\prime}$ of the same facet $f_{1}$, they end up in unique points $x_{2},x_{2}^{\prime}$ (respectively) of a unique facet $f_{2}$ such that $\lvert\!\lvert x_{2}\rvert\!\rvert/\lvert\!\lvert x_{1}\rvert\!\rvert=\lvert\!\lvert x_{2}^{\prime}\rvert\!\rvert/\lvert\!\lvert x_{1}^{\prime}\rvert\!\rvert$. This gives us the following lemma (proved in Appendix),

For the rest of the section, we will assume all paths of the semantics $\mathcal{M}_{\mathcal{H}}$ of a planar PPCD $\mathcal{H}$ start at $(q_{0},x_{0})$ and $x_{0}\in f_{0}$, where $f_{0}$ is a facet in $\cup_{q\in Q}\mathbb{F}(\textit{Inv}(q))$.

We now define the quotient of a planar PPCD $\mathcal{H}$, which is a finite WDTMC having the same convergence properties as $\mathcal{M}_{\mathcal{H}}$. Here we consider the set of states as $Q\times\cup_{q\in Q}\mathbb{F}(\textit{Inv}(q))$ instead of $Q\times\mathcal{X}$ and use Lemma 4 to define the probabilistic edges and their weights.

The above definition is well-defined, that is, the choice of $x_{1}$ and $x_{2}$ do not matter due to Lemma 4.

We will eventually prove that a planar PPCD $\mathcal{H}$ is absolutely (almost surely) stable if and only if its quotient WDTMC $\mathcal{H}^{\textit{red}}$ is absolutely (almost surely) convergent. First, let us show that for every infinite path $\sigma$ of $\mathcal{M}_{\mathcal{H}}$, there is a path $\pi$ in $\mathcal{H}^{\textit{red}}$ with same weight and vice versa.