Non-equilibrium Solutions of Dynamic Networks: A Hybrid System Approach

Abstract

Many dynamic networks can be analyzed through the framework of equilibrium problems. While traditionally, the study of equilibrium problems is solely concerned with obtaining or approximating equilibrium solutions, the study of equilibrium problems not in equilibrium provides valuable information into dynamic network behavior. One approach to study such non-equilibrium solutions stems from a connection between equilibrium problems and a class of parametrized projected differential equations. However, there is a drawback of this approach: the requirement of observing distributions of demands and costs. To address this problem we develop a hybrid system framework to model non-equilibrium solutions of dynamic networks, which only requires point observations. We demonstrate stability properties of the hybrid system framework and illustrate the novelty of our approach with a dynamic traffic network example.

Appendices

Appendix 1: Common Definitions and Theorems for VI, EVI, and PrDE

Definition 13

Some classifications of the mapping F [17]. Given that X is a Hilbert space of arbitrary dimension, $\mathbb{K} \subset X$ is a non-empty, closed, and convex set, then a mapping $F: \mathbb{K} \rightarrow X$ is said to be

• 1. Pseudomonotone on $\mathbb{K}$ if

  $$\displaystyle{\begin{array}{ll} \langle F(x),y - x\rangle \geq 0 \Rightarrow \langle F(y),y - x\rangle \geq 0&\forall x,y \in \mathbb{K}\end{array} }$$

• 2. Strictly pseudomonotone on $\mathbb{K}$ if

  $$\displaystyle{\begin{array}{ll} \langle F(x),y - x\rangle \geq 0 \Rightarrow \langle F(y),y - x\rangle > 0&\forall x\neq y \in \mathbb{K}\end{array} }$$

• 3. Strongly pseudomonotone of degree α on $\mathbb{K}$ if for some η > 0,

  $$\displaystyle{\begin{array}{ll} \langle F(x),y - x\rangle \geq 0 \Rightarrow \langle F(y),y - x\rangle \geq \eta \| x - y\|^{\alpha }&\forall x,y \in \mathbb{K}\end{array} }$$

Definition 14

Monotone attractor. Let X be a Hilbert space of arbitrary dimension, $\mathbb{K} \subset X$ be a non-empty, closed, and convex set, and $F: \mathbb{K} \rightarrow X$ a Lipschitz continuous mapping. Then

• 1. A point $x^{{\ast}}\in \mathbb{K}$ is a local monotone attractor for a PrDE if there exists a neighborhood V of x ^∗ such that the function ϕ(τ): =  ∥ x(τ) − x ^∗ ∥  _X is non-increasing with respect to τ for any solution x(τ) of a PrDE starting in V.

• 2. A point $x^{{\ast}}\in \mathbb{K}$ is a global monotone attractor for a PrDE if condition X is satisfied for any $x(\tau ) \in \mathbb{K}$.

Definition 15

Stability of equilibria. Let X be a Hilbert space of arbitrary dimension, $\mathbb{K} \subset X$ be a non-empty, closed, and convex set, and $F: \mathbb{K} \rightarrow X$ a Lipschitz continuous mapping. If $x^{{\ast}}\subset \mathbb{K}$ is an equilibrium of a PrDE, B(x, r) is a ball of radius r centered on $x: \mathbb{R}^{+} \rightarrow \mathbb{K}$ (a non-equilibrium solution to a PrDE), then

• 1. The point x ^∗ is exponentially stable if there exists ε > 0 and μ > 0 such that $\forall x \in B(x^{{\ast}},\epsilon )$ and $\forall \tau \geq 0$, we have that ∥ x(τ) − x ^∗ ∥  _X  ≤ ∥ x(0) − x ^∗ ∥  _X exp(−μ τ).

• 2. The point x ^∗ is a finite-time attractor if there exists ε > 0 such that $\forall x \in B(x^{{\ast}},\epsilon )$ and $\forall \tau \geq 0$, there exists T: = T(x) < ∞, where x(τ) = x ^∗ for all τ ≥ T.

• 3. The point x ^∗ is globally exponentially stable, or a global finite-time attractor if X, or respectively Y hold for any $x \in \mathbb{K}$.

Theorem 2

Let $\mathbb{K} \subset X$ be a non-empty, closed, and convex set, $F: \mathbb{K} \rightarrow X$ a Lipschitz continuous mapping, and x ^∗ an equilibrium of a PrDE.

• 1. If F is locally (strictly) pseudomonotone around x ^∗ , then x ^∗ is a local (strictly) monotone attractor.

• 2. If F is (strictly) pseudomonotone on $\mathbb{K}$ , then x ^∗ is a global (strictly) monotone attractor.

Theorem 3

Let $\mathbb{K} \subset X$ be a non-empty, closed, and convex set, $F: \mathbb{K} \rightarrow X$ a Lipschitz continuous mapping, and x ^∗ an equilibrium of a PrDE.

• 1. If F is strongly pseudomonotone around x ^∗ , then x ^∗ is a locally exponentially stable.

• 2. If F is strongly pseudomonotone with degree α < 2 around x ^∗ , then x ^∗ is a local finite-time attractor.

• 3. If F is strongly pseudomonotone on $\mathbb{K}$ , then x ^∗ is a globally exponentially stable.

• 4. If F is strongly pseudomonotone with degree α < 2 on $\mathbb{K}$ ,around x ^∗ , then x ^∗ is a global finite-time attractor.

Appendix 2: Strongly Pseudomonotone of Degree α < 2

Here we show that the mapping F from the example in section “A Dynamic Traffic Network Example” is strongly pseudomonotone of degree $\frac{3} {2}$.

Proof

To begin, recall that

$$\displaystyle{F(x) = (2\sqrt{x_{1 } - x_{1 }^{{\ast}}} + x_{2} - x_{2}^{{\ast}},x_{ 2} - x_{2}^{{\ast}})^{T}}$$

with the constraint set,

$$\displaystyle{\mathbb{K} =\{ x \in L^{2}([0,110], \mathbb{R}^{2})\vert 0 \leq x_{ i} \leq 100,x_{1} + x_{2} =\rho \}.}$$

To show F is strongly pseudomonotone of degree $\frac{3} {2}$, we use the following identity:

$$\displaystyle{x_{1} - y_{1} = -(x_{2} - y_{2})\ \text{ for all }\ x,y \in \mathbb{K}.}$$

It follows that

$$\displaystyle{\langle F(x)-F(y),x-y\rangle = (2\sqrt{x_{1 } - x_{1 }^{{\ast}}}+x_{2}-2\sqrt{y_{1 } - x_{1 }^{{\ast}}}-y_{2})(x_{1}-y_{1})+(x_{2}-y_{2})(x_{2}-y_{2}).}$$

Equivalently, replacing x ₂ − y ₂ through the identity above, we have that

$$\displaystyle{\langle F(x)-F(y),x-y\rangle = (2\sqrt{x_{1 } - x_{1 }^{{\ast}}}-2\sqrt{y_{1 } - x_{1 }^{{\ast}}}-(x_{1}-y_{1}))(x_{1}-y_{1})+(x_{1}-y_{1})(x_{1}-y_{1}),}$$

and

$$\displaystyle{\langle F(x) - F(y),x - y\rangle = (2\sqrt{x_{1 } - x_{1 }^{{\ast}}}- 2\sqrt{y_{1 } - x_{1 }^{{\ast}}})(x_{1} - y_{1}).}$$

Because the square root function is subadditive, it follows that

$$\displaystyle{\langle F(x) - F(y),x - y\rangle \geq (2\sqrt{x_{1 } - y_{1}})(x_{1} - y_{1}) = 2(x_{1} - y_{1})^{\frac{3} {2} }.}$$

Finally, the proof is complete upon noting that

$$\displaystyle{\eta \|x - y\|^{\alpha } =\eta \sqrt{(x_{1 } - y_{1 } )^{2 } - (x_{2 } - y_{2 } )^{2}}^{\alpha } =\eta \sqrt{2}^{\alpha }(x_{ 1} - y_{1})^{\alpha }.}$$

Thus, we have that F is strongly pseudomonotone of degree $\alpha = \frac{3} {2}$ with $\eta = \sqrt{2}^{2-\alpha }$.

Appendix 3: Stability of a Hybrid System Non-equilibrium Solution

To demonstrate the stability properties of a hybrid system non-equilibrium solution, consider a mapping F that is strongly pseudomonotone of degree α < 2 with constant η, and the jump rules:

$$\displaystyle{G_{j}(t_{j}^{-},x_{\delta }(t_{ j}^{-})) = P_{ K_{j+1}}(x_{\delta }(t_{j}^{-}) - x_{\delta }^{{\ast}}(t_{ j}^{-}) + x_{\delta }^{{\ast}}(t_{ j}^{-}))}$$

and

$$\displaystyle{H_{j}(\theta _{j}) =\theta ^{j}.}$$

It follows that δ can be selected sufficiently small so that for some t ^∗ ∈ [0, T],

$$\displaystyle{\|x_{\delta } - x^{{\ast}}\|_{ L^{2}([t^{{\ast}},T],\mathbb{R}^{q})} <\epsilon \ \text{ for \ any }\ \epsilon > 0.}$$

Proof

The proof here follows the same approach for showing finite time attraction to an equilibrium of a projected differential equation [9, 21]. To begin, let Δ: = Δ _m be a uniform division of [0, T] for some fixed m, with division points t _j , so that | t _j+1 − t _j  |  = δ. Taking t > t _j we have that

$$\displaystyle{ \begin{array}{ll} \|x_{\delta }(t) - x^{{\ast}}(t_{j+1})\|_{\mathbb{R}^{q}}^{2-\alpha }&\leq \| x_{\delta }(t_{j}) - x^{{\ast}}(t_{j+1})\|_{\mathbb{R}^{q}}^{2-\alpha }- (2-\alpha ) \frac{\eta }{2}(t - t_{j}). \end{array} }$$

(16)

From the jump rule defined by (15), we have that

$$\displaystyle{ \|x_{\delta }(t_{j}) - x^{{\ast}}(t_{ j+1})\|_{\mathbb{R}^{q}} \leq \| x_{\delta }(t_{j}^{-}) - x^{{\ast}}(t_{ j})\|_{\mathbb{R}^{q}}. }$$

(17)

Since 2 −α > 0 and the power function is increasing we get

$$\displaystyle{ \begin{array}{ll} \|x_{\delta }(t) - x^{{\ast}}(t_{j+1})\|_{\mathbb{R}^{q}}^{2-\alpha }&\leq \| x_{\delta }(t_{j}^{-}) - x^{{\ast}}(t_{j})\|_{\mathbb{R}^{q}}^{2-\alpha }- (2-\alpha ) \frac{\eta }{2}(t - t_{j}),\\ \\ & \leq \| x_{\delta }(t_{j-1}) - x^{{\ast}}(t_{j})\|_{\mathbb{R}^{q}}^{2-\alpha }- (2-\alpha ) \frac{\eta }{2}(t - t_{j-1}). \end{array} }$$

(18)

Continuing in this fashion, we finally arrive at

$$\displaystyle{ \begin{array}{ll} \|x_{\delta }(t) - x^{{\ast}}(t_{j+1})\|_{\mathbb{R}^{q}} & \leq {\bigl (\| x_{\delta }(0^{-}) - x^{{\ast}}(0)\|_{\mathbb{R}^{q}}^{2-\alpha }- (2-\alpha ) \frac{\eta }{2}t\bigr )}^{ \frac{1} {2-\alpha }}.\end{array} }$$

(19)

Thus t ^∗ is taken such that:

$$\displaystyle{ t^{{\ast}}\geq \frac{1} {(2-\alpha ) \frac{\eta }{2}}\|x_{\delta }(0^{-}) - x^{{\ast}}(0)\|_{ \mathbb{R}^{q}}^{2-\alpha }. }$$

(20)

Thus on the subinterval [t _k , t _k+1] that contains t ^∗, we have necessarily that

$$\displaystyle{ \|x_{\delta }(t) - x^{{\ast}}(t_{ k+1})\|_{\mathbb{R}^{q}} = 0\ \text{ for }\ t \geq t^{{\ast}}. }$$

(21)

Furthermore, since the jump rule maps x ^∗(t _j ) → x ^∗(t _j+1) for all j,

$$\displaystyle{ \|x_{\delta }(t) - x^{{\ast}}(t_{ i+1})\|_{\mathbb{R}^{q}} = 0, }$$

(22)

for all t ≥ t ^∗ on each interval $[t_{i},t_{i+1}]\;\;\forall i > k$. Thus by Lebesgue’s dominated convergence theorem, we have that δ can be selected so that

$$\displaystyle{ \|x_{\delta } - x^{{\ast}}\|_{ L^{2}([t^{{\ast}},T],\mathbb{R}^{q})} \leq \epsilon, }$$

(23)

for any ε > 0.