Stability notions for a class of nonlinear systems with measure controls

Abstract

We consider the problem of stability in a class of differential equations which are driven by a differential measure associated with inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive Lyapunov-based conditions on the system’s vector fields for asymptotic stability under a specific class of inputs. These conditions are based on the stability margin of the Lebesgue-integrable and the measure-driven components of the system. For more general inputs which do not necessarily lead to asymptotic stability, we then derive conditions such that the maximum norm of the resulting trajectory is bounded by some function of the total variation of the input, which generalizes the notion of integral input-to-state stability in measure-driven systems.

Appendices

Appendix A: Solution of MDEs

We study the existence of solution of system (5) and the content of this Appendix is primarily borrowed from [12]. Before arriving at the main result, we first recall a definition that is essential for the development to follow.

Definition 4

Let $u:[t_0,T] \rightarrow R^m$ be an $RCBV$ function and set

$$\begin{aligned} U(t) := \frac{t-t_0+\mathrm{var}_u([t_0,t])}{T-t_0+\mathrm{var}_u([t_0,T])}, \quad t\in [t_0,T]. \end{aligned}$$

The canonical graph completion $\varphi :[0,1] \rightarrow [t_0,T]\times \mathbb {R}^m$ of $u(\cdot )$ is defined as:

$$\begin{aligned} \varphi (s):={\left\{ \begin{array}{ll} (t,u(t)) &{}\quad \text {if } s = U(t)\\ (t,u(t)+\frac{s-U(t)}{U(t^+)- U(t^-)}u(t^+)-u(t^-)) &{}\quad \text{ if } s \in (U(t^-), U(t^+)).\end{array}\right. } \end{aligned}$$

As an illustration, we have sketched the function $\varphi (s)$ in Fig. 1 corresponding to a piecewise-constant input $u(\cdot )$ over a compact interval.

Fig. 1

Canonical graph completion of a step function $u:[0,2] \rightarrow \{0,1\}$ with $u(t) = 0$ for $0 \le t <1$, and $u(t)=1$ for $1 \le t \le 2$

We now define a coordinate transformation using the canonical graph completion of the input. Let $\varphi := (\varphi _0, \varphi _1, \dots , \varphi _m)$ denote the graph completion of $u(\cdot )$ and consider the following auxiliary $(n+1)$-dimensional Cauchy problem

$$\begin{aligned} \frac{\mathrm{d}y_0}{\mathrm{d}s}&= \frac{d\varphi _0}{ds}, \quad y_0(t_0) = 0; ,\end{aligned}$$

(38a)

$$\begin{aligned} \frac{\mathrm{d}y}{\mathrm{d}s}&= f(y(s)) \frac{\mathrm{d}\varphi _0}{\mathrm{d}s} + \sum _{j=1}^m g_j(y(s)) \frac{\mathrm{d}\varphi _j}{\mathrm{d}s}, \quad y(t_0) = x_0 \end{aligned}$$

(38b)

where $\tilde{y}:=(y_0,y): [0,1] \rightarrow [t_0,T] \times \mathbb {R}^n$. Since the graph completions (by definition) are Lipschitz continuous functions, the solutions of (38) are studied in the classical sense. The following result establishes the link between the solutions of auxiliary system (38) and (6).

Theorem 3

([12, Theorem 2.2]) Let $u:[t_0,T] \rightarrow \mathbb {R}^m$ be an RCBV function. An RCBV function $x(\cdot )$ is called a solution of (6) if and only if there exists a solution $\tilde{y} = (y_0,y)$ of (38) corresponding to the canonical graph completion $\varphi (\cdot )$ such that

$$\begin{aligned} x(t) = y (U(t)) \end{aligned}$$

for almost every $t \in (t_0,T)$.

A straightforward consequence of this result is that there exists a unique solution to system (5) whenever there exists a unique solution to system (38) in the classical sense. Thus, if $f(\cdot )$, $g(\cdot )$ are continuously differentiable, then there exists a maximal interval over which the solution of (38), and hence (5), is well defined. This interval may include accumulation of the discontinuity points of $u(\cdot )$ because the graph completions of discontinuous inputs are Lipschitz continuous and the solution of (38) is still well defined in that case. However, it is important to note that the forward completeness of the vector fields $f(\cdot )$ and $g(\cdot )$ is not sufficient to guarantee that the solution of (38b) is forward complete. While working with singularly continuous and scalar inputs, a counterexample has been given in [41, Section8]. In the same paper in Theorem 2, sufficient conditions were presented (which roughly state that $f(\cdot )$ satisfies a linear growth condition and $g(\cdot ) \in \mathcal {C}^1(\mathbb {R}^n, \mathbb {R}^n)$ with $\frac{\partial g}{\partial x}$ uniformly bounded) which guarantee the existence and uniqueness of the solutions to (38) with $u(\cdot )$ being continuous. Later, it was shown in [5, Theorem 2.1] that these conditions are sufficient for existence of global (in time) solutions of (38b). Without imposing such strong conditions on system vector fields, it is nonetheless assumed in this paper that there exists a unique solution to system (38) for every Lipschitz continuous function $\varphi (\cdot )$.

Appendix B: Generalized comparison lemma

Lemma 1

(Comparison lemma for MDEs) Consider a continuous locally BV function $u:[t_0,T] \rightarrow \mathbb {R}^m$ with $du$ as the associated differential measure and another continuous locally BV function $V:[t_0,T] \rightarrow \mathbb {R}$ that satisfies

$$\begin{aligned} \mathrm{d}V \le -aV \mathrm{d}t + bV \sum _{j=1}^m \mathrm{d}u_j + c \sum _{j=1}^m \mathrm{d}u_j. \end{aligned}$$

(39)

Then for each $t \in [t_0,T]$

$$\begin{aligned} V(t) \le \mathrm{e}^{-a(t-t_0) + b \sum _{j=1}^m \mu _j((t_0,t))} V(t_0) + c \, \mathrm{e}^{b |\mu |_{(t_0,t)}}|\mu |_{(t_0,t)} \end{aligned}$$

where $\mu$ is the Lebesgue–Stieltjes measure associated with $u(\cdot )$.

Proof

We split the proof in two steps:

Step 1: Consider a continuous BV function $W:[t_0,T]\rightarrow \mathbb {R}$ such that

$$\begin{aligned} \mathrm{d}W = - aW \mathrm{d}t + bW \sum _{j=1}^m \mathrm{d}u_j +c \sum _{j=1}^m \mathrm{d}u_j \end{aligned}$$

(40)

with $W(t_0) = V(t_0)$. Let us consider a sequence $\{u^k\}_{k=1}^\infty$ of continuously differentiable functions converging pointwise to $u(\cdot )$ and having the property that

$$\begin{aligned} \lim _{k\rightarrow \infty }\mathrm{var}_{u_k}([t_0,T]) = \mathrm{var}_{u}([t_0,T]). \end{aligned}$$

Let $W^k(\cdot )$ represent the solution of (40) obtained by replacing $\mathrm{d}u_j$ with $\dot{u}_j^k \mathrm{d}t$, $j = 1,\dots ,m$. Then

$$\begin{aligned} W^k(t) = \Psi (t,t_0)W^k(t_0)+ c\sum _{j=1}^m \int _{t_0}^t \Psi (t,s)\dot{u}_j^k(s)\mathrm{d}s \end{aligned}$$

where $\Psi (t,s) = \mathrm{e}^{-a(t-s)+b\sum _{j=1}^m (u_j^k(t)-u_j^k(s))}$ and $W^k(t_0) = W(t_0)$ for each $k \ge 1$. Using the inequalities $\mathrm{e}^{-a(t-s)} \le 1$ and $|u^k(t)-u^k(s)|_1 \le \int _{t_0}^t|\dot{u}^k(\tau )|_1 \mathrm{d}\tau$ for each $s \in [t_0,t]$, $t < t_1$, we obtain

$$\begin{aligned} W^k(t) \le \mathrm{e}^{-a(t-t_0)+b\sum _{j=1}^m (u_j^k(t)-u_j^k(t_0))} W^k(t_0)+c \,\mathrm{e}^{b\Vert \dot{u}^k_{(t_0,t)}\Vert _1}\Vert \dot{u}^k_{(t_0,t)}\Vert _1. \end{aligned}$$

Noting that, ${\Vert \dot{u}^k_{(t_0,t)}\Vert _1}$ denotes the total variation of $\dot{u}_k(\cdot )$ over the interval $(t_0,t)$, so by construction ${\Vert \dot{u}^k_{(t_0,t)}\Vert _1}$ converges to $|\mu |_{(t_0,t)}$. Hence,

$$\begin{aligned} W(t) \le \mathrm{e}^{-a(t-t_0)+b\sum _{j=1}^m \mu _j((t_0,t))}W(t_0)+c\mathrm{e}^{b|\mu |_{(t_0,t)}}|\mu |_{(t_0,t)}. \end{aligned}$$

Step 2: We now show that $V(t) \le W(t)$ for each $t \in [t_0,T]$, from which the desired result follows. Indeed, if $V$ and $W$ satisfy (39) and (40), then $\overline{W}:= V-W$ satisfies

$$\begin{aligned} \mathrm{d}\overline{W} \le -a\overline{W} \mathrm{d}t + b \overline{W} \mathrm{d}u \end{aligned}$$

where $\overline{W} (t_0) = 0$. Using the same arguments as in the proof of Theorem 1, we obtain

$$\begin{aligned} \overline{W}(t) \le \mathrm{e}^{-a(t-t_0)+b\mu ((t_0,t))}\overline{W}(t_0) \end{aligned}$$

so that $\overline{W}(t) \le 0$, or equivalently $V(t) \le W(t)$. $\square$