ME 583 Review. Based on Nonlinear Systems (Hassan K. Khalil) book.

# Introduction

Dynamical system can be represented by a finite number of coupled ODEs: $\displaylines{\dot{x} = f(t, x, u) \\ y = h(t, x, u) }$ When $$f$$ does not depend explicitly on $$u$$, the state equation becomes: $\displaylines{\dot{x} = f(t, x) }$ Furthermore, the system is said to be autonomous or time invariant if $$f$$ does not depend explicitly on $$t$$: $\displaylines{\dot{x} = f(x) }$

Compared to Linear systems, nonlinear systems have some unique phenomenons:

1. Finite escape time:
• Linear system: state can only go to infinity in infinite time.
• Nonlinear system: can go in finite time.
2. Multiple isolated equilibriums:
• Linear system: can have only one isolated equi. point.
• Nonlinear system: state may converge to one of several steady-state operating points, depending on the initial state of the system.
3. Limit cycles:
• Linear system: must have a pair of eigenvalues on the imaginary axis to oscillate, which is nonrobust (unstable to perturbations).
• Nonlinear system: can go into an oscillation of fixed amplitude and frequency, irrespective of the initial state.
4. Subharmonic, harmonic, or almost-periodic oscillations:
• Linear system: produces an output of the same frequency under a periodic input.
• Nonlinear system: can oscillate with frequencies that are submultiples or multiples. It may even generate an almost-periodic oscillation.
5. Chaos:
• Linear system: deterministic steady-state behavior.
• Nonlinear system: can have more complicated steady-state behavior that is not equilibrium, periodic oscillation, or almost-periodic oscillation.
6. Multiple modes of behavior: Nonlinear system may exhibit multiple models of behavior based on type of excitations. When the property of excitation change smoothly, the behavior mode can have discontinuous jump.

# Second-Order System

Consider a second-order autonomous system: $$$\displaylines{ \begin{split} \dot{x}_1 = f_1(x_1, x_2) \\ \dot{x}_2 = f_2(x_1, x_2) \end{split} } \label{second}$$$

## Qualitative Behavior of Linear Systems

For a second-order LTI system, $$\eqref{second}$$ becomes: $\displaylines{\dot{x} = Ax }$ and the solution given $$x(0) = x_0$$ is $\displaylines{x(t) = M e^{J t} M^{-1} x_0 }$ where $$J$$ is the Jordan form of $$A$$ and $$M$$ is a real nonsingular matrix s.t. $$M^{-1} A M = J$$. $$J$$ can have three forms depending on the eigenvalues of $$A$$.

### Case 1

$$\lambda_1 \ne \lambda_2 \ne 0$$, $$J = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}$$

The change of coordinates $$z = M^{-1} x$$ transforms the system into two decoupled first-order DE: $\displaylines{\dot{z}_1 = \lambda_1 z_1 \\ \dot{z}_2 = \lambda_2 z_2 }$

1. $$\lambda_1 < 0, \lambda_2 < 0$$: The equi. point $$x=0$$ is stable. The phase portrait in $$z_1$$-$$z_2$$ plane

2. $$\lambda_1 > 0, \lambda_2 > 0$$: $$x=0$$ is unstable. Change the arrow direction in the above image to get the phase portrait.

3. $$\lambda_1 > 0 > \lambda_2$$: $$x=0$$ is a saddle point.

### Case 2

$$\lambda_1 = \lambda_2 \in \mathbb{R}$$, $$J = \begin{bmatrix} \lambda_1 & k \\ 0 & \lambda_2 \end{bmatrix}$$, $$k$$ is either 0 or 1

The phase portrait for $$k=0$$ and $$k=1$$ respectively:

### Case 3

$$\lambda_{1,2} = \alpha \pm j \beta$$, $$J = \begin{bmatrix} \alpha & -\beta \\ \beta & \alpha \end{bmatrix}$$

The phase portrait for $$\alpha < 0$$, $> 0$, $$\alpha = 0$$ respectively:

$$x=0$$ is referred as a stable focus if $$\alpha < 0$$, unstable focus if $$\alpha > 0$$, and center if $$\alpha = 0$$

### Case 4

$$\lambda_1 \lambda_2 = 0$$

$$A$$ has a nontrivial null space and the system has a equilibrium subspace.

## Periodic Orbits

Consider the second-order autonomous system $$$\displaylines{ \dot{x} = f(x) } \label{sys}$$$ where $$f(x)$$ is cont. diff..

Poincare-Bendixson Criterion: Consider system $$\eqref{sys}$$ and let $$M$$ be a closed bounded subset of the plane s.t.

• $$M$$ contains no equi. points, or contains only one equi. point s.t. the Jacobian matrix $$\partial f / \partial x$$ at this point has eigenvalues with positive real parts.
• Every trajectory starting in $$M$$ stays in $$M$$ for all future time

Then, $$M$$ contains a periodic orbit of $$\eqref{sys}$$.

Bendixson Criterion: If, on a simply connected region $$D$$ of the plane, $$\nabla \cdot f$$ is not identically zero and does not change sign, then system $$\eqref{sys}$$ has no periodic orbits lying entirely in $$D$$

# Fundamental Properties

## Definition

• Connected set is a set that can not be partitioned into two open nonempty sets
• Compact set: closed and bounded
• Domain: open and connected set
• Locally Lipschitz (LL) on a domain $$D \in \mathbb{R}^n$$ if each point of $$D$$ has a neighborhood $$D_0$$ s.t. $$f$$ satisfies the Lipschitz condition for all points in $$D_0$$ with some Lipschitz const. $$L_0$$.
• Globally Lipschitz (GL): Lipschitz on $$\mathbb{R}^n$$ with a uniform Lipschitz const..

## Existence and Uniqueness

Theorem 1 (Local Existence and Uniqueness): Let $$f(t, x)$$ be piecewise cont. in $$t$$ and satisfy the Lipschitz condition $\displaylines{\| f(t, x) - f(t, y) \| \le L \| x - y \| }$ $$\forall x, y\in B = \{ x\in \mathbb{R}^n \mid \| x - x_0 \| \le r \}$$, $$\forall t \in [t_0, t_1]$$. Then there exists some $$\delta > 0$$ s.t. the state equation $$\dot{x} = f(t, x)$$ with $$x(t_0) = x_0$$ has a unique solution over $$[t_0, t_0 + \delta]$$.

We have some lemmas below to prove Lipschitz condition by $$\partial f / \partial x$$.

Lemma 1: Let $$f : [a, b] \times D \rightarrow \mathbb{R}^m$$ be cont. on $$D\subseteq \mathbb{R}^n$$. Suppose that $$[\partial f / \partial x]$$ exists and is cont. on $$[a,b] \times D$$. If, for a convex set $$W \subseteq D$$, there is a const. $$L \ge 0$$ s.t. $\displaylines{\left\| \frac { \partial f } { \partial x } ( t , x ) \right\| \leq L }$ on $$[a,b] \times W$$, then $\displaylines{\left\| f ( t , x ) - f ( t , y ) \right\| \leq L \| x - y \| }$ for all $$t \in [a,b]$$, $$x\in W$$, and $$y\in W$$.

Lemma 2: If $$f(t,x)$$ and $$[\partial f / \partial x](t,x)$$ are cont. on $$[a,b]\times D$$, for $$D\in \mathbb{R}^n$$, then $$f$$ is LL in $$x$$ on $$[a,b] \times D$$.

Lemma 3: If $$f(t,x)$$ and $$[\partial f / \partial x](t,x)$$ are cont. on $$[a,b] \times \mathbb{R}^n$$, then $$f$$ is GL in $$x$$ on $$[a,b]\times \mathbb{R}^n$$ iff $$[\partial f / \partial x]$$ is uniformly bounded (UB) on $$[a,b]\times \mathbb{R}^n$$.

Theorem 2 (Global Existence and Uniqueness): Let $$f(t,x)$$ be piecewise cont. in $$t$$ and satisfy $\displaylines{\| f(t, x) - f(t, y) \| \le L \| x - y \| }$ $$\forall x, y \in \mathbb{R}^n$$, $$\forall t \in [t_0, t_1]$$. Then, the state equation $$\dot{x} = f(t, x)$$, with $$x(t_0) = x_0$$, has a unique solution over $$[t_0, t_1]$$

Theorem 3: Global existence and uniqueness theorem that requires $$f$$ to be only LL:

Let $$f(t,x)$$ be piecewise cont. in $$t$$ and LL in $$x$$ for all $$t \ge t_0$$ and all $$x$$ in $$D\subset \mathbb{R}^n$$. Let $$W$$ be a compact subset of $$D$$, $$x_0 \in W$$, and suppose every solution of $$$\displaylines{ \dot{x} = f(t,x) \qquad x(t_0) = x_0 } \label{init}$$$ lies entirely in $$W$$. Then, there is a unique solution that is defined for all $$t \ge t_0$$.

## Continuous Dependence on Initial Conditions and Parameters

The solution of $$\eqref{init}$$ must depend cont. on the initial state $$x_0$$, the initial time $$t_0$$, and the right-hand side function $$f(t, x)$$.

Theorem 4: Let $$f(t, x)$$ be piecewise cont. in $$t$$ and Lipschitz in $$x$$ on $$[t_0, t_1] \times W$$ with a Lipschitz const. $$L$$, where $$W \subset \mathbb{R}^n$$ is an open connected set. Let $$y(t)$$ and $$z(t)$$ be the solution of $\displaylines{\dot{y} = f(t, y), \qquad y(t_0) = y_0 \\ \dot{z} = f(t, z) + g(t, z), \qquad z(t_0) = z_0 }$ s.t. $$y(t), z(t) \in W$$ for all $$t \in [t_0, t)_1]$$. Suppose that $\displaylines{\|g(t, x)\| \le \mu,\quad \forall (t, x) \in [t_0, t_1] \times W }$ for some $$\mu >0$$, Then $\displaylines{\| y ( t ) - z ( t ) \| \leq \left\| y _{ 0 } - z_ { 0 } \right\| e^{L \left( t - t _{ 0 } \right)} + \frac { \mu } { L } \left( e^{ L \left( t - t_ { 0 } \right) } - 1 \right) }$ $$\forall t \in [t_0, t_1]$$.

And the next theorem shows the continuity of solutions in terms of initial states and parameters.

Theorem 5: Let $$f(t, x, \lambda)$$ be cont. in $$(t, x, \lambda)$$ and LL in $$x$$ (uniformly in $$t$$ and $$\lambda$$) on $$[t_0, t_1] \times D \times \{ \|\lambda - \lambda_0 \| \le c \}$$, where $$D \subset \mathbb{R}^n$$ is an open connected set. Let $$y(t, \lambda_0)$$ be a solution of $$\dot{x} = f(t, x, \lambda_0)$$ with $$y(t_0, \lambda_0) = y_0 \in D$$. Suppose $$y(t, \lambda_0)$$ is defined and belongs to $$D$$ for all $$t \in [t_0, t_1]$$. Then, given $$\epsilon > 0$$, there is $$\delta >0$$ s.t. if $\displaylines{\| z_0 - y_0 \| < \delta, \qquad \| \lambda - \lambda_0 \| < \delta }$ then there is a unique solution $$z(t, \lambda)$$ of $$\dot{x} = f(t, x, \lambda)$$ defined on $$[t_0, t_1]$$, with $$z(t_0, \lambda) = z_0$$, and $$z(t, \lambda)$$ satisfies $\displaylines{\| z(t, \lambda) - y(t, \lambda_0) \| < \epsilon, \quad \forall t\in [t_0, t_1] }$

## Sensitivity Equations

Suppose that $$f(t, x, \lambda)$$ is cont. in $$(t, x, \lambda)$$ and has cont. first partial derivatives w.r.t. $$x$$ and $$\lambda$$ for all $$(t, x, \lambda) \in [t_0, t_1] \times \mathbb{R}^n \times \mathbb{R}^p$$. Let $$\lambda_0$$ be a nominal value of $$\lambda$$, and suppose that the nominal state equation $\displaylines{\dot{x} = f(t,x,\lambda_0), \qquad x(t_0) = x_0 }$ has a unique solution $$x(t, \lambda_0)$$ over $$[t_0, t_1]$$. We know that for all $$\lambda$$ sufficiently close to $$\lambda_0$$, the state equation $\displaylines{\dot{x} = f(t,x,\lambda), \qquad x(t_0) = x_0 }$ has a unique solution $$x(t, \lambda)$$ over $$[t_0, t_1]$$ that is close to the nominal solution $$x(t, \lambda_0)$$. The cont. diff. of $$f$$ w.r.t. $$x$$ and $$\lambda$$ implies the additional property that the solution $$x(t, \lambda)$$ is diff. w.r.t. $$\lambda$$ near $$\lambda_0$$. $\displaylines{x ( t , \lambda ) = x _{ 0 } + \int_ { t _{ 0 } } ^ { t } f ( s , x ( s , \lambda ) , \lambda ) d s }$ Take partial derivatives w.r.t. $$\lambda$$ yields $\displaylines{x_ { \lambda } ( t , \lambda ) = \int _{ t_ { 0 } } ^ { t } \left[ \frac { \partial f } { \partial x } ( s , x ( s , \lambda ) , \lambda ) x _ { \lambda } ( s , \lambda ) + \frac { \partial f } { \partial \lambda } ( s , x ( s , \lambda ) , \lambda ) \right] d s }$ where $$x_\lambda (t_0, \lambda) = 0$$. Differentiating w.r.t. $$t$$ yields \displaylines{\begin{aligned} \frac{\partial}{\partial t} x_\lambda (t, \lambda) &= \left.\frac{\partial f(t, x, \lambda)}{\partial x} \right|_{x = x(t, \lambda)} x_\lambda (t, \lambda) + \left.\frac{\partial f(t, x, \lambda)}{\partial \lambda} \right|_{x = x(t, \lambda)} \\ &= A(t, \lambda) x_\lambda (t, \lambda) + B(t, \lambda) \end{aligned} } For $$\lambda$$ sufficiently close to $$\lambda_0$$, the matrix $$A(t, \lambda)$$ and $$B(t, \lambda)$$ are defined on $$[t_0, t_1]$$. Hence, $$x_\lambda(t, \lambda)$$ is defined on the same interval. Let $$S(t) = x_\lambda(t, \lambda_0)$$, then $$S(t)$$ is the unique solution of the equation $$$\displaylines{ \dot{S}(t) = A(t, \lambda_0) S(t) + B(t, \lambda_0), \qquad S(t_0) = 0 } \label{sens}$$$ $$S(t)$$ is called the sensitivity function, and $$\eqref{sens}$$ is called the sensitivity equation.

# Lyapunov Stability

## Autonomous System

Consider the autonomous system $$$\displaylines{ \dot{x} = f(x) } \label{as}$$$ where $$f:D\rightarrow \mathbb{R}^n$$ is a LL map from $$D\subset \mathbb{R}^n$$ into $$\mathbb{R}^n$$. $$\bar{x}$$ is an equilibrium point of the system if $$f(\bar{x}) = 0$$. Without loss of generality, we can assume the equi. point is at the origin.

Definition 1: The equi. point $$x=0$$ of $$\eqref{as}$$ is

• stable if, for each $$\epsilon >0$$, there is $$\delta > 0$$ s.t. $\displaylines{\| x(0) \| < \delta \Rightarrow \| x(t) \| < \epsilon, \quad\forall t \ge 0 }$

• unstable if it is not stable

• asymptotically stable (AS) if it is stable and $$\delta$$ can be chosen s.t. $\displaylines{\| x(0) \| < \delta \Rightarrow \lim_{t\to \infty} x(t) = 0 }$

Theorem 1: Let $$x=0$$ be an equi. point of $$\eqref{as}$$ and $$D\subset \mathbb{R}^n$$ be a domain containing $$x=0$$. Let $$V : D \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. \displaylines{\begin{aligned} V(0) &= 0 \\ V(x) &> 0 \quad \forall x \in D -\{0\} \\ \dot{V}(x) &\le 0 \quad \forall x \in D \end{aligned} } which is called a Lyapunov function. Then, $$x=0$$ is stable. Moreover, if $\displaylines{\dot{V}(x) < 0 \quad \forall x \in D - \{0\} }$ then $$x=0$$ is AS.

Theorem 2: Let $$x=0$$ be an equi. point of $$\eqref{as}$$. Let $$V : \mathbb{R}^n \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. $\displaylines{V(0) = 0 \\ V(x) > 0 \quad \forall x \ne 0 \\ \| x \| \rightarrow \infty \Rightarrow V(x) \rightarrow \infty \\ \dot{V}(x) < 0 \quad \forall x \ne 0 }$ then $$x=0$$ is globally asymptotically stable (GAS).

A function $$V(x)$$ satisfying the condition $$V(x) \rightarrow \infty$$ as $$\| x \| \rightarrow \infty$$ is said to be radially unbounded (RU).

If the origin is a GAS, then it must be the unique equi. point.

Theorem 3: Let $$x=0$$ be an equi. point of $$\eqref{as}$$. Let $$V: D \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. $$V(0) = 0$$ and $$V(x_0) > 0$$ for some $$x_0$$ with arbitrarily small $$\| x_0 \|$$. Define a set $$U=\{x \in B_r \mid V(x) > 0\}$$ and suppose $$\dot{V}(x) > 0$$ in $$U$$. Then $$x=0$$ is unstable.

## Invariance Principle

Definition 2: A point $$p$$ is said to be a positive limit point of $$x(t)$$ if there is a sequence $$\{ t_n \}$$, with $$t_n \to \infty$$ as $$n\to \infty$$, s.t. $$x(t_n) \to p$$ as $$n \to \infty$$.

Definition 3: A set $$M$$ is said to be an invariant set w.r.t. $$\eqref{as}$$ if $\displaylines{x(0) \in M \Rightarrow x(t) \in M, \quad \forall t \in \mathbb{R} }$ it is positively invariant set if $\displaylines{x(0) \in M \Rightarrow x(t) \in M, \quad \forall t \ge 0 }$

Lemma 1: If a solution $$x(t)$$ of $$\eqref{as}$$ is bounded and belongs to $$D$$ for $$t \ge 0$$, then its positive limit set $$L^+$$ is a nonempty, compact, invariant set. Moreover, $$x(t)$$ approaches $$L^+$$ as $$t \to \infty$$.

Theorem 4: Let $$\Omega \subset D$$ be a compact set that is PI w.r.t. $$\eqref{as}$$. Let $$V : D \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. $$\dot{V}(x) \le 0$$ in $$\Omega$$. Let $$E=\{x \in \Omega \mid \dot{V}(x)=0 \}$$. Let $$M$$ be the largest invariant set in $$E$$. Then every solution starting in $$\Omega$$ approaches $$M$$ as $$t \to \infty$$.

Corollary 1: Let $$x=0$$ be an equi. point of $$\eqref{as}$$. Let $$V : D \rightarrow \mathbb{R}$$ be a cont. diff. PD function on a domain $$D$$ containing the origin, s.t. $$\dot{V}(x) \le 0$$ in $$D$$. Let $$S = \{ x \in D \mid \dot{V}(x) = 0 \}$$ and suppose that no solution can stay identically in $$S$$, other than the trivial solution $$x(t) \equiv 0$$. Then the origin is AS.

Corollary 2: Let $$x=0$$ be an equi. point of $$\eqref{as}$$. Let $$V : \mathbb{R}^n \rightarrow \mathbb{R}$$ be a cont. diff., RU, PD function s.t. $$\dot{V}(x) \le 0$$ for all $$x \in \mathbb{R}^n$$. Let $$S = \{ x \in \mathbb{R}^n \mid \dot{V}(x) = 0 \}$$ and suppose that no solution can stay identically in $$S$$, other than the trivial solution $$x(t) \equiv 0$$. Then the origin is GAS.

## LTI Systems and Linearization

Theorem 5: The equi. point $$x = 0$$ of $$\dot{x} = Ax$$ is stable iff all eigenvalues of $$A$$ satisfy $$\operatorname{Re}\lambda_i \le 0$$ and for every eigenvalues with $$\operatorname{Re}\lambda_i = 0$$ and algebraic multiplicity $$q_i \ge 2$$, $$\operatorname{rank}(A - \lambda_i I) = n - q_i$$, where $$n$$ is the dimension of $$x$$. The equi. point $$x=0$$ is (globally) AS iff all eigenvalues of $$A$$ satisfy $$\operatorname{Re}\lambda_i < 0$$

Theorem 6: A matrix $$A$$ is Hurwitz ($$\operatorname{Re}\lambda_i \le 0$$) iff for any given PSD $$Q$$ there exists a PSD that satisfies the Lyapunov equation: $\displaylines{PA + A^T P = -Q }$ Moreover, if $$A$$ is Hurwitz, then $$P$$ is the unique solution.

Theorem 7 (Lyapunov's indirect method): Let $$x=0$$ be an equi. point for $$\eqref{as}$$, where $$f:D\rightarrow \mathbb{R}^n$$ is cont. diff. and $$D$$ is a neighborhood of the origin. Let $$A = \left. \frac{\partial f}{\partial x} \right|_{x=0}$$, then

• The origin is AS if $$\operatorname{Re}\lambda_i < 0$$ for all eigenvalues of $$A$$.
• The origin is unstable if $$\operatorname{Re}\lambda_i > 0$$ for at least one eigenvalue of $$A$$.

## Comparison Functions

Definition 4: A cont. function $$\alpha: [0, a) \rightarrow [0, \infty)$$ is said to belong to $$\mathcal{K}$$ if it is strictly increasing and $$\alpha(0) = 0$$. It is said to belong to $$\mathcal{K}_\infty$$ if $$a = \infty$$ and $$\alpha(r) \rightarrow \infty$$ as $$r \rightarrow \infty$$.

Definition 5: A cont. function $$\beta: [0, a) \times [0, \infty) \rightarrow [0, \infty)$$ is said to belong to $$\mathcal{KL}$$ if, for each fixed $$s$$, the mapping $$\beta(r, s)$$ belongs to $$\mathcal{K}$$ w.r.t. $$r$$ and, for each fixed $$r$$, the mapping $$\beta(r, s)$$ is decreasing w.r.t. $$s$$ and $$\beta(r, s) \rightarrow 0$$ as $$s \rightarrow \infty$$.

Lemma 2: Let $$\alpha_1$$ and $$\alpha_2$$ be $$\mathcal{K}$$ functions on $$[0, a)$$, $$\alpha_3$$ and $$\alpha_4$$ be $$\mathcal{K}_\infty$$ functions, and $$\beta$$ be a $$\mathcal{KL}$$ function. Denote the inverse of $$\alpha_i$$ by $$\alpha_i^{-1}$$. Then

• $$\alpha_1^{-1}$$ is defined on $$[0, \alpha_1(a))$$ and belongs to $$\mathcal{K}$$
• $$\alpha_3^{-1}$$ is defined on $$[0, \infty)$$ and belongs to $$\mathcal{K}_\infty$$
• $$\alpha_1 \circ \alpha_2$$ belongs to $$\mathcal{K}$$
• $$\alpha_3 \circ \alpha_4$$ belongs to $$\mathcal{K}_\infty$$
• $$\sigma(r, s) = \alpha_1( \beta(\alpha_2(r), s))$$ belongs to $$\mathcal{KL}$$

Lemma 3: Let $$V : D \rightarrow \mathbb{R}$$ be a cont. PD function defined on a domain $$D \subset \mathbb{R}^n$$ that contains the origin. Let $$B_r \subset D$$ for some $$r > 0$$. Then, there exist $$\mathcal{K}$$ functions $$\alpha_1$$ and $$\alpha_2$$, defined on $$[0, r]$$, s.t. $\displaylines{\alpha_1(\|x\|) \le V(x) \le \alpha_2(\|x\|) }$ for all $$x\in B_r$$. If $$D = \mathbb{R}^n$$, the functions $$\alpha_1$$ and $$\alpha_2$$ will be defined on $$[0, \infty)$$ and the foregoing inequality will hold for all $$x\in \mathbb{R}^n$$. Moreover, if $$V(x)$$ is RU, then $$\alpha_1$$ and $$\alpha_2$$ can be chosen to belong to $$\mathcal{K}_\infty$$

Lemma 4: Consider the scalar autonomous DE $\displaylines{\dot{y} = -\alpha(y), \qquad y(t_0) = y_0 }$

where $$\alpha$$ is a LL $$\mathcal{K}$$ function defined on $$[0,a)$$. For all $$0 \le y_0 \le a$$, this equation has a unique solution $$y(t)$$ defined for all $$t \ge t_0$$. Moreover, $\displaylines{y(t) = \sigma(y_0, t_0) }$ where $$\sigma$$ is a $$\mathcal{KL}$$ function defined on $$[0, a) \times [0, \infty)$$.

## Nonautonomous System

Consider the nonautonomous system $$$\displaylines{ \dot{x} = f(t, x) } \label{nas}$$$ where $$f: [0, \infty) \times D \rightarrow \mathbb{R}^n$$ is piecewise cont. in $$t$$ and LL in $$x$$ on $$[0, \infty) \times D$$, and $$D \subset \mathbb{R}^n$$ is a domain that contains the origin. The origin is an equi. point at $$t=0$$ if $\displaylines{f(t, 0) = 0, \quad \forall t \ge 0 }$

Definition 6: The equi. point $$x=0$$ of $$\eqref{nas}$$ is

• stable if, for each $$\epsilon >0$$, there is $$\delta = \delta(\epsilon, t_0) > 0$$ s.t. $$$\displaylines{ \left\| x \left( t_ { 0 } \right) \right\| < \delta \Rightarrow \| x ( t ) \| < \varepsilon , \quad \forall t \geq t _ { 0 } \ge 0 } \label{nastable}$$$

• uniformly stable (US) if, for each $$\epsilon >0$$, there is $$\delta = \delta(\epsilon) > 0$$, independent of $$t_0$$ s.t. $$\eqref{nastable}$$ is satisfied.

• unstable if it is not stable

• AS if it is stable and there is a const. $$c=c(t_0) > 0$$ s.t. $$$\displaylines{ \| x(t_0) \| < c \Rightarrow \lim_{t\to \infty} x(t) = 0 } \label{naas}$$$

• uniformly asymptotically stable (UAS) if it is US and there exists $$c>0$$, independent of $$t_0$$ s.t. $$\eqref{naas}$$ is satisfied.

• globally uniformly asymptotically stable (GUAS) if it is US, $$\delta(\epsilon)$$ can be chosen to satisfy $$\lim_{\epsilon \to \infty} \delta(\epsilon) = \infty$$, and, for each pair of $$\eta > 0$$ and $$c > 0$$, there is $$T = T(\eta, c) > 0$$ s.t. $\displaylines{\| x ( t ) \| < \eta , \quad \forall t \geq t_ { 0 } + T ( \eta , c ) , \quad \forall \left\| x \left( t _ { 0 } \right) \right\| < c }$

Definition 7: The equi. point $$x=0$$ of $$\eqref{nas}$$ is

• US iff there exist a $$\mathcal{K}$$ function $$\alpha$$ and a const. $$c>0$$, independent of $$t_0$$, s.t. $\displaylines{\| x ( t ) \| \leq \alpha \left( \left\| x \left( t_ { 0 } \right) \right\| \right) , \quad \forall t \geq t _{ 0 } \geq 0 ,\ \forall \left\| x \left( t_ { 0 } \right) \right\| < c }$

• UAS iff there exist a $$\mathcal{KL}$$ function $$\beta$$ and a const. $$c>0$$, independent of $$t_0$$, s.t. $$$\displaylines{ \| x ( t ) \| \leq \beta \left( \left\| x \left( t_ { 0 } \right) \right\| , t - t _{ 0 } \right) , \quad \forall t \geq t_ { 0 } \geq 0 ,\ \forall \left\| x \left( t _ { 0 } \right) \right\| < c } \label{uaskl}$$$

• GUAS iff $$\eqref{uaskl}$$ is satisfied for any initial state $$x(t_0)$$

Definition 8: The equi. point $$x=0$$ of $$\eqref{nas}$$ is exponentially stable (ES) if there exist const. $$c > 0$$, $$k > 0$$ and $$\lambda > 0$$ s.t. $\displaylines{\| x ( t ) \| \leq k \left\| x \left( t _{ 0 } \right) \right\| e ^ { - \lambda \left( t - t_ { 0 } \right) } , \quad \forall \left\| x \left( t _{ 0 } \right) \right\| < c }$ and globally exponentially stable (GES) if it holds for any initial state $$x(t_0)$$

Theorem 8: Let $$x=0$$ be an equi. point of $$\eqref{nas}$$ and $$D\subset \mathbb{R}^n$$ be a domain containing $$x=0$$. Let $$V : [0, \infty) \times D \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. $\displaylines{W_1(x) \le V(t, x) \le W_2(x) \\ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(t, x) \le 0 }$ $$\forall t \ge 0$$ and $$\forall x \in D$$, where $$W_1(x)$$ and $$W_2(x)$$ are cont. PD functions on $$D$$. Then, $$x=0$$ is US.

Theorem 9: Suppose the assumptions of Theorem 8 are satisfied with inequality strengthened to $\displaylines{\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(t, x) \le -W_3(x) }$ $$\forall t \ge 0$$ and $$\forall x \in D$$, where $$W_3(x)$$ is a cont. PD function on $$D$$. Then, $$x=0$$ is UAS. Moreover, if $$r$$ and $$c$$ are chosen s.t. $$B_r = \{ \|x\| \le r \} \subset D$$ and $$c < \min_{\|x\| = r} W_1(x)$$, then every trajectory starting in $$\{ x \in B_r \mid W_2(x) \le c\}$$ satisfies $\displaylines{\| x ( t ) \| \leq \beta \left( \left\| x \left( t_ { 0 } \right) \right\| , t - t _{ 0 } \right) , \quad \forall t \geq t_ { 0 } \geq 0 }$ for some class $$\mathcal{KL}$$ function $$\beta$$. Finally, if $$D = \mathbb{R}^n$$ and $$W_1(x)$$ is RU, then $$x=0$$ is GUAS.

Theorem 10: Let $$x=0$$ be an equi. point of $$\eqref{nas}$$ and $$D\subset \mathbb{R}^n$$ be a domain containing $$x=0$$. Let $$V : [0, \infty) \times D \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. $$$\displaylines{ { k _{ 1 } \| x \| ^ { a } \leq V ( t , x ) \leq k_ { 2 } \| x \| ^ { a } } \\ { \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - k _{ 3 } \| x \| ^ { a } } } \label{es}$$$ $$\forall t \ge 0$$ and $$\forall x \in D$$, where $$k_1, k_2, k_3, a$$ are positive const.. Then $$x=0$$ is ES. If the assumptions hold globally, then $$x=0$$ is GES.

## LTV Systems and Linearization

The stability for the LTV system $$$\displaylines{ \dot{x}(t) = A(t) x } \label{ltv}$$$ can be completely characterized in terms of the state transition matrix $\displaylines{x(t) = \Phi(t, t_0) x(t_0) }$

Theorem 11: The equi. point $$x=0$$ of $$\eqref{ltv}$$ is (globally) UAS iff the state transition matrix satisfies $\displaylines{\left\| \Phi \left( t , t _{ 0 } \right) \right\| \leq k e ^ { - \lambda \left( t - t_ { 0 } \right) } , \quad \forall t \geq t _ { 0 } \geq 0 }$ for some positive const. $$k$$ and $$\lambda$$

Theorem 12: Let $$x=0$$ be the ES equi. point of $$\eqref{ltv}$$. Suppose $$A(t)$$ is cont. and bounded. Let $$Q(t)$$ be a cont., bounded, PD, symmetric matrix. Then, there is a cont. diff. bounded, PD, symmetric matrix $$P(t)$$ that satisfies $\displaylines{-\dot { P } ( t ) = P ( t ) A ( t ) + A ^ { T } ( t ) P ( t ) + Q ( t ) }$ and $$V(t,x) = x^T P(t) x$$ is a Lyapunov function for the system that satisfies $$\eqref{es}$$

Theorem 13: Let $$x=0$$ be an equi. point for $$\eqref{nas}$$, where $$f: [0, \infty) \times D \rightarrow \mathbb{R}^n$$ is cont. diff., $D = { x |x|_2 < r }$, and the Jacobian matrix $f / x$ is bounded and Lipschitz on $$D$$, uniformly in $t$. Let $\displaylines{A(t) = \left. \frac{\partial f}{\partial x} (t, x) \right|_{x=0} }$

Then, the origin is ES for the nonlinear system iff it is an ES for the linear system $$\dot{x} = A(t) x$$

## Converse Theorems

Theorem 14: Let $$x=0$$ be an equi. point for $$\eqref{nas}$$, where $$f: [0, \infty) \times D \rightarrow \mathbb{R}^n$$ is cont. diff., $D = { x |x| < r }$, and the Jacobian matrix $f / x$ is bounded on $$D$$, uniformly in $$t$$. Let $$k, \lambda, r_0$$ be positive const. with $$r_0 < r/ k$$. Let $D_0 = { x |x| < r_0 }$. Assume that the trajectories of the system satisfy $\displaylines{\| x ( t ) \| \leq k \left\| x \left( t _{ 0 } \right) \right\| e ^ { - \lambda \left( t - t_ { 0 } \right) } , \quad \forall x \left( t _{ 0 } \right) \in D_ { 0 } , \ \forall t \geq t _{ 0 } \geq 0 }$ Then, there is a function $$V: [0, \infty) \times D_0 \rightarrow \mathbb{R}$$ that satisfies the inequalities $\displaylines{c _{ 1 } \| x \| ^ { 2 } \leq V ( t , x ) \leq c_ { 2 } \| x \| ^ { 2 } \\ \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - c _{ 3 } \| x \| ^ { 2 } \\ \left\| \frac { \partial V } { \partial x } \right\| \leq c_ { 4 } \| x \| }$ for some positive const. $$c_1, c_2, c_3, c_4$$. Moreover, if $$r=\infty$$ and the origin is GES, then $$V(t,x)$$ is defined and satisfies the above inequalities on $$\mathbb{R}^n$$. Furthermore, if the system is autonomous, $$V$$ can be chosen independent of $$t$$

Theorem 15: Let $$x=0$$ be an AS equil. point for $$\eqref{nas}$$ where $$f: [0, \infty) \times D \rightarrow \mathbb{R}^n$$ is cont. diff., $D = { x |x| < r }$, and the Jacobian matrix $f / x$ is bounded on $$D$$, uniformly in $$t$$. Let $$\beta$$ be a $$\mathcal{KL}$$ function and $$r_0$$ be a positive const. s.t. $$\beta(r_0, 0) < r$$. Let $$D_0 = \{x\in \mathbb{R}^n \mid \|x\| < r_0 \}$$. Assume that the trajectory of the system satisfies $\displaylines{\| x ( t ) \| \leq \beta \left( \left\| x \left( t _{ 0 } \right) \right\| , t - t_ { 0 } \right) , \quad \forall x \left( t _{ 0 } \right) \in D_ { 0 } ,\ \forall t \geq t _{ 0 } \geq 0 }$ Then, there is a cont. diff. function $$V: [0, \infty) \times D_0 \rightarrow \mathbb{R}$$ that satisfies $\displaylines{\alpha _{ 1 } ( \| x \| ) \leq V ( t , x ) \leq \alpha_ { 2 } ( \| x \| ) \\ \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - \alpha _{ 3 } ( \| x \| ) \\ \left\| \frac { \partial V } { \partial x } \right\| \leq \alpha_ { 4 } ( \| x \| ) }$ where $$\alpha_{1,2,3,4}$$ are $$\mathcal{K}$$ functions defined on $$[0, r_0]$$. If the system is autonomous, $$V$$ can be chosen independent of $$t$$

Theorem 16: Let $$x=0$$ be an AS equil. point for $$\eqref{as}$$ where $$f: D \rightarrow \mathbb{R}^n$$ is LL and $$D \subset \mathbb{R^n}$$ is a domain contains the origin. Let $$R_A \subset D$$ be the region of attraction of $$x=0$$. Then there is a smooth, PD function $$V(x)$$ and a cont. PD function $$W(x)$$, both defined for all $$x\in R_A$$, s.t. $\displaylines{\lim_{x \rightarrow \partial R_ { A }} V ( x ) \rightarrow \infty \\ \frac { \partial V } { \partial x } f ( x ) \leq - W ( x ) , \quad \forall x \in R _{ A } }$ and for any $$c>0$$, $$\{V(x) \le c\}$$ is a compact subset of $$R_A$$. When $$R_A = \mathbb{R}^n$$, $$V(x)$$ is RU.

## Boundedness

Definition 9: The solution of $$\eqref{nas}$$ are

• uniformly bounded (UB) if there exists a positive const. $$c$$, independent of $$t_0 \ge 0$$, and for every $$a \in (0, c)$$, there is $$\beta = \beta(a) > 0$$, independent of $$t_0$$, s.t. $$\left\| x \left( t _{ 0 } \right) \right\| \leq a \Rightarrow \| x ( t ) \| \leq \beta , \quad \forall t \geq t_ { 0 }$$
• globally uniformly bounded (GUB) if UB holds for $$c=\infty$$
• uniformly ultimately bounded (UUB) with ultimate bound $$b$$ if there exist positive const. $$b$$ and $$c$$, independent of $$t_0 \ge 0$$, and for every $$a\in (0, c)$$, there is $$T = T(a, b) \ge 0$$, independent of $$t_0$$, s.t. $$\left\| x \left( t _{ 0 } \right) \right\| \leq a \Rightarrow \| x ( t ) \| \leq b , \quad \forall t \geq t_ { 0 } + T$$
• globally uniformly ultimately bounded (GUUB) if UUB holds for $$c=\infty$$

Theorem 17: Let $$D \subset \mathbb{R}^n$$ be a domain containing $$x=0$$ and $$V : [0, \infty) \times D \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. $\displaylines{\alpha _{ 1 } ( \| x \| ) \leq V ( t , x ) \leq \alpha_ { 2 } ( \| x \| ) \\ \frac { \partial V } { \partial t } + \frac { \partial V } { \partial x } f ( t , x ) \leq - W _{ 3 } ( x ) , \quad \forall \|x\| \ge \mu > 0 }$ $$\forall t \ge 0$$ and $$\forall x \in D$$, where $$\alpha_{1,2}$$ are $$\mathcal{K}$$ functions and $$W_3(x)$$ is a cont. PD function. Take $$r > 0$$ s.t. $$B_r \subset D$$ and suppose that $$\mu < \alpha_2^{-1}(\alpha_1(r))$$, then, there exists a $$\mathcal{KL}$$ function $$\beta$$ and $$\forall x(t_0)$$ that satisfies $$\|x(t_0)\| \le \alpha_2^{-1}(\alpha_1(r))$$, there is $$T \ge 0$$ (dependent on $$x(t_0)$$ and $$\mu$$) s.t. the solution of $$\eqref{nas}$$ satisfies $\displaylines{\| x ( t ) \| \leq \beta \left( \left\| x \left( t_ { 0 } \right) \right\| , t - t _{ 0 } \right) , \forall t_ { 0 } \leq t \leq t _{ 0 } + T \\ \| x ( t ) \| \leq \alpha_ { 1 } ^ { - 1 } \left( \alpha _{ 2 } ( \mu ) \right) , \forall t \geq t_ { 0 } + T }$ Moreover, if $$D = \mathbb{R}^n$$ and $$\alpha_1$$ belongs to $$\mathcal{K}_\infty$$, then it holds $$\forall x(t_0)$$

## Input-to-State Stability

Consider the system $$$\displaylines{ \dot{x} = f(t, x, u) } \label{nasinput}$$$ where $$f: [0, \infty) \times \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$$ is piecewise cont. in $$t$$ and LL in $$x$$ and $$u$$. The input $$u(t)$$ is a piecewise cont., bounded function of $$t$$ for all $$t \ge 0$$. Suppose the unforced system $$\dot{x} = f(t, x, 0)$$ has a GUAS equi. point at the origin. What can we say about the system $$\eqref{nasinput}$$ in the presence of a bounded input $$u(t)$$.

For the LTI system: $\displaylines{\dot{x} = Ax + Bu }$ with a Hurwitz matrix $$A$$, we can write the solution as $\displaylines{x(t)=e^{\left(t-t_{0}\right) A} x\left(t_{0}\right)+\int_{t_{0}}^{t} e^{(t-\tau) A} B u(\tau) d \tau }$ and use the bound $$\left\|e^{\left(t-t_{0}\right) A}\right\| \leq k e^{-\lambda\left(t-t_{0}\right)}$$ to estimate the solution by \displaylines{\begin{aligned} \|x(t)\| & \leq k e^{-\lambda\left(t-t_{0}\right)}\left\|x\left(t_{0}\right)\right\|+\int_{t_{0}}^{t} k e^{-\lambda(t-\tau)}\|B\|\|u(\tau)\| d \tau \\ & \leq k e^{-\lambda\left(t-t_{0}\right)}\left\|x\left(t_{0}\right)\right\|+\frac{k\|B\|}{\lambda} \sup _{t_{0} \leq \tau \leq t}\|u(\tau)\| \end{aligned} } This estimate shows that the zero-input response decays to zero exponentially fast, while the zero-state response is bounded for every bounded input (BIBS).

Definition 10: The system $$\eqref{nasinput}$$ is said to be input-to-state stable (ISS) if there exist a $$\mathcal{KL}$$ function $$\beta$$ and a $$\mathcal{K}$$ function $$\gamma$$ s.t. for any initial state $$x(t_0)$$ and any bounded input $$u(t)$$, the solution $$x(t)$$ exists for all $$t \ge t_0$$ and satisfies $\displaylines{\|x(t)\| \leq \beta\left(\left\|x\left(t_{0}\right)\right\|, t-t_{0}\right)+\gamma\left(\sup _{t_{0} \leq \tau \leq t}\|u(\tau)\|\right) }$

Theorem 18: Let $$V : [0, \infty) \times \mathbb{R}^n \rightarrow \mathbb{R}$$ be a cont. diff. function s.t. \displaylines{\begin{aligned} \alpha_{1}(\|x\|) & \leq V(t, x) \leq \alpha_{2}(\|x\|) \\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x} f(t, x, u) &\leq-W_{3}(x), \quad \forall\|x\| \geq \rho(\|u\|)>0 \end{aligned} } $$\forall (t,x,u) \in [0, \infty) \times \mathbb{R}^n \times \mathbb{R}^m$$, where $$\alpha_{1,2}$$ are $$\mathcal{K}_\infty$$ functions, $$\rho$$ is $$\mathcal{K}$$ function, and $$W_3(x)$$ is a cont. PD function on $$\mathbb{R}^n$$. Then, the system $$\eqref{nasinput}$$ is ISS with $$\gamma = \alpha_1^{-1} \circ \alpha_2 \circ \rho$$

Lemma 5: Suppose $$f(t,x,u)$$ is cont. diff. and GL in $$(x,u)$$, uniformly in $$t$$. If the unforced system $$\dot{x} = f(t, x, 0)$$ has a GES equi. point at the origin, then the system $$\eqref{nasinput}$$ is ISS.

Consider the cascade system: \begin{align} dot{x}_{1}&=f_{1}\left(t, x_{1}, x_{2}\right) \label{cascade1} \\ \dot{x}_{2}&=f_{2}\left(t, x_{2}\right) \label{cascade2} \end{align} where $$f_1 : [0, \infty) \times \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \rightarrow \mathbb{R}^{n_1}$$ and $$f_2 : [0, \infty) \times \mathbb{R}^{n_2} \rightarrow \mathbb{R}^{n_2}$$ are piecewise cont. in $$t$$ and LL in $$x$$. Suppose both $$\dot{x}_1 = f_1(t, x_1, 0)$$ and $$\dot{x}_{2}=f_{2}\left(t, x_{2}\right)$$ have GUAS equi. point at their respective origins.

Lemma 6: If the system $$\eqref{cascade1}$$, with $$x_2$$ as input, is ISS and the origin of $$\eqref{cascade2}$$ is GUAS, then the origin of the cascade system is GUAS.

# Input-Output Stability

## $$\mathcal{L}$$ Stability

Consider a system with input-output relation represented by $\displaylines{y = H u }$ Where $$H$$ is a mapping from $$u$$ to $$y$$, $$u : [0, \infty) \rightarrow \mathbb{R}^m$$.

Define the space $$\mathcal{L}_p^m$$ for $$1\le p \le \infty$$ as the set of all piecewise cont. functions $$u : [0, \infty) \rightarrow \mathbb{R}^m$$ s.t. $\displaylines{\|u\|_{\mathcal{L}_{p}}=\left(\int_{0}^{\infty}\|u(t)\|^{p} d t\right)^{1 / p}<\infty }$

Specifically, for $$p=2$$ and $$p=\infty$$, the space are defined respectively as \displaylines{\begin{aligned} \|u\|_{\mathcal{L}_{2}} &= \sqrt{\int_{0}^{\infty} u^{T}(t) u(t) d t}<\infty \\ \| u \|_ { \mathcal { L } _{ \infty } } &= \sup_ { t \geq 0 } \| u ( t ) \| < \infty \end{aligned} } Define the extended space $$\mathcal{L}_e^m$$ as $\displaylines{\mathcal{L}_{e}^{m}=\left\{u | u_{\tau} \in \mathcal{L}^{m}, \forall \tau \in[0, \infty)\right\} }$ where $$u_\tau$$ is a truncation of $$u$$ defined by $\displaylines{u_{\tau}(t)=\left\{\begin{array}{cc}{u(t),} & {0 \leq t \leq \tau} \\ {0,} & {t>\tau}\end{array}\right. }$

Definition 1: A mapping $$H : \mathcal{L}_e^m \rightarrow \mathcal{L}_e^q$$ is $$\mathcal{L}$$ stable if there exist a $$\mathcal{K}$$ function $$\alpha$$, defined on $$[0, \infty)$$, and a nonnegative constant $$\beta$$ s.t. $\displaylines{\left\|(H u)_{\tau}\right\|_{\mathcal{L}} \leq \alpha\left(\left\|u_{\tau}\right\|_{\mathcal{L}}\right)+\beta }$ for all $$u \in \mathcal{L}_e^m$$ and $$\tau \in [0, \infty)$$. It is finite-gain $$\mathcal{L}$$ stable if there exist nonnegative const. $$\gamma$$ and $$\beta$$ s.t. $\displaylines{\left\|(H u)_{\tau}\right\|_{\mathcal{L}} \leq \gamma \left\|u_{\tau}\right\|_{\mathcal{L}} + \beta }$ for all $$u \in \mathcal{L}_e^m$$ and $$\tau \in [0, \infty)$$.

Note that the definition of $$\mathcal{L}_\infty$$ stability is same as BIBO stability.

Definition 2: A mapping $$H : \mathcal{L}_e^m \rightarrow \mathcal{L}_e^q$$ is small-signal $$\mathcal{L}$$ stable (small-signal finite-gain $$\mathcal{L}$$ stable) if there is a positive const. $$r$$ s.t. inequality in definition 1 is satisfied for all $$u \in \mathcal{L}_e^m$$ with $$\sup_{0 \le t \le \tau} \|u(t)\| \le r$$

## $$\mathcal{L}$$ Stability of State Models

Consider the system: \displaylines{ \begin{aligned} \dot{x} &=f(t, x, u), \quad x(0)=x_{0} \\ y &=h(t, x, u) \end{aligned} } \label{statemodel} where $$x\in \mathbb{R}^n$$, $$u\in \mathbb{R}^m$$, $$y\in \mathbb{R}^q$$, $$f:[0, \infty) \times D \times D_u \rightarrow \mathbb{R}^n$$ is piecewise cont. in $$t$$ and LL in $$(x,u)$$; $$u:[0, \infty) \times D \times D_u \rightarrow \mathbb{R}^q$$ is piecewise cont. in $$t$$ and cont. in $$(x,u)$$; $$D \subset \mathbb{R}^n$$ is a domain that contains $$x=0$$, and $$D_u \subset \mathbb{R}^m$$ is a domain that contains $$u = 0$$. Suppose $$x=0$$ is an equi. point of the unforced system $$$\displaylines{ \dot{x} = f(t,x,0) } \label{unforced}$$$ Theorem 1: Consider the system $$\eqref{statemodel}$$ and take $$r > 0$$ and $$r_u > 0$$ s.t. $$\{ \|x\| \le r \} \subset D$$ and $$\{ \|u\| \le r_u \} \subset D_u$$. Suppose that

• $$x=0$$ is an ES equi. ponit of $$\eqref{unforced}$$, and there is a $$V(t, x)$$ that satisfies $\displaylines{c_{1}\|x\|^{2} \leq V(t, x) \leq c_{2}\|x\|^{2} \\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x} f(t, x, 0) \leq -c_{3}\|x\|^{2} \\ \left\|\frac{\partial V}{\partial x}\right\| \leq c_{4}\|x\| }$ for all $$(t,x) \in [0, \infty) \times D$$ for some positive const. $$c_{1,2,3,4}$$

• $$f$$ and $$h$$ satisfy the inequalities \begin{align} \|f(t, x, u)-f(t, x, 0)\| \leq L\|u\| \\ \|h(t, x, u)\| \leq \eta_{1}\|x\|+\eta_{2}\|u\| \label{hbound} \end{align} for all $$(t,x,u) \in [0, \infty) \times D \times D_u$$ for some nonnegative const. $$L, \eta_{1,2}$$

Then, for each $$x_0$$ with $$\|x_0\| \le r \sqrt{c_1 / c_2}$$, the system $$\eqref{statemodel}$$ is small-signal finite-gain $$\mathcal{L}_p$$ stable for each $$p \in [1, \infty]$$. In particular, for each $$u \in \mathcal{L}_{pe}$$ with $$\sup_{0 \le t \le \tau} \| u(t) \| \le \min \{ r_u, c_1 c_3 r / (c_2 c_4 L) \}$$, the output $$y(t)$$ satisfies $$$\displaylines{ \left\|y_{\tau}\right\|_{\mathcal{L}_{p}} \leq \gamma \left\|u_{\tau}\right\|_{\mathcal{L}_{p}} + \beta } \label{outputgain}$$$ for all $$\tau \in [0, \infty)$$, with $\displaylines{\gamma=\eta_{2}+\frac{\eta_{1} c_{2} c_{4} L}{c_{1} c_{3}}, \quad \beta=\eta_{1}\left\|x_{0}\right\| \sqrt{\frac{c_{2}}{c_{1}}} \rho, \text { where } \rho=\left\{\begin{array}{ll} {1,} & {\text { if } p=\infty} \\ {\left(\frac{2 c_{2}}{c_{3} p}\right)^{1 / p},} & {\text { if } p \in[1, \infty)} \end{array}\right. }$ Furthermore, if the origin is GES and all the assumptions hold globally (with $$D = \mathbb{R}^n$$ and $$D_u = \mathbb{R}^m$$), then, for each $$x_0 \in \mathbb{R}^n$$, the system $$\eqref{statemodel}$$ if finite-gain $$\mathcal{L}_p$$ stable for each $$p \in [1, \infty)$$.

Corollary 1: Suppose that in some neighborhood of $$(x=0, u=0)$$, the function $$f(t,x,u)$$ is cont. diff., the Jacobian matrices $$\partial f / \partial x$$ and $$\partial f / \partial u$$ are bounded, uniformly in $$t$$, and $$h(t,x,u)$$ satisfies $$\eqref{hbound}$$. If the origin is an ES equi. point of $$\eqref{unforced}$$, then there is a const. $$r_0 > 0$$ s.t. for each $$x_0$$ with $$\|x_0\| < r_0$$, the system $$\eqref{statemodel}$$ is small-signal finite-gain $$\mathcal{L}_p$$ stable for each $$p \in [1, \infty]$$. Furthermore, if all the assumptions hold globally and the origin is a GES equi. point of $$\eqref{unforced}$$, then for each $$x_0 \in \mathbb{R}^n$$, the system $$\eqref{statemodel}$$ if finite-gain $$\mathcal{L}_p$$ stable for each $$p \in [1, \infty]$$

Corollary 2: The LTI system \displaylines{ \begin{aligned} \dot{x} &=A x+B u \\ y &=C x+D u \end{aligned} } \label{ltiinput} is finite-gain $$\mathcal{L}_p$$ stable for each $$p \in [1, \infty]$$ if $$A$$ is Hurwitz. Moreover, $$\eqref{outputgain}$$ is satisfied with $\displaylines{\gamma=\|D\|_{2}+\frac{2 \lambda_{\max }^{2}(P)\|B\|_{2}\|C\|_{2}}{\lambda_{\min }(P)}, \quad \beta=\rho\|C\|_{2}\left\|x_{0}\right\| \sqrt{\frac{\lambda_{\max }(P)}{\lambda_{\min }(P)}}, \text { where } \rho=\left\{\begin{array}{ll} {1,} & {\text { if } p=\infty} \\ {\left(\frac{2 \lambda_\max(P)}{p}\right)^{1 / p},} & {\text { if } p \in[1, \infty)} \end{array}\right. }$ and $$P$$ is the solution of the Lyapunov equation $$PA + A^TP = - I$$

Theorem 2: Consider the system $$\eqref{statemodel}$$ and take $$r > 0$$ s.t. $$\{ \|x\| \le r \} \subset D$$. Suppose that

• $$x=0$$ is an UAS equi. ponit of $$\eqref{unforced}$$, and there is a $$V(t, x)$$ that satisfies $\displaylines{\alpha_{1}(\|x\|) \leq V(t, x) \leq \alpha_{2}(\|x\|) \\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x} f(t, x, 0) \leq -\alpha_{3}(\|x\|) \\ \left\|\frac{\partial V}{\partial x}\right\| \leq \alpha_4(\|x\|) }$ for all $$(t,x) \in [0, \infty) \times D$$ for some $$\mathcal{K}$$ functions $$\alpha_{1,2,3,4}$$

• $$f$$ and $$h$$ satisfy the inequalities \begin{align} \|f(t, x, u)-f(t, x, 0)\| \leq \alpha_5(\|u\|) \\ \|h(t, x, u)\| \leq \alpha_6(\|x\|) + \alpha_7(\|u\|) + \eta \label{hbound2} \end{align} for all $$(t,x,u) \in [0, \infty) \times D \times D_u$$ for some $$\mathcal{K}$$ functions $$\alpha_{5,6,7}$$, and a nonnegative const. $$\eta$$

Then, for each $$x_0$$ with $$\|x_0\| \le \alpha_2^{-1}(\alpha_1(r))$$, the system $$\eqref{statemodel}$$ is small-signal $$\mathcal{L}_\infty$$ stable.

Corollary 3: Suppose that in some neighborhood of $$(x=0, u=0)$$, the function $$f(t,x,u)$$ is cont. diff., the Jacobian matrices $$\partial f / \partial x$$ and $$\partial f / \partial u$$ are bounded, uniformly in $$t$$, and $$h(t,x,u)$$ satisfies $$\eqref{hbound2}$$. If the origin is an UAS equi. point of $$\eqref{unforced}$$, then the system $$\eqref{statemodel}$$ is small-signal $$\mathcal{L}_\infty$$ stable.

Theorem 3: Consider the system $$\eqref{statemodel}$$ with $$D = \mathbb{R}^n$$ and $$D_u = \mathbb{R}^m$$. Suppose that

• The system $$\dot{x} =f(t, x, u), \quad x(0)=x_{0}$$ is ISS
• $$h$$ satisfies $$\eqref{hbound2}$$

Then, for each $$x_0 \in \mathbb{R}^n$$, the system $$\eqref{statemodel}$$ is $$\mathcal{L}_\infty$$ stable.

## $$\mathcal{L}_2$$ Gain

Theorem 4: Consider the system $$\eqref{ltiinput}$$ where $$A$$ is Hurwitz. Let $$G(s) = C (sI - A)^{-1} B + D$$. Then, the $$\mathcal{L}_2$$ gain of the system is $$\sup_{\omega \in \mathbb{R}} \| G(j \omega) \|_2$$

Theorem 5: Consider the time-invariant nonlinear system \displaylines{ \begin{aligned} \dot{x} &=f(x)+G(x) u, \quad x(0)=x_{0} \\ y &=h(x) \end{aligned} } \label{tinonlinear}

where $$f(x)$$ is LL, and $$G(x), h(x)$$ are cont. over $$\mathbb{R}^n$$. The matrix $$G \in \mathbb{R}^{n \times m}$$ and $$h : \mathbb{R}^n \rightarrow \mathbb{R}^q$$. $$f(0)=0, h(0)=0$$. Let $$\gamma$$ be a positive number and suppose there is a cont. diff. PSD function $$V(x)$$ that satisfies the Hamilton-Jacobi inequality $\displaylines{\mathcal{H}(V, f, G, h, \gamma) \stackrel{\text { def }}{=} \frac{\partial V}{\partial x} f(x)+\frac{1}{2 \gamma^{2}} \frac{\partial V}{\partial x} G(x) G^{T}(x)\left(\frac{\partial V}{\partial x}\right)^{T}+\frac{1}{2} h^{T}(x) h(x) \leq 0 }$ for all $$x \in \mathbb{R}^n$$. Then, for each $$x_0 \in \mathbb{R}^n$$, the system $$\eqref{tinonlinear}$$ is finite-gain $$\mathcal{L}_2$$ stable and its $$\mathcal{L}_2$$ gain is less than or equal to $$\gamma$$.

Corollary 4: Suppose the assumption of Theorem 5 are satisfied on a domain $$D \subset \mathbb{R}^n$$ that contains the origin. Then, for any $$x_0 \in D$$ and any $$u \in \mathcal{L}_{2e}$$ for which the solution $$x$$ of $$\eqref{tinonlinear}$$ satisfies $$x(t) \in D$$ for all $$t \in [0, \tau]$$, we have $\displaylines{\left\|y_{\tau}\right\|_{\mathcal{L}_{2}} \leq \gamma\left\|u_{\tau}\right\|_{\mathcal{L}_{2}}+\sqrt{2 V\left(x_{0}\right)} }$

Lemma 1: Suppose the assumption of Theorem 5 are satisfied on a domain $$D \subset \mathbb{R}^n$$ that contains the origin, $$f(x)$$ is a cont. diff function, and $$x=0$$ is an AS equi. point of $$\dot{x} = f(x)$$. Then, there is $$k_1 > 0$$ s.t. for each $$x_0$$ with $$\| x_0 \| \le k_1$$, the system $$\eqref{tinonlinear}$$ is small-signal finite-gain $$\mathcal{L}_2$$ stable with $$\mathcal{L}_2$$ gain less than or equal to $$\gamma$$

Lemma 2: Suppose the assumption of Theorem 5 are satisfied on a domain $$D \subset \mathbb{R}^n$$ that contains the origin, $$f(x)$$ is a cont. diff function, and no solution of $$\dot{x} = f(x)$$ can stay identically in $$S = \{ x\in D | h(x) =0 \}$$ other than $$x(t) \equiv 0$$. Then, the origin of $$\dot{x} = f(x)$$ is AS and there is $$k_1 > 0$$ s.t. for each $$x_0$$ with $$\| x_0 \| \le k_1$$, the system $$\eqref{tinonlinear}$$ is small-signal finite-gain $$\mathcal{L}_2$$ stable with $$\mathcal{L}_2$$ gain less than or equal to $$\gamma$$

## Feedback Systems

Consider two systems $$H_1 : \mathcal{L}_e^m \rightarrow \mathcal{L}_e^q$$ and $$H_2 : \mathcal{L}_e^q \rightarrow \mathcal{L}_e^m$$. Suppose both systems are finite-gain $$\mathcal{L}$$ stable, that is $\displaylines{{\left\|y_{1 \tau}\right\|_{\mathcal{L}} \leq} {\gamma_{1}\left\|e_{1 \tau}\right\|_{\mathcal{L}}+\beta_{1}, \quad \forall e_{1} \in \mathcal{L}_{e}^{m}, \forall \tau \in[0, \infty)} \\ {\left\|y_{2 \tau}\right\|_{\mathcal{L}}} {\leq \gamma_{2}\left\|e_{2 \tau}\right\|_{\mathcal{L}}+\beta_{2}, \quad \forall e_{2} \in \mathcal{L}_{e}^{q}, \forall \tau \in[0, \infty)} }$ Suppose further that the feedback system is well defined: for every pair of inputs $$u_1 \in \mathcal{L}_e^m$$ and $$u_2 \in \mathcal{L}_e^q$$, there exist unique outputs $$e_1, y_2 \in \mathcal{L}_e^m$$ and $$e_2, y_1 \in \mathcal{L}_e^q$$. Define $\displaylines{u=\begin{bmatrix}{u_{1}} \\ {u_{2}}\end{bmatrix}, \quad y=\begin{bmatrix}{y_{1}} \\ {y_{2}}\end{bmatrix}, \quad e=\begin{bmatrix} {e_{1}} \\ {e_{2}}\end{bmatrix} }$ The question is whether the feedback connection, when viewed as a mapping from $$u$$ to $$e$$ or a mapping from $$u$$ to $$y$$, is finite-gain $$\mathcal{L}$$ stable. The two statements are equivalent.

Theorem 6: The feedback connection is finite-gain $$\mathcal{L}$$ stable if $$\gamma_1 \gamma_2 < 1$$.