On systematic criteria for the global stability of nonlinear systems via the Koopman operator framework

For multi-index notations $\displaystyle\alpha=(\alpha_{1},...,\alpha_{n})\in\mathbb{N}^{n}$, we define $\displaystyle|\alpha|=\alpha_{1}+\cdots+\alpha_{n}$ and $\displaystyle z^{\alpha}=z_{1}^{\alpha_{1}}\cdots z_{n}^{\alpha_{n}}$. The complex conjugate and real part of a complex number $\displaystyle a$ is denoted by $\displaystyle\bar{a}$ and $\displaystyle\Re(a)$, respectively. The Jacobian matrix of the vector field $\displaystyle F$ at $\displaystyle z$ is given by $\displaystyle JF(z)$. The complex polydisc centered at $\displaystyle 0$ and of radius $\displaystyle\rho>0$ is defined by

$$\displaystyle\mathbb{D}^{n}(\rho)=\left\{z\in\mathbb{C}^{n}:|z_{1}|<\rho,\cdots,|z_{n}|<\rho\right\}$$

and $\displaystyle\partial\mathbb{D}^{n}(\rho)$ and $\displaystyle\left(\partial\mathbb{D}(\rho)\right)^{n}$ is its boundary and distinguished boundary respectively. In particular, $\displaystyle\mathbb{D}^{n}$ denotes the unit polydisc (i.e. with $\displaystyle\rho=1$).

The Hardy space of holomorphic functions on the polydisc $\displaystyle\mathbb{D}^{n}(\rho)$ is the space

$$\displaystyle\mathbb{H}^{2}(\mathbb{D}^{n}(\rho))=\left\{f:\mathbb{D}^{n}(\rho)\rightarrow\mathbb{C},\mbox{holomorphic}:\|f\|^{2}<\infty\right\},$$

where

$$\displaystyle\|f\|^{2}=\lim_{r\rightarrow 1^{-}}\int_{(\partial\mathbb{D}(\rho))^{n}}|f\left(r\omega\right)|^{2}dm_{n}(\omega)$$

and $\displaystyle m_{n}$ is the normalized Lebesgue measure on $\displaystyle(\partial\mathbb{D}(\rho))^{n}$. The space is equipped with an inner product defined by

$$\displaystyle\left\langle f,g\right\rangle=\int_{(\partial\mathbb{D}(\rho))^{n}}f\left(\omega\right)\bar{g}\left(\omega\right)dm_{n}(\omega),$$

so that the set of monomials $\displaystyle\left\{z^{\alpha}:\alpha\in\mathbb{N}^{n}\right\}$ is a standard orthonormal basis on $\displaystyle\mathbb{H}^{2}(\mathbb{D}^{n}(\rho))$. In the sequel, the monomials will be denoted by $\displaystyle e_{k}(z)=z^{\alpha(k)}$, where the map $\displaystyle\alpha:\mathbb{N}\to\mathbb{N}^{n}$, $\displaystyle k\mapsto\alpha(k)$ refers to the lexicographic order¹¹1that is, $\displaystyle e_{k_{1}}<e_{k_{2}}$ if $\displaystyle|\alpha(k_{1})|<|\alpha(k_{2})|$, or if $\displaystyle|\alpha(k_{1})|=|\alpha(k_{2})|$ and $\displaystyle\alpha_{j}(k_{1})>\alpha_{j}(k_{2})$ for the smallest $\displaystyle j$ such that $\displaystyle\alpha_{j}(k_{1})\neq\alpha_{j}(k_{2})$. For $\displaystyle f$ and $\displaystyle g$ in $\displaystyle\mathbb{H}^{2}(\mathbb{D}^{n}(\rho))$, with $\displaystyle f=\sum_{k\in\mathbb{N}}f_{k}e_{k}$ and $\displaystyle g=\sum_{k\in\mathbb{N}}g_{k}e_{k}$, the isomorphism

$$\displaystyle\sum_{k\in\mathbb{N}}f_{k}e_{k}\mapsto(f_{k})_{k\geq 0}$$

between $\displaystyle\mathbb{H}^{2}(\mathbb{D}^{n}(\rho))$ and the $\displaystyle l^{2}$-space allows to rewrite the norm and the inner product as

$$\displaystyle\|f\|^{2}=\sum_{k\in\mathbb{N}}\rho^{2|\alpha(k)|}|f_{k}|^{2}\quad\mbox{and}\quad\left\langle f,g\right\rangle=\sum_{k\in\mathbb{N}}\rho^{2|\alpha(k)|}f_{k}\,\bar{g}_{k}.$$

By using the change of variables $\displaystyle\phi(z)=z^{\prime}=z/\rho$ on $\displaystyle\mathbb{D}^{n}(\rho)$ , the map $\displaystyle f\mapsto f^{\prime}=f\circ\phi$ defines an isometry between the two Hardy spaces $\displaystyle\mathbb{H}^{2}(\mathbb{D}^{n}(\rho))$ and $\displaystyle\mathbb{H}^{2}(\mathbb{D}^{n})$ where $\displaystyle\|f^{\prime}\|^{2}_{\mathbb{H}^{2}(\mathbb{D}^{n})}=\sum_{k\in\mathbb{N}}|f_{k}|^{2}$. For more details on the Hardy space, we refer the reader to [9, 10, 11].

The Koopman operator is defined here as the composition operator on $\displaystyle\mathbb{H}^{2}(\mathbb{D}^{n}(\rho))$ with symbol $\displaystyle\varphi^{t}$ (see e.g. [4, 5, 12]).

Under a contraction assumption on the flow $\displaystyle\varphi^{t}$, one can prove the boundedness and the strong continuity of the Koopman semigroup. In this work, we focus on the evolution of the evaluation functionals $\displaystyle k_{z}$ of the Hardy space (see [14] for the technical details), so that the above properties are not required.

Moreover, the expression of the Koopman generator in the basis of monomials can be obtained from the Taylor expansion

$$F_{l}(z)=\sum_{|\alpha|\geq 1}a_{l,\alpha}\,z^{\alpha}=\sum_{k=1}^{\infty}a_{l,k}\,z^{\alpha(k)}$$

(2)

of the vector field (with a slight abuse of notation, we will use two different conventions for the subscripts of the Taylor coefficients, i.e. $\displaystyle a_{l,k}=a_{l,\alpha(k)}$). It is shown in [14] that

$$\left\langle L_{F}e_{k},e_{j}\right\rangle=\begin{cases}\sum_{l=1}^{n}\alpha_{l}(k)\,a_{l,(\alpha(j)-\alpha(k))_{l}}&\textrm{if }|\alpha(j)|\geq|\alpha(k)|\\ 0&\textrm{if }|\alpha(j)|<|\alpha(k)|.\end{cases}$$

(3)

with

$$(\alpha(j)-\alpha(k))_{l}=(\alpha_{1}(j)-\alpha_{1}(k),\cdots,\alpha_{l}(j)-\alpha_{l}(k)+1,\cdots,\alpha_{n}(j)-\alpha_{n}(k)).$$

In particular, for monomials $\displaystyle e_{k}$ and $\displaystyle e_{j}$ of same total degree $\displaystyle|\alpha(j)|=|\alpha(k)|$, we have

$$\left\langle L_{F}e_{k},e_{j}\right\rangle=\begin{cases}\sum_{l=1}^{n}\,\alpha_{l}(j)\,a_{l,\alpha(l)}&\textrm{if }j=k\\ \alpha_{l}(k)\,a_{l,\alpha(r)}&\textrm{if }\alpha(j)=(\alpha_{1}(k),\cdots,\alpha_{l}(k)-1,\cdots,\\ &\qquad\qquad\qquad\qquad\alpha_{r}(k)+1,\cdots,\alpha_{n}(k)),\\ 0&\textrm{otherwise}.\end{cases}$$

(4)

We now present an intermediate result that we will use to prove our main stability results. It is adapted from [14], where a switched system was considered instead of (1).

The proof follows on similar lines as in [14].

Let us consider a dynamical system with a polynomial vector field

$$\dot{z}_{l}=F_{l}(z)=\sum_{k=1}^{r}a_{l,k}\,z^{\alpha(k)},\quad l=1,\dots,n.$$

(8)

We first define the following quantities associated with the polynomial vector field.

• •

  Let $\displaystyle d$ be the maximal degree of the polynomials $\displaystyle F_{l}$, i.e.

  $$d=\max_{k\in\mathbb{N}}\left\{|\alpha(k)|:a_{l,k}\neq 0\textrm{ for some }l\right\}=|\alpha(r)|$$

• •

  Let $\displaystyle K$ be the number of nonzero terms (without counting the term containing the monomial $\displaystyle z_{l}$ in $\displaystyle F_{l}$), i.e.

  $$K=\sum_{l=1}^{n}\#\left\{k\neq l:a_{l,k}\neq 0\right\}$$

  (9)

  where $\displaystyle\#$ is the cardinal of a set.

• •

  Let $\displaystyle S$ be the maximal polynomial coefficient over all components of the vector field (again discarding the terms containing the monomial $\displaystyle z_{l}$ in $\displaystyle F_{l}$), i.e.

  $$S=\max_{l=1,\cdots,n}\max_{\begin{subarray}{c}k=1,\cdots,r\\ k\neq l\end{subarray}}\left|a_{l,k}\right|.$$

• •

  Let $\displaystyle R$ be the minimal real part of the diagonal entries of $\displaystyle JF(0)$, i.e.

  $$R=\min_{l=1,\cdots,n}\left|\Re\left(a_{l,l}\right)\right|.$$

Then we have the following result.

See Appendix A for the proof.

In this section, we provide a result for dynamics with analytic vector fields, which we rewrite as

$$\dot{z}_{l}=F_{l}(z)=\sum_{k=1}^{\infty}a_{l,k}\,z^{\alpha(k)},\quad l=1,\dots,n,$$

(10)

under the assumption that the Jacobian matrix $\displaystyle JF(0)$ is diagonal.

We first define the following quantities associated with the Taylor expansion (2) of the vector field.

• •

  Let $\displaystyle L_{\mu}$ be the discounted (infinite) sum of Taylor coefficients of the vector field, i.e.

  $$L_{\mu}=\sum_{l=1}^{n}\sum_{k=1}^{\infty}\mu^{|\alpha(k)|}\left|a_{l,k}\right|.$$

  (11)

  Note that $\displaystyle L_{\mu}$ might not be a convergent series for all $\displaystyle\mu$, but is always convergent for $\displaystyle\mu\leq 1$ under Assumption 1.

• •

  Let $\displaystyle R$ be the minimal real part of the diagonal entries of $\displaystyle JF(0)$, i.e.

  $$R=\min_{l=1,\cdots,n}\left|\Re\left(a_{l,l}\right)\right|.$$

We have the following result.

See Appendix B for the proof.

Consider the vector field

$$F(z_{1},z_{2})=\begin{cases}a\left(z_{1}-\frac{1}{ac}z_{2}\right)\\ a\left(z_{2}-\frac{1}{ac}z_{1}+bz_{1}^{2}\right),\end{cases}$$

(13)

where $\displaystyle a=-1/4$, $\displaystyle c=8$ and $\displaystyle b=-1/50$. The dynamics admit the equilibria $\displaystyle(0,0)$ and $\displaystyle\left(-75,150\right)$ so that $\displaystyle(0,0)$ is the unique equilibrium point on the polydisc $\displaystyle\mathbb{D}^{n}\left(\mu\right)$ with $\displaystyle\mu<75$. The Jacobian matrix $\displaystyle JF(0)$ has negative eigenvalues $\displaystyle a-1/c=-3/8$ and $\displaystyle a+1/c=-1/8$, and is diagonalizable by the matrix $\displaystyle P=\begin{pmatrix}1&-1\\ 1&1\end{pmatrix}$. Using the change of coordinates $\displaystyle\widehat{z}=P^{-1}z$, we have

$$\widehat{F}(\widehat{z}_{1},\widehat{z}_{2})=\begin{cases}(a-\frac{1}{c})\widehat{z}_{1}+\frac{a^{2}b}{2}\left(\widehat{z}_{1}^{2}-2\widehat{z}_{1}\widehat{z}_{2}+\widehat{z}_{2}^{2}\right)\\ (a+\frac{1}{c})\widehat{z}_{2}+\frac{a^{2}b}{2}\left(\widehat{z}_{1}^{2}-2\widehat{z}_{1}\widehat{z}_{2}+\widehat{z}_{2}^{2}\right).\end{cases}$$

(14)

The dynamics in the new variables is forward invariant in the polydisc $\displaystyle\mathbb{D}^{n}\left(\widehat{\rho}\right)$, if we assume that $\displaystyle\widehat{\rho}=\rho\in]1,\mu[$ since

• •

  $\displaystyle|\widehat{z}_{1}|=\rho\Rightarrow\Re\left(\bar{\widehat{z}}_{1}\widehat{F}_{1}(\widehat{z})\right)=-3\rho^{2}/8-\rho^{2}\Re\left(\widehat{z}_{1}\right)/1600+\rho^{2}\Re\left(\widehat{z}_{2}\right)/800-\Re\left(\bar{\widehat{z}}_{1}\widehat{z}_{2}^{2}\right)/1600<0$ as $\displaystyle\rho<75$ and

• •

  $\displaystyle|\widehat{z}_{2}|=\rho\Rightarrow\Re\left(\bar{\widehat{z}}_{2}\widehat{F}_{2}(\widehat{z})\right)=-\rho^{2}/8-\rho^{2}\Re\left(\widehat{z}_{2}\right)/1600+\rho^{2}\Re\left(\widehat{z}_{1}\right)/800-\Re\left(\bar{\widehat{z}}_{2}\widehat{z}_{1}^{2}\right)/1600<0$ as $\displaystyle\rho<75$.

For the vector field $\displaystyle\widehat{F}$, we compute $\displaystyle\widehat{d}=2$, $\displaystyle\widehat{K}=6$, $\displaystyle\widehat{S}=a^{2}|b|=1/800$ and $\displaystyle\widehat{R}=|a+1/c|=1/8$, so that $\displaystyle\widehat{K}\widehat{S}/\widehat{R}=3/50<1.$ Hence, it follows from Theorem 1 that the nonlinear system (14) is GAS on $\displaystyle\mathbb{D}^{2}(\widehat{\rho})$ with $\displaystyle\widehat{\rho}<50/3$. Finally, this implies that (13) is GAS on $\displaystyle P(\mathbb{D}^{2}(\widehat{\rho}))\supset\mathbb{D}^{2}(\widehat{\rho})=\mathbb{D}^{2}(\rho)$ since $\displaystyle\|P\|_{\infty}=2>1$ and with $\displaystyle\rho=\widehat{\rho}<50/3$.

Consider the vector field

$$F(z_{1},z_{2})=\begin{cases}a\left(z_{1}-\dfrac{2z_{2}^{2}}{c-z_{2}}\right)\\ a\left(z_{2}-\dfrac{bz_{1}^{2}}{(d-z_{1})^{2}}\right),\end{cases}$$

(15)

where $\displaystyle a=-1$, $\displaystyle b=4$, $\displaystyle c=30$ and $\displaystyle d=20$. The origin $\displaystyle(0,0)$ is the unique equilibrium point on the polydisc $\displaystyle\mathbb{D}^{n}\left(\mu\right)$ with $\displaystyle\mu=10$. If we assume that $\displaystyle\rho\in]1,\mu[$, $\displaystyle\mathbb{D}^{n}\left(\rho\right)$ is invariant with respect to the flow since

• •

  $\displaystyle|z_{1}|=\rho\Rightarrow\Re\left(\bar{z}_{1}F_{1}(z)\right)=-\rho^{2}+2\Re\left(\dfrac{\bar{z}_{1}z_{2}^{2}}{30-z_{2}}\right)<0$ since $\displaystyle 1>\dfrac{2\rho}{30-\rho}$ and it follows that

  $$\displaystyle\rho^{2}>\dfrac{2\rho^{3}}{30-\rho}>\dfrac{2\rho^{3}}{\left|30-|z_{2}|\right|}>2\left|\dfrac{\bar{z}_{1}z_{2}^{2}}{30-z_{2}}\right|\geq 2\left|\Re\left(\dfrac{\bar{z}_{1}z_{2}^{2}}{30-z_{2}}\right)\right|$$

• •

  $\displaystyle|z_{2}|=\rho\Rightarrow\Re\left(\bar{z}_{2}F_{2}(z)\right)=-\rho^{2}+4\Re\left(\dfrac{\bar{z}_{2}z_{1}^{2}}{(20-z_{1})^{2}}\right)<0$ since $\displaystyle 1>\dfrac{4}{(20-\rho)^{2}}$ and it follows that

  $$\rho^{2}>\dfrac{4\rho^{2}}{|20-\rho|^{2}}>\dfrac{4\rho^{2}}{\left|20-|z_{1}|\right|^{2}}>\left|\dfrac{4\bar{z}_{2}z_{1}}{(20-z_{1})^{2}}\right|\geq\left|\Re\left(\dfrac{4\bar{z}_{2}z_{1}}{(20-z_{1})^{2}}\right)\right|.$$

As

$$\displaystyle F_{1}(z)=-z_{1}+2\sum_{k=0}^{\infty}\dfrac{z_{2}^{k+2}}{30^{k+1}}\text{ and }F_{2}(z)=-z_{2}+4\sum_{k=0}^{\infty}\dfrac{(k+1)z_{1}^{k+2}}{20^{k+2}},$$

we obtain

$$\displaystyle L_{\mu}=\left(10+2\sum_{k=0}^{\infty}\dfrac{10^{k+2}}{30^{k+1}}\right)+\left(10+4\sum_{k=0}^{\infty}\dfrac{(k+1)10^{k+2}}{20^{k+2}}\right)=73/3,$$

and $\displaystyle R=1$. Hence, it follows from Theorem 2 that the nonlinear system (15) is GAS on $\displaystyle\mathbb{D}^{2}(\rho)$ with $\displaystyle\rho<30/73$.