Bounds for Invariance Pressure

TL;DR

This paper establishes bounds for invariance pressure in control systems, providing explicit formulas for hyperbolic linear systems and exploring applications to Lie groups and inner control sets.

Contribution

It introduces bounds for invariance pressure in control sets and derives an explicit formula for hyperbolic linear control systems, extending to Lie groups and inner control sets.

Findings

Upper bounds for invariance pressure in control sets with nonempty interior.

Lower bounds for invariance pressure in sets with finite volume.

Explicit formula for hyperbolic linear control systems.

Abstract

This paper provides an upper for the invariance pressure of control sets with nonempty interior and a lower bound for sets with finite volume. In the special case of the control set of a hyperbolic linear control system in R^{d} this yields an explicit formula. Further applications to linear control systems on Lie groups and to inner control sets are discussed.

Equations

$$\dot{x}(t)=F(x(t),\omega (t)),\omega \in \mathcal{U},$$

$${\mathcal{O}}_{\le t}^{+}(x):=\{y\in M;\text{mbox}thereares\in [0,t]\text{mbox}and\omega \in \mathcal{U}\text{mbox}with\phi (s,x,\omega )=y\},$$

$${\mathcal{O}}_{\le t}^{-}(x):=\{y\in M;\text{mbox}thereares\in [0,t]\text{mbox}and\omega \in \mathcal{U}\text{mbox}with\phi (s,y,\omega )=x\},$$

$${\mathcal{O}}^{+}(x):=\bigcup _{t>0}{\mathcal{O}}_{\le t}^{+}(x)\text{ and }{\mathcal{O}}^{-}(x):=\bigcup _{t>0}{\mathcal{O}}_t^{-}(x).$$

$$\frac{Dz}{dt}(t)=A(t)z(t)+B(t)\mu (t),\mu \in L^{\infty }(\mathbb{R},{\mathbb{R}}^m),$$

$${\varphi }^{x,\omega }(0,\lambda ,\mu )=\lambda ,{\varphi }^{x,\omega }(t,\lambda ,\mu )=D{\phi }_t(x,\omega )(\lambda ,\mu ),t\in \mathbb{R},$$

$${\varphi }_s^{x,\omega }(t,{\varphi }^{x,\omega }(s,\lambda ,\mu ),{\Theta }_s\mu )={\varphi }^{x,\omega }(t+s,\lambda ,\mu ),$$

$${\varphi }_s^{x,\omega }(t,{\varphi }^{x,\omega }(s,\lambda ,0),0)={\varphi }^{x,\omega }(t+s,\lambda ,0).$$

$${\varphi }^{x,\omega }(\tau ,{\lambda }_1,\mu )={\lambda }_2.$$

$$\underset{t\to \infty }{\lim }\frac{1}{t}\log \Vert {\varphi }^{x,\omega }(t,\lambda ,0)\Vert =\underset{n\to \infty }{\lim }\frac{1}{n\tau }\log \Vert {\varphi }^{x,\omega }(n\tau ,\lambda ,0)\Vert ,\lambda \in T_{x}M.$$

$$L_j(\omega ,x)=\left\{\lambda \in T_{x}M;\underset{t\to \pm \infty }{\lim }\frac{1}{t}\log \Vert {\varphi }^{x,\omega }(t,\lambda ,0)\Vert ={\rho }_j(\omega ,x)\right\},j=1,\dots ,r,$$

$$T_{x_0}M=L_1(\omega ,x)\oplus \cdots \oplus L_r(\omega ,x).$$

$$a_{\tau }(f,K,Q):=\inf \left\{\sum _{\omega \in \mathcal{S}}{e}^{(S_{\tau }f)(\omega )};\mathcal{S}(\tau ,K,Q)\text{-spanning}\right\}.$$

$$P_{inv}(f,K,Q):=\underset{\tau \to \infty }{\mathrm{limsup}}\frac{1}{\tau }\log a_{\tau }(f,K,Q).$$

$$P_{out}(f,K,Q):=\underset{\epsilon \to 0}{\lim }{P}_{inv}(f,K,N_{\epsilon }(Q)),$$

$$a_{\tau }(f,K,Q)=\inf \left\{\sum _{\omega \in \mathcal{S}}{e}^{(S_{\tau }f)(\omega )};\mathcal{S}\text{ finite and }(\tau ,K,Q)\text{-spanning}\right\}.$$

$$\left|P_{inv}(f,K,Q)-P_{inv}(g,K,Q)\right|\le {\Vert f-g\Vert }_{\infty }\text{ for }f,g\in C(U,\mathbb{R}),$$

$$\frac{{\sum }_{i=1}^n{a}_i}{{\sum }_{i=1}^n{b}_i}\ge \underset{i=1,...n}{\min }\frac{a_i}{b_i}\ge \underset{i\in \mathbb{N}}{\inf }\frac{a_i}{b_i},$$

$$P_{inv}(f,K,Q)=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n\tau }\log a_{n\tau }(f,K,Q)\text{ for all }f\in C(U,\mathbb{R}).$$

$$\underset{k\to \infty }{\mathrm{limsup}}\frac{1}{{\tau }_k}\log a_{{\tau }_k}(\tilde{f},K,Q)$$

$$=\underset{k\to \infty }{\mathrm{limsup}}\frac{n_k+1}{n_k}\frac{1}{(n_k+1)\tau }\log a_{(n_k+1)\tau }(\tilde{f},K,Q)$$

$$\le \underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n\tau }\log a_{n\tau }(\tilde{f},K,Q).$$

$$P_{inv}(f,K,Q)\le \underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n\tau }\log a_{n\tau }(f,K,Q).$$

$$P_{inv}(f,K_1,Q)=P_{inv}(f,K_2,Q).$$

$$P_{inv}(f,K,Q)\le \underset{(T,x,\omega )}{\inf }\left\{\sum _{j=1}^{r(x,\omega )}\max \{0,d_j(x,\omega ){\rho }_j(x,\omega )\}+\frac{1}{T}{\int }_0^{T}f(\omega (s))ds\right\},$$

$$h_{inv}(K,Q)=P_{inv}(0,K,Q)\le \underset{(T,x,\omega )}{\inf }\left\{\sum _{j=1}^{r(x,\omega )}\max \{0,d_j(x,\omega ){\rho }_j(x,\omega )\}\right\}.$$

$$S_0>\sum _{j=1}^r\max (0,d_j{\rho }_j)),$$

$$\Vert \omega -{\omega }_0{\Vert }_{[0,n\tau ]}\le C{b}_0\sqrt{d},$$

$$\frac{1}{n\tau }\log \#{\mathcal{S}}_n\le S_0+\epsilon ,$$

$$C{b}_0\sqrt{d}<\delta .$$

$$\text{allowdisplaybreaks}\frac{1}{n\tau }\log a_{n\tau }(f,K,Q)$$

Full text

Bounds for Invariance Pressure

Fritz Colonius

Institut für Mathematik

  

Universität Augsburg

  

Augsburg

  

Germany

  

Alexandre J. Santana and João A. N. Cossich

Departamento de Matemática

  

Universidade Estadual de Maringá

Maringá

  

Brazil

**Abstract: **This paper provides an upper for the invariance pressure of control sets with nonempty interior and a lower bound for sets with finite volume. In the special case of the control set of a hyperbolic linear control system in $\mathbb{R}^{d}$ this yields an explicit formula. Further applications to linear control systems on Lie groups and to inner control sets are discussed.

Keywords: Invariance pressure, invariance entropy, control sets

AMS subject classification. 93C15, 37B40, 94A17

1 Introduction

The notion of invariance pressure generalizes invariance entropy by adding potentials $f$ on the control range. It has been introduced and analyzed in Colonius, Cossich, Santana [6, 7]. Zhong and Huang [20] show that invariance pressure can be characterized as a dimension-like notion within the framework due to Pesin. A basic reference for invariance entropy is Kawan’s monograph [18]; here also the relation to minimal data rates is explained which gives the main motivation from applications. Further references include the seminal paper Nair, Evans, Mareels and Moran [19] as well as Colonius and Kawan [9] and da Silva and Kawan [13], [14]. In the latter paper, robustness properties in the hyperbolic case are proved. Huang and Zhong [16] show that several generalized notions of invariance entropy fit into the dimension-theoretic framework due to Pesin.

The main results of the present paper are upper and lower bounds for the invariance pressure of compact subsets $K$ in a control set $D$ with nonvoid interior and compact closure. For hyperbolic linear control systems in $\mathbb{R}^{d}$ this yield a formula for the invariance pressure. We also give applications for inner control sets and for certain linear systems on Lie groups. Invariance entropy of these systems has been analyzed by da Silva [11].

Section 2 collects results on linearization of control systems and on the notion of invariance pressure. Upper and lower bounds for invariance pressure are given in Sections 3 and 4, respectively. Section 5 presents a formula for the invariance pressure of linear control systems in $\mathbb{R}^{d}$ and Section 6 discusses applications linear systems on Lie groups and for inner control sets.

2 Preliminaries

In this section we first recall basic notions for control systems on manifolds and their linearization  Then the concepts of invariance pressure and outer invariance pressure are presented as well as some of their properties.

2.1 Control systems and linearization

Throughout the paper, $M$ will denote a smooth manifold, that is, a connected, second-countable, topological Hausdorff manifold endowed with a $C^{\infty}$ differentiable structure. A continuous-time control system on a smooth manifold $M$ is a family of ordinary differential equations

[TABLE]

on $M$ which is parametrized by measurable functions $\omega:\mathbb{R}\rightarrow\mathbb{R}^{m}$, $\omega(t)\in U\subset\mathbb{R}^{m}$ almost everywhere, called controls forming the set $\mathcal{U}$ of admissible control functions, where $U\subset\mathbb{R}^{m}$ is a compact set, the control range. The function $F:M\times\mathbb{R}^{m}\rightarrow TM$ is a $C^{1}$-map such that for each $u\in U$, $F_{u}(\cdot):=F(\cdot,u)$ is a smooth vector field on $M$. For each $x\in M$ and $\omega\in\mathcal{U}$, we suppose that there exists an unique solution $\varphi(t,x,\omega)$ which is defined for all $t\in\mathbb{R}$. We usually refer to the solution $\varphi(\cdot,x,\omega)$ as a trajectory of $x$ with control function $\omega$ and write $\varphi_{t}(x,\omega)=\varphi(t,x,\omega)$ where convenient.

We need several notions characterizing controllability properties of subsets of the state space $M$ of system (1).

For $x\in M$ and $t>0$, the set of points reachable from $x$ up to time $t$ and the set of points controllable to $x$ within time $t$ are given by

[TABLE]

and

[TABLE]

respectively. The positive and negative orbit from $x\in M$ are

[TABLE]

A key concept of this paper is presented in the following definition.

Definition 1**.**

A subset $D$ of $M$ is a control set if

(i) for each $x\in D$, there exists $\omega\in\mathcal{U}$ with $\varphi(\mathbb{R}_{+},x,\omega)\subset D$ (controlled invariance);

(ii) for each $x\in D$ one has $D\subset\overline{\mathcal{O}^{+}(x)}$ (approximate controllability);

(iii) $D$ is maximal with these properties.

If for all $t>0$ the sets $\mathcal{O}_{\leq t}^{-}(x)$ and $\mathcal{O}_{\leq t}^{+}(x)$ have nonempty interior, we say that system (1) is locally accessible from $x\in M$. Of main interest are control sets with nonvoid interior which are locally accessible from all $x\in\mbox{int}D$. Then $\mathrm{int}D\subset\mathcal{O}^{+}(x)$ for all $x\in D$, cf. Colonius and Kliemann [10, Lemma 3.2.13].

Next we recall some basic concepts and results on linearization of a control system.

Definition 2**.**

For a control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ the linearized system is given by

[TABLE]

where $A(t):=\nabla F_{\omega(t)}(\varphi(t,x,\omega))\ \mbox{ and }\ B(t):=D_{2}F(\varphi(t,x,\omega),\omega(t))$.

The derivative on the left-hand side of (2) is the covariant derivative of $z(\cdot)$ along $\varphi(\cdot,x,\omega)$ and $D_{2}$ is the derivative with respect to second component. A solution of (2) corresponding to $\mu\in L^{\infty}(\mathbb{R},\mathbb{R}^{m})$ with initial value $\lambda\in T_{x}M$ is a locally absolutely continuous vector field $z=\phi^{x,\omega}(\cdot,\lambda,\mu):\mathbb{R}\rightarrow TM$ along $\varphi(\cdot,x,\omega)$ with $z(0)=\lambda$, satisfying the differential equation (2) for almost all $t\in\mathbb{R}$.

The next proposition presents some properties of linearized systems.

Proposition 3**.**

Let $(\omega(\cdot),\varphi(\cdot,x,\omega))$ be a control-trajectory pair with corresponding linearization (2). Then the following statements hold:

(i) For all $\tau>0$ the mapping $\varphi_{\tau}:M\times L^{\infty}([0,\tau],\mathbb{R}^{m})\rightarrow M,(x,\omega)\mapsto\varphi(\tau,x,\omega)$ is continuously (Fréchet) differentiable.

(ii) For every initial value $\lambda\in T_{x}M$ and every $\mu\in L^{\infty}(\mathbb{R},\mathbb{R}^{m})$ there exists a unique solution $\phi^{x,\omega}(\cdot,\lambda,\mu):\mathbb{R}\rightarrow TM$ of (2) satisfying

[TABLE]

for $(\lambda,\mu)\in T_{x}M\times L^{\infty}(\mathbb{R},\mathbb{R}^{m})$, where $D$ stands for the total derivative of $\varphi_{t}:M\times L^{\infty}(\mathbb{R},\mathbb{R}^{m})\rightarrow M$ which consists of the derivative $d_{x}\varphi_{t}(\cdot,\omega):T_{x}M\rightarrow T_{\varphi(t,x,\omega)}M$ in the first, and the Fréchet derivative of $\varphi_{t}(x,\cdot):L^{\infty}(\mathbb{R},\mathbb{R}^{m})\rightarrow T_{\varphi(t,x,\omega)}M$ in the second component.

(iii) For every $\tau>0$ the map $\phi^{x,\omega}(\tau,\cdot,\cdot):T_{x}M\times L^{\infty}([0,\tau],\mathbb{R}^{m})\rightarrow T_{\varphi(\tau,x,\omega)}M$ is linear and continuous.

(iv) For each $t\in\mathbb{R}$ abbreviate $\phi_{t}^{x,\omega}:=\phi^{\varphi(t,x,\omega),\omega(t+\cdot)}.$ Then for all $t,s\in\mathbb{R}$, $\lambda\in T_{x}M$ and $\mu\in L^{\infty}(\mathbb{R},\mathbb{R}^{m})$,

[TABLE]

and, in particular,

[TABLE]

The next definition introduces the notion of regularity of a control-trajectory pair.

Definition 4**.**

Consider some $(x,\omega,\tau)\in M\times\mathcal{U}\times(0,\infty)$ and let $y:=\varphi(\tau,x,\omega)$. Then we call the linearization along $(\omega(\cdot),\varphi(\cdot,x,\omega))$ controllable on $[0,\tau]$ if for each $\lambda_{1}\in T_{x}M$ and $\lambda_{2}\in T_{y}M$ there exists $\mu\in L^{\infty}([0,\tau],\mathbb{R}^{m})$ with

[TABLE]

In this case, we say that the control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ is regular on $[0,\tau]$.

A control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ is called $\tau$-periodic, $\tau\geq 0$, if $(\varphi(t+\tau,x,\omega),\omega(t+\tau))=(\varphi(t,x,\omega),\omega(t))$ for all $t\in\mathbb{R}$, or equivalently if $\varphi(\tau,x,\omega)=x$ and $\Theta_{\tau}\omega=\omega$, where $(\Theta_{\tau}\omega)(t)=\omega(t+\tau),t\in\mathbb{R}$, is the $\tau$-shift on $\mathcal{U}$. A periodic regular control-trajectory pair enjoys the property described in the following proposition (cf. [18, Proposition 1.30]).

Proposition 5**.**

Let $(\omega(\cdot),\varphi(\cdot,x,\omega))$ be a $\tau$-periodic control-trajectory pair which is regular on $[0,\tau]$. Then there exists $C>0$ such that for every $\lambda\in T_{x}M$ there is $\mu\in L^{\infty}([0,\tau],\mathbb{R}^{m})$ with $\phi^{x,\omega}(\tau,\lambda,\mu)=0_{x}$ and $\|\mu\|_{[0,\tau]}\leq C|\lambda|$, where $\|\cdot\|_{[0,\tau]}$ denotes the $L^{\infty}$-norm.

For a $\tau$-periodic control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ the Floquet or Lyapunov exponents are given by

[TABLE]

These limits exist and the Lyapunov exponents are denoted by $\rho_{1}(\omega,x),\dotsc,\rho_{r}(\omega,x)$ with $1\leq r:=r(\omega,x)\leq d=\dim M$. The Lyapunov spaces are given by

[TABLE]

with dimensions $d_{j}(\omega,x)$. They yield the decomposition

[TABLE]

2.2 Invariance pressure

In this subsection we recall the concepts of invariance and outer invariance pressure introduced in Colonius, Cossich and Santana [6, 7] and some of their properties.

A pair $(K,Q)$ of nonempty subsets of $M$ is called admissible if $K$ is compact and for each $x\in K$ there exists $\omega\in\mathcal{U}$ such that $\varphi(\mathbb{R}_{+},x,\omega)\subset Q$. For an admissible pair $(K,Q)$ and $\tau>0$, a $(\tau,K,Q)$-spanning set $\mathcal{S}$ is a subset of $\mathcal{U}$ such that for all $x\in K$ there is $\omega\in\mathcal{S}$ with $\varphi(t,x,\omega)\in Q$ for all $t\in[0,\tau]$. Denote by $C(U,\mathbb{R})$ the set of continuous function $f:U\rightarrow\mathbb{R}$ which we call potentials.

For a potential $f\in C(U,\mathbb{R})$ denote $(S_{\tau}f)(\omega):=\int_{0}^{\tau}f(\omega(t))dt$ and

[TABLE]

The invariance pressure $P_{inv}(f,K,Q)$ of control system (1) is defined by

[TABLE]

Given an admissible pair $(K,Q)$ such that $Q$ is closed in $M$, and a metric $\varrho$ on $M$ which is compatible with the Riemannian structure, we define the outer invariance pressure of $(K,Q)$ by

[TABLE]

where $N_{\varepsilon}(Q)=\{y\in M;\ \exists\ x\in Q\mbox{ with }\varrho(x,y)<\varepsilon\}$ denotes the $\varepsilon$-neighborhood of $Q$.

Note that $P_{out}(f,K,Q)\leq P_{inv}(f,K,Q)\leq\infty$ for every admissible pair $(K,Q)$ and all potentials $f$. For the potential $f=\mathbf{0}$, this reduces to the notion of invariance entropy, $P_{inv}(\mathbf{0},K,Q)=h_{inv}(K,Q)$ and $P_{out}(\mathbf{0},K,Q)=h_{out}(K,Q)$, cf. Kawan [18].

The next proposition presents some properties of the function $P_{inv}(\cdot,K,Q):C(U,\mathbb{R})\rightarrow\mathbb{R}$, cf. [7, Proposition 3.4].

Proposition 6**.**

The following assertions hold for an admissible pair $(K,Q)$, functions $f,g\in C(U,\mathbb{R})$ and $c\in\mathbb{R}$:

(i) $P_{inv}(f,K,Q)\leq P_{inv}(g,K,Q)$ and $P_{out}(f,K,Q)\leq P_{out}(g,K,Q)$ for $f\leq g$.

(ii) $P_{inv}(f+c,K,Q)=P_{inv}(f,K,Q)+c$.

(iii) $h_{inv}(K,Q)+\min_{u\in U}f(u)\leq P_{inv}(f,K,Q)\leq h_{inv}(K,Q)+\max_{u\in U}f(u).$

Remark 7**.**

If there is no countable $(\tau,K,Q)$-spanning set, then $a_{\tau}(f,K,Q)=\infty$ (see Kawan [18, Example 2.3] for an example). In particular, if $P_{inv}(f,K,Q)<\infty$, then for every $\tau>0$ there are countable $(\tau,K,Q)$-spanning sets. On the other hand if for all $\tau>0$ there is a countable $(\tau,K,Q)$-spanning set, $a_{\tau}(f,K,Q)=\infty$ is also possible. Proposition 6 (iii) shows that $P_{inv}(f,K,Q)<\infty$ if and only if $h_{inv}(f,K,Q)<\infty$. If every $(\tau,K,Q)$-spanning set $\mathcal{S}$ contains a finite $(\tau,K,Q)$-spanning subset $\mathcal{S}^{\prime}$, then

[TABLE]

This follows, since all summands satisfy $e^{(S_{\tau}f)(\omega)}>0$, and hence the summands in $\mathcal{S}\setminus\mathcal{S}^{\prime}$ can be omitted. This situation occurs e.g. if $Q$ is open, where compactness of $K$ may be used. For the outer invariance entropy one considers $(\tau,K,N_{\varepsilon}(Q))$-spanning sets, $\varepsilon>0$, and hence here it is also sufficient to consider finite $(\tau,K,N_{\varepsilon}(Q))$-spanning sets. For the inner invariance pressure of discrete time systems, one considers sets which are $(\tau,K,\mathrm{int}Q)$-spanning. Here again finite spanning sets are sufficient (the proof given in [6, Propositon 5] for the case $K=Q$ easily extends to admissible pairs $(K,Q)$).

Remark 8**.**

The Lipschitz continuity property

[TABLE]

holds if $P_{inv}(f,K,Q),P_{inv}(g,K,Q)<\infty$. In fact, in this case, there are for every $\tau>0$ countable $(\tau,K,Q)$-spanning sets $\mathcal{S}$ with $\sum_{\omega\in\mathcal{S}}e^{(S_{\tau}f)(\omega)}<\infty$. Then the arguments in [6, Proposition 13(iii)] can be applied in this situation observing that the elementary lemma [6, Lemma 12], on which the proof is based, is valid not only for finite but also for infinite sequences: Let $a_{i}\geq 0,b_{i}>0,i\in\mathbb{N}$. Then for all $n\in\mathbb{N}$

[TABLE]

and one may take the limit for $n\rightarrow\infty$.

The following proposition shows that in the definition of invariance pressure we can take the limit superior over times which are integer multiples of some fixed time step $\tau>0$.

Proposition 9**.**

The invariance pressure satisfies for every $\tau>0$

[TABLE]

Proof.

Let $(\tau_{k})_{k\geq 1}$, $\tau_{k}\in(0,\infty)$ and $\tau_{k}\rightarrow\infty$. Then for every $k\geq 1$ there exists $n_{k}\geq 1$ such that $n_{k}\tau\leq\tau_{k}\leq(n_{k}+1)\tau$ and $n_{k}\rightarrow\infty$ for $k\rightarrow\infty$. Since $\tilde{f}(u):=f(u)-\inf f\geq 0,u\in U$, it follows that $a_{\tau_{k}}(\tilde{f},K,Q)\leq a_{(n_{k}+1)\tau}(\tilde{f},K,Q)$ and consequently $\frac{1}{\tau_{k}}\log a_{\tau_{k}}(\tilde{f},K,Q)$ is less than or equal to $\frac{1}{n_{k}\tau}\log a_{(n_{k}+1)\tau}(\tilde{f},K,Q)$. Hence

[TABLE]

This shows that $P_{inv}(f-\inf f,K,Q)\leq\limsup_{n\rightarrow\infty}\frac{1}{n\tau}\log a_{n\tau}(f-\inf f,K,Q)$, and by Proposition 6 (iii) we obtain

[TABLE]

The converse inequality is obvious. ∎

For the proof of the following proposition see [7, Corollary 4.3].

Proposition 10**.**

Let $K_{1},K_{2}$ be two compact sets with nonempty interior contained in a control set $D\subset M$. Then $(K_{1},Q)$ and $(K_{2},Q)$ are admissible pairs and for all $f\in C(U,\mathbb{R})$ we have

[TABLE]

3 An upper bound on control sets

Our goal in this section is to obtain an upper bound for the invariance pressure of a control set. We consider a smooth control system (1) on a Riemannian manifold $(M,g)$ under our standard assumptions.

In the following theorem, given a periodic control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$, the different Lyapunov exponents at $(x,\omega)$ are denoted by $\rho_{1}(x,\omega),\dotsc,\rho_{r}(x,\omega),r=r(x,\omega)$, with Lyapunov spaces of dimensions $d_{1}(x,\omega),\dotsc,d_{r}(x,\omega)$, respectively.

Theorem 11**.**

Let $D\subset M$ be a control set with nonempty interior and compact closure for control system (1). Then for every compact set $K\subset D$ and every set $Q\supset D$, the pair $(K,Q)$ is admissible and for all potentials $f\in C(U,\mathbb{R})$ the invariance pressure satisfies

[TABLE]

where the infimum is taken over all $(T,x,\omega)\in(0,\infty)\times\mathrm{int}D\times\mathcal{U}$ such that the control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ is $T$-periodic and regular and the values $\omega(t),t\in[0,T]$, are in a compact subset of $\mathrm{int}U$.

Remark 12**.**

For $f\equiv 0$, the statement of the theorem reduces to Kawan [17, Theorem 4.4],

[TABLE]

Proof.

The theorem will follow by inspection of the proof given in [17, Theorem 4.4] for invariance entropy and by indicating the complementary arguments needed for invariance pressure. Note first that by Proposition 10 we can choose $K$ as an arbitrary compact subset of $D$ with nonvoid interior. Let $(\omega_{0}(\cdot),\varphi(\cdot,x_{0},\omega_{0}))$ be a $T$-periodic and regular control-trajectory pair as in the statement of the theorem. Then fix real numbers $\varepsilon>0$ and

[TABLE]

where $d_{j}=d_{j}(x_{0},\omega_{0})$ and $\rho_{j}(x_{0},\omega_{0}),j=1,\dotsc,r$. An ingenious and lengthy construction provides a compact set $K=\mathrm{cl}(B_{b_{0}}(x_{0}))\subset D$ containing $x_{0}$ in the interior with the following properties: For some $\tau=kT,k\in\mathbb{N}$, and arbitrary $n\in\mathbb{N}$ one finds a set $\mathcal{S}_{n}$ of $(n\tau,K,Q)$-spanning controls $\omega\in\mathcal{S}_{n}$ satisfying

[TABLE]

where $C>0$ is a constant and $b_{0}>0$ can be taken arbitrarily small (see [17, formula (4.17)]: the elements of $\mathcal{S}_{n}$ are $n$-fold concatenations of the controls denoted there by $u_{x}$). The cardinality $\#\mathcal{S}_{n}$ of $\mathcal{S}_{n}$ is bounded by

[TABLE]

cf. [17, estimate on middle of p. 745].

In order to get a bound for the invariance pressure we need the following additional arguments: Let $f\in C(U,\mathbb{R})$ be a potential. Since $f$ is defined on the compact set $U$, its uniform continuity implies that there exists $\delta>0$ such that $\left\|u-v\right\|<\delta$ implies $|f(u)-f(v)|<\varepsilon$. Take $b_{0}>0$ small enough such that

[TABLE]

By (6) every $\omega\in\mathcal{S}_{n}$ satisfies $|\omega(t)-\omega_{0}(t)|\leq\|\omega-\omega_{0}\|_{[0,n\tau]}<\delta$ for almost all $t\in[0,n\tau]$. Hence it follows that $|f(\omega(t))-f(\omega_{0}(t))|<\varepsilon$ for almost all $t\in[0,n\tau]$.

Now we can estimate

[TABLE]

For the last inequality we have used (7) and $T$-periodicity of $\omega_{0}$. By Proposition 9 this implies

[TABLE]

Since $\varepsilon>0$ can be chosen arbitrarily small and $S_{0}$ arbitrarily close to $\sum\limits_{j=1}^{r}\max(0,d_{j}\rho_{j})$, the assertion of the theorem follows. ∎

Remark 13**.**

In Kawan [18, Section 5.2] and da Silva and Kawan [13, Section 3.2] one finds more information on regular periodic control-trajectory pairs.

4 A lower bound

Again we consider a smooth control system (1) on a Riemannian manifold $(M,g)$ under our standard assumptions. Thus for each $t\geq 0$ and each control $\omega\in\mathcal{U}$ the map $\varphi_{t,\omega}:M\rightarrow M$ is a diffeomorphism.

Theorem 14**.**

Let $(K,Q)$ be an admissible pair where both $K$ and $Q$ have positive and finite volume. Then for every $f\in C(U,\mathbb{R})$

[TABLE]

where both infima are taken over all $(x,\omega)\in K\times\mathcal{U}$ with $\varphi([0,\tau],x,\omega)\subset Q$.

Proof.

First observe that by Remark 7 we may assume that for all $\tau>0$ there exists a countable $(\tau,K,Q)$-spanning set, since otherwise $P_{inv}(f,K,Q)=\infty$, and the infimum in $a_{\tau}(f,K,Q)$ my be taken over all countable $(\tau,K,Q)$-spanning sets $\mathcal{S}$. For each $\omega$ in a countable $(\tau,K,Q)$-spanning set $\mathcal{S}$ define

[TABLE]

Thus $K={\textstyle\bigcup\nolimits_{\omega\in\mathcal{S}}}K_{\omega}$. Since $Q$ is Borel measurable, each set $K_{\omega}$ is measurable as the countable intersection of measurable sets,

[TABLE]

Then

[TABLE]

where the infimum is taken over all $(x,\omega)\in K\times\mathcal{U}$ with $\varphi([0,\tau],x,\omega)\subset Q$. Abbreviating with the same infima

[TABLE]

we find

[TABLE]

Since this holds for every countable $(\tau,K,Q)$-spanning set $\mathcal{S}$, we find

[TABLE]

implying

[TABLE]

By Liouville’s formula

[TABLE]

and hence the assertion of the theorem follows:

[TABLE]

∎

5 Linear control systems

In this section we prove a formula for the invariance pressure of linear control systems in $\mathbb{R}^{d}$. They have the form

[TABLE]

where $A\in\mathbb{R}^{d\times d}$ and $B\in\mathbb{R}^{d\times m}$.

For system (9) there exists a unique control set $D$ with nonvoid interior, if, without control constraint, the system is controllable and the control range $U$ is a compact neighborhood of the origin in $\mathbb{R}^{m}$. It is convex, and it is bounded if and only if $A$ is hyperbolic, i.e., there is no eigenvalue of $A$ with vanishing imaginary part (cf. Hinrichsen and Pritchard [15, Theorems 6.2.22 and 6.2.23], Colonius and Kliemann [10, Example 3.2.16]). The state space $\mathbb{R}^{d}$ can be decomposed into the direct sum of the stable subspace $E^{s}$ and the unstable subspace $E^{u}$ which are the direct sums of all generalized real eigenspaces for the eigenvalues $\lambda$ with $\operatorname{Re}\lambda<0$ and $\operatorname{Re}\lambda>0$, resp. Let $\pi:\mathbb{R}^{d}\rightarrow E^{u}$ be the projection along $E^{s}$. We obtain the following estimates.

Lemma 15**.**

Consider a linear control system in $\mathbb{R}^{d}$ of the form (9) and assume that the pair $(A,B)$ is controllable, that $A$ is hyperbolic and the control range $U$ is a compact neighborhood of the origin. Let $D$ be the unique control set with nonvoid interior. Then for every compact set $K\subset D$ with nonempty interior every potential $f\in C(U,\mathbb{R})$ satisfies

[TABLE]

where the first infimum is taken over all $(T^{\prime},x^{\prime},\omega^{\prime})\in\mathbb{R}_{+}\times\pi K\times\mathcal{U}$ with $\pi\varphi([0,T^{\prime}],x^{\prime},\omega^{\prime})\allowbreak\subset\pi D$ and the second infimum is taken over all $(T,x,\omega)\in[0,\infty)\times D\times\mathcal{U}$ such that the control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ is $T$-periodic and contained in $\mathrm{int}D$ and the values $\omega(t),t\in[0,\tau]$, are in a compact subset of $\mathrm{int}U$.

Proof.

The hypotheses imply (see [10], Example 3.2.16) that $0\in\mathrm{int}D\subset\mathbb{R}^{d}$ and the Lebesgue measure of $K$ and $D$ (which coincides with the volume) is finite and positive. Theorem 11 yields

[TABLE]

where the infimum is taken over all $T>0$ and all $(x,\omega)\in\mathrm{int}D\times\mathcal{U}$ such that the control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ is $T$-periodic and the values $\omega(t),t\in[0,T]$, are in a compact subset of $\mathrm{int}U$. By Floquet theory it follows (cf. [7, Proposition 20]) that for all $T$-periodic $(\omega(\cdot),\varphi(\cdot,x,\omega))$

[TABLE]

where the sum is over the $r$ eigenvalues $\lambda_{j}$ of $A$ with multiplicities $d_{j}$. Hence

[TABLE]

where the infimum is taken over all $(T,x,\omega)$ as in (10). This proves second inequality.

Hence it remains to prove the first inequality. By Theorem 14

[TABLE]

where both infima in the second line are taken over all pairs $(x,\omega)\in K\times\mathcal{U}$ with $\varphi([0,\tau],x,\omega)\subset D$ and both infima in the third line are taken over all $(T,x,\omega)\in(0,\infty)\times K\times\mathcal{U}$ with $\varphi([0,T],x,\omega)\subset D$. Liouville’s formula (8) (or direct inspection) shows

[TABLE]

where the sum is over the $r$ eigenvalues $\lambda_{j}$ of $A$ with multiplicities $d_{j}$.

Step 1: Suppose that $\operatorname{Re}\lambda_{j}>0$ for all $j$. Then

[TABLE]

where the infimum is taken over all $(T,x,\omega)\in(0,\infty)\times K\times\mathcal{U}$ with $\varphi([0,T],x,\omega)\subset D$.

**Step 2: **Next we treat the general case, where also eigenvalues with negative real part are allowed. Recall that $\pi:\mathbb{R}^{d}\rightarrow E^{u}$ denotes the projection onto the unstable subspace $E^{u}$ along the stable subspace $E^{s}$.

Since these subspaces are $A$-invariant, this defines a semi-conjugacy between system (9) and the system on $E^{u}$ given by

[TABLE]

with trajectories $\pi\varphi(\cdot,x^{\prime},\omega^{\prime})$, and the sets $K$ and $D$ are mapped to $\pi K$ and $\pi D$, resp. Then $\pi K$ and $\pi D$ have positive volume and form an admissible pair (cf. Kawan [18, proof of Theorem 3.1]) . One easily proves that (cf. [6, Proposition 10])

[TABLE]

since every $(\tau,K,D)$-spanning set yields a $(\tau,\pi K,\pi D)$-spanning set. Similarly as in Step 1, Theorem 14 applied to system (11) implies that

[TABLE]

where the infimum is taken over all $(T^{\prime},x^{\prime},\omega^{\prime})\in\mathbb{R}_{+}\times\pi K\times\mathcal{U}$ with $\pi\varphi([0,T^{\prime}],x^{\prime},\omega^{\prime})\subset\pi D$. ∎

Next we show that the two infima in the proposition above actually coincide and again use hyperbolicity of $A$ in a crucial way. This provides the announced formula for the invariance pressure.

Theorem 16**.**

Consider a linear control system in $\mathbb{R}^{d}$ of the form (9) and assume that the pair $(A,B)$ is controllable, the matrix $A$ is hyperbolic and the control range $U$ is a compact neighborhood of the origin. Let $D$ be the unique control set with nonvoid interior. Then for every compact set $K\subset D$ with nonempty interior every potential $f\in C(U,\mathbb{R})$ satisfies

[TABLE]

Proof.

Let $\varepsilon>0$ and consider a control $\omega_{0}\in\mathcal{U}$ satisfying

[TABLE]

where the second infimum is taken over all triples $(T^{\prime},x^{\prime},\omega^{\prime})\in\mathbb{R}_{+}\times\pi K\times\mathcal{U}$ with $\pi\varphi([0,T],x^{\prime},\omega^{\prime})\subset\pi D$. Observe that there is a control value $u_{0}$ with

[TABLE]

Then there is a control

[TABLE]

with

[TABLE]

Claim: For every $T>0$ and every control $\omega\in\mathcal{U}$ there exists $x_{1}\in\mathbb{R}^{d}$ with $\varphi(T,x_{1},\omega)=x_{1}$.

In fact, hyperbolicity of $A$ implies that the matrix $I-e^{AT}$ is invertible, and hence there is a unique solution $x(T,\omega)$ of

[TABLE]

Now the variation-of-constants formula shows the claim:

[TABLE]

Applying this to $T_{0}$ and $\omega_{1}$ we find a point $x_{1}:=x(T_{0},\omega_{1})=\varphi(T_{0},x_{1},\omega_{1})$. Since $\omega_{1}\in\mathrm{int}\mathcal{U}_{\left|[0,T_{0}]\right.}$it follows that a neighborhood of $x_{1}$ can be reached in time $T_{0}$ from $x_{1}$. This follows, since by controllability, the map

[TABLE]

is a linear surjective map, hence maps open sets to open sets, and the same is true for the map

[TABLE]

Analogously, $x_{1}$ can be reached from every point in a neighborhood of $x_{1}$ in time $T_{0}$. Hence in the intersection of these two neighborhoods every point can be steered in time $2T_{0}$ into every other point. This shows that $x_{1}$ is in the interior of the (unique) control set $D$, and the corresponding trajectory $\varphi(t,x_{1},\omega_{1}),t\in[0,T_{0}]$, remains in the interior of $D$. Extending $\omega_{1}(t),t\in[0,T_{0}]$, to a $T_{0}$-periodic control $\omega_{2}$ we find that the control-trajectory pair $(\omega_{2}(\cdot),\varphi(\cdot,x_{1},\omega_{2}))$ is $T_{0}$-periodic, the trajectory is contained in $\mathrm{int}D$ and the values $\omega_{2}(t+T_{0})=\omega_{1}(t),t\in[0,T_{0}]$, are in a compact subset of $\mathrm{int}U$. It follows that

[TABLE]

where the second infimum is taken over all $(T,x,\omega)\in(0,\infty)\times D\times\mathcal{U}$ such that the control-trajectory pair $(\omega(\cdot),\varphi(\cdot,x,\omega))$ is $T$-periodic, the trajectory is contained in $\mathrm{int}D$ and the values $\omega(t),t\in[0,T]$, are in a compact subset of $\mathrm{int}U$.

Together with the inequalities in Lemma 15 this implies

[TABLE]

Since $\varepsilon>0$ is arbitrary, assertion (12) follows. ∎

Remark 17**.**

The proof of the Claim above follows arguments in the proof of da Silva and Kawan [14, Theorem 20].

Remark 18**.**

Theorem 16 improves [7, Theorem 6.2], where it had to be assumed that the minimum of $f(u),u\in U$, is attained in an equilibrium.

6 Further applications

In this section, we apply Theorem 11 to linear control systems on Lie groups and to inner control sets.

6.1 Control sets and equilibrium pairs

Given a control system (1), a pair $(u_{0},x_{0})\in U\times M$ is called an equilibrium pair if $F(x_{0},u_{0})=0$, or equivalently, $\varphi(t,x_{0},\bar{u}_{0})=x_{0}$ for all $t\in\mathbb{R}$, where $\bar{u}_{0}(t)\equiv u_{0}$.

If $(u_{0},x_{0})$ is an equilibrium pair, the linearized system is an autonomous linear control system in $T_{x_{0}}M$ and the Lyapunov exponents at $(u_{0},x_{0})$ in the direction $\lambda\in T_{x_{0}}M\backslash\{0_{x_{0}}\}$ coincide with the real parts of the eigenvalues of $\nabla F_{u_{0}}(x_{0}):T_{x_{0}}M\rightarrow T_{x_{0}}M$. Then regularity, i.e., controllability of the linearized system, can be checked by Kalman’s rank condition.

Corollary 19**.**

Let $D\subset M$ be a control set with nonempty interior and let $f\in C(U,\mathbb{R})$. Suppose that there is a regular equilibrium pair $(u_{0},x_{0})\in\mathrm{int}U\times\mathrm{int}D$. Then for every compact set $K\subset D$ and every set $Q\supset D$ we have

[TABLE]

where $n_{\lambda}$ is the algebraic multiplicity of the eigenvalue $\lambda$ in the spectrum $\sigma(\nabla F_{u_{0}}(x_{0}))$.

Proof.

Since $(u_{0},x_{0})$ is a regular equilibrium pair, the control-trajectory pair $(\varphi(\cdot,x_{0},\bar{u}_{0}),\bar{u}_{0}(\cdot))$ is $T$-periodic and regular for every $T>0$. By Theorem 11 we obtain

[TABLE]

∎

6.2 Control sets of linear control systems on Lie groups

In this subsection we consider linear control systems on a connected Lie group $G$ introduced in Ayala and San Martin [2] and Ayala and Tirao [4].

They are given by a family of ordinary differential equations on $G$ of the form

[TABLE]

where the drift vector field $\mathcal{X}$, called the linear vector field, is an infinitesimal automorphism, i.e., its solutions are a family of automorphisms of the group, and the $X_{j}$ are right invariant vector fields. Note that the linear control systems of the form (9) are a special case with $G=$ $\mathbb{R}^{d}$.

Their controllability properties have been analyzed in da Silva [12], Ayala, da Silva and Zsigmond [3] and Ayala and da Silva [1]. In particular, the existence and uniqueness of control sets for general systems of the form (13) has been analyzed in [3]. If [math] is in the interior of the control range $U$ and the reachable set $\mathcal{O}^{+}(e)$ from the neutral element $e$ is open (this holds e.g. if $e\in\mathrm{int}\mathcal{O}^{+}(e)$), then there exists a control set $D$ containing $e$ in the interior. For semisimple or nilpotent Lie groups $G$ sufficient conditions for boundedness of $C$ are given in [3, Theorem 3.9], and [3, Corollary 3.12] shows uniqueness of the control set with nonvoid interior if $G$ is decomposable.

Along with system (13) comes an associated derivation $\mathcal{D}$ of the Lie algebra $\mathfrak{g}$ of $G$ which is given by

[TABLE]

Corollary 20**.**

Consider the linear control system (13) on a Lie group $G$. Suppose that $D$ is a control set with $e_{G}\in\mathrm{int}D$ and compact closure $\overline{D}$ and let $K\subset D\subset Q$. Let $f\in C(U,\mathbb{R})$ be a potential. If the equilibrium pair $(0,e_{G})\in\mathrm{int}U\times\mathrm{int}D$ is regular, then

[TABLE]

If furthermore $K$ has positive Haar measure and $f(0)=\min_{u\in U}f$, then

[TABLE]

Proof.

Note that the right hand side of the system is given by $F(x,u)=\mathcal{X}(x)+\sum_{i=1}^{m}u_{i}X_{i}(x)$ and hence $F_{0}(x):=F(x,0)=\mathcal{X}(x)$. Let $g$ the Riemannian metric on $G$ defined as in the proof of Theorem 11 and $\nabla$ the Levi-Civita connection. Let $(\phi,U)$ be a local coordinate neighborhood of $e_{G}$ and pick a left invariant vector field $Y$ in the Lie algebra $\mathfrak{g}$ of $G$. Then we can express $\mathcal{X}$ in terms of $(\phi,U)$ by

[TABLE]

Note that since $\mathcal{X}(e_{G})=0$, then $y_{i}(e_{G})=0$ for every $i\in\{1,\cdots,d\}$, hence

[TABLE]

Since $\nabla$ is symmetric, we have

[TABLE]

Since this holds for every $Y\in\mathfrak{g}$, we have $\nabla F_{0}(e_{G})=\mathcal{D}$. By Corollary 19 we obtain

[TABLE]

Now, suppose that $K$ has positive Haar measure. By da Silva [11, Theorem 4.3], we know that

[TABLE]

Define $\tilde{f}(u)=f(u)-f(u_{0}),u\in U$. Since $\tilde{f}\geq 0$ Proposition 6(i) implies that

[TABLE]

Proposition 6(ii) implies $P_{inv}(\tilde{f},K,Q)=P_{inv}(f,K,Q)-\inf f$, hence this yields

[TABLE]

∎

6.3 Inner control sets

This section presents an application of Theorem 11 to the class of inner control sets as defined (with small changes) in Kawan [18, Definition 2.6]. This nomenclature refers to a control set $D\subset M$ for which there exists an decreasing family of compact and convex sets $\{U_{\rho}\}_{\rho\in[0,1]}$ in $\mathbb{R}^{m}$ (i.e., $U_{\rho_{2}}\subset U_{\rho_{1}}$ for $\rho_{1}<\rho_{2}$), such that for every $\rho\in[0,1]$ system (1)ρ with control range $U_{\rho}$ (instead of $U$ in (1)) has a control set $D_{\rho}$ with nonvoid interior and compact closure, and the following conditions are satisfied:

(i) $U=U_{0}$ and $D=D_{1}$;

(ii) $\overline{D_{\rho_{2}}}\subset\mathrm{int}D_{\rho_{1}}$ whenever $\rho_{1}<\rho_{2}$;

(iii) for every neighborhood $W$ of $\overline{D}$ there is $\rho\in[0,1)$ with $\overline{D_{\rho}}\subset W$.

Corollary 21**.**

Consider an inner control set $D$ of control system (1). Let $(\omega_{0}(\cdot),\varphi(\cdot,x_{0},\omega_{0}))$ be a regular $T$-periodic control-trajectory pair with $x_{0}\in\overline{D}$ and $\omega_{0}\in\mathcal{U}_{1}$. Then

[TABLE]

holds, where $\lambda_{1},\dotsc,\lambda_{r}$ are the Lyapunov exponents at $(x_{0},\omega_{0})$ with corresponding multiplicities $d_{1},\dotsc,d_{r}$.

Proof.

Note that the definition of inner control sets implies that for every $\rho\in[0,1)$ the set $\overline{D}$ is a compact subset of $D_{\rho}$. By Theorem 11 it follows that the outer invariance pressure $P_{out}^{\rho}(f,\overline{D},\overline{D_{\rho}})$ for system (1)ρ satisfies

[TABLE]

Now for given $\varepsilon>0$ we may choose $\rho\in[0,1)$ such that $\overline{D_{\rho}}\subset N_{\varepsilon}(\overline{D})$. Then

[TABLE]

The first two inequalities follow from $U_{\rho}\subset U_{0}$ and $\overline{D_{\rho}}\subset N_{\varepsilon}(\overline{D})$. Since $P_{out}(f,\overline{D})=\lim_{\varepsilon\rightarrow 0}P_{out}(f,\overline{D},N_{\varepsilon}(\overline{D}))$, the assertion follows. ∎

6.4 Example

Consider the following linear control system in $\mathbb{R}^{d}$,

[TABLE]

and assume that $\omega(t)\in U:=[-1,1]+u_{0}$ for some $u_{0}\in(-1,1)$. In this case, $0\in\mathrm{int}U$ and $(A,B)$ is controllable, and $A$ is hyperbolic with eigenvalues given by $\lambda_{\pm}=1\pm i$. There exists a unique control set $D\subset\mathbb{R}^{2}$ such that $(0,0)\in\mathrm{int}D$, and $\overline{D}$ is compact.

We may interpret the control functions $\omega(t)$ and also $u_{0}$ as external forces acting on the system. Take $f\in C(U,\mathbb{R})$ as $f(u):=|u-u_{0}|$, then $(S_{\tau}f)(\omega)$ represents the impulse of $\omega-u_{0}$ until time $\tau$. For a subset $K\subset D$ a $(\tau,K,D)$-spanning set $\mathcal{S}$ represents a set of external forces $\omega$ that cause the system to remain in $D$ when it starts in $K$. By Theorem 16 we obtain for a compact subset $K\subset D$ with nonzero Lebesgue measure that

[TABLE]

Here $P_{inv}(f,K,Q)$ represent the exponential growth rate of the amount of total impulse required of the external forces $\omega-u_{0}$ acting on the system to remain in $D$ as time tends to infinity. The minimum of $f$ is attained in $u=u_{0}$, which does not correspond to an equilibrium if $u_{0}\not=0$. Hence [7, Theorem 6.2] (cf. Remark 18) could not be applied in this case.