Nagumo-Type Characterization of Forward Invariance for Constrained Systems

TL;DR

This paper introduces a Nagumo-type condition for ensuring forward invariance of constrained systems, extending classical results to more general boundary cases with additional geometric assumptions.

Contribution

It generalizes Nagumo's invariance condition to differential inclusions on closed sets, providing necessary and sufficient conditions under specific geometric assumptions.

Findings

The proposed condition reduces to classical Nagumo condition on interior boundaries.

Additional assumptions ensure equivalence between necessary and sufficient conditions.

Examples demonstrate the importance of geometric assumptions for invariance.

Abstract

This paper proposes a Nagumo-type invariance condition for differential inclusions defined on closed constraint sets. More specifically, given a closed set to render forward invariant, the proposed condition restricts the system's dynamics, assumed to be locally Lipschitz, on the boundary of the set restricted to the interior of the constraint set. In particular, when the boundary of the set is entirely within the interior of the constraint set, the proposed condition reduces to the well-known Nagumo condition, known to be necessary and sufficient for forward invariance in this case. This being said, the proposed condition is only necessary in the general setting. As a result, we provide a set of additional assumptions relating the constrained system to the set to render forward invariant, and restricting to the geometry at the intersection between the two sets, so that the equivalence holds. The importance of the proposed assumptions is illustrated via examples.

Equations

$$\dot{x}=f(x)x\in {\mathbb{R}}^n$$

$$f(x)\in T_K(x)\forall x\in \partial K,$$

$$\dot{x}\in F(x)x\in {\mathbb{R}}^n$$

$$F(x)\subset T_K(x)\forall x\in \partial K.$$

$$\dot{x}\in F(x)x\in C\subset {\mathbb{R}}^n.$$

$$F(x)\subset T_K(x)\forall x\in \partial K\cap \mathrm{int}(C)$$

$$\begin{array}{rl}(x_1-x_2{)}^{\top }F(x_1)\subset & (x_1-x_2{)}^{\top }F(x_2)+k|x_1-x_2{|}^2\mathbb{B}\\ & \forall (x_1,x_2)\in \mathcal{N}\times \mathcal{N}.\end{array}$$

$$\begin{array}{rl}F(x_1)\subset & F(x_2)+k|x_1-x_2|\mathbb{B}\forall (x_1,x_2)\in \mathcal{N}\times \mathcal{N}.\end{array}$$

$$\Sigma :\dot{x}\in F(x)x\in C\subset {\mathbb{R}}^n,$$

$$T_K(x):=\left\{v\in {\mathbb{R}}^n:\exists t_i\to 0^{+},\exists v_i\to v,x+t_i{v}_i\in K\right\}.$$

$$\begin{array}{rl}& C_K(x):=\\ & \left\{v\in {\mathbb{R}}^n:\forall t_i\to 0^{+},\forall x_i{\to }_{K}x,\exists v_i\to v,x_i+t_i{v}_i\in K\right\},\end{array}$$

$$T_K^a(x):=\left\{v\in {\mathbb{R}}^n:\forall t_i\to 0^{+},\exists v_i\to v,x+t_i{v}_i\in K\right\}.$$

$$C_K(x)\subset T_K^a(x)\subset T_K(x)\forall x\in \text{cl}(K).$$

$$\begin{array}{rl}& D_K(x):=\\ & \left\{v\in {\mathbb{R}}^n:\forall t_i\to 0^{+},\forall v_i\to v,x+t_i{v}_i\in \text{int}(K)\forall i\text{large}\right\}.\end{array}$$

$$x+a(v+e)\subset \text{int}(K)\forall a\in (0,\alpha ],\forall e\in ϵ\mathbb{B}.$$

$$F(x)\subset T_K(x)\forall x\in \partial K\cap \text{int}(C)$$

$$F(x)\subset T_K(x)\forall x\in \partial K.$$

$$F(x)\subset T_{K\cap C}(x)\forall x\in \partial K\cap \text{int}(C)$$

$$F(x)\subset T_{K\cap C}(x)\forall x\in \partial K\cap \partial C,$$

$$F(x)\subset T_{K\cap C}(x)\forall x\in \partial (K\cap C).$$

$${\text{Proj}}_{\partial K}(\varphi (t))\cap \text{int}(C)\ne \varnothing \forall t\in (0,T]\subset \mathrm{dom}\varphi .$$

$$\mathcal{P}:=\{x\in \partial K\cap \partial C:\mathcal{N}(x)\cap (C\backslash K)\ne \varnothing \forall \mathcal{N}(x)\},$$

$$\begin{array}{rl}{F}_{\varphi }(x):=\{& v\in {\mathbb{R}}^n:\exists \{t_i{\}}_{i=1}^{\infty }\subset {\mathbb{R}}_{\ge 0}\text{with}\underset{i\to \infty }{\lim }{t}_i=0\\ & \text{ and }\underset{i\to \infty }{\lim }\frac{\varphi (t_i)-x}{t_i}=v\},\end{array}$$

$$F_{\varphi }(x)\subset T_{\partial K\cap C}(x)\backslash T_{\partial K\backslash \text{int}(C)}(x).$$

$$F_{\varphi }(x)\subset T_{\partial K\cap C}(x)\forall x\in \mathcal{P}.$$

$$T_{\partial K}(x)\cap T_C(x)=T_{\partial K\cap C}(x)\forall x\in \mathcal{P}.$$

$$F_{\varphi }(x)\cap T_{\partial K\backslash \text{int}(C)}(x)=\varnothing \forall x\in \mathcal{P},$$

$$T_{\partial K\cap C}(x)$$

$$x_2-x_1\in v_1-v_2+\alpha (|x_1-x_2|)\mathbb{B}$$

$$\forall (x_1,x_2)\in ((\partial K\backslash C)\cap \mathcal{N}(x))\times ((\partial C\backslash \partial K)\cap \mathcal{N}(x))$$

$$\begin{array}{rl}(a+b)(a-b)& =a^2-b^2\forall a,b\in \mathbb{R},\\ (a+b{)}^{\top }(a-b)& =a^{\top }a-b^{\top }b\forall a,b\in {\mathbb{R}}^n.\end{array}$$

Full text

1 Introduction

Forward invariance of sets for dynamical systems refers to the property whereby all solutions starting in a given set remain in it over their entire domain of definition. Set invariance plays a central role in control theory, as it underlies the concept of safety. Safety, or conditional invariance [21], is understood as forward invariance of a set that contains the initial conditions while being disjoint from the unsafe region [1, 37, 27]. Hence, forward invariance is particularly important in safety-critical applications, where entering the unsafe region can correspond to failures or a loss of model fidelity. Applications span a wide range of domains, including transportation networks [13], power systems [40], and robotics [38, 17, 16]. These examples highlight the fundamental role of characterizing forward invariance for both verification and design of modern control-system applications. Furthermore, forward invariance provides a cornerstone for establishing more elaborate dynamical behaviors. In particular, invariance analysis is instrumental in establishing asymptotic stability via relaxed Lyapunov conditions, including invariance principle [32] and Matrosov’s theorem [33]. Beyond safety and stability, extensions of invariance theory have been developed to encompass related concepts such as quasi-invariance, conditional quasi-invariance [22], and contractivity [6].

Characterizing forward invariance typically requires conditions that guarantee the property without explicitly computing the system’s trajectories. A relevant context where the problem is not fully covered yet concerns dynamical systems, modeled by differential inclusions, subject to state constraints. That is, the solutions are well defined in this case as long as they are contained within the set where the differential inclusion is defined. The main motivation behind studying constrained differential inclusions is that they represent a key component of hybrid systems [18]. Indeed, for general hybrid systems, the continuous-time dynamics is usually defined on a closed set, called flow set. Constrained differential inclusions are also common when modeling physical systems such as energy and power systems [10], and they appear naturally in the context of singular and complementarity systems [19].

1.1 Background

The earliest characterization of forward invariance is due to Nagumo [25], who established a necessary-and-sufficient condition for the existence of (at least) a single solution that remains in a closed set $K$ from every initial condition in $K$. This property, sometimes referred to as weak forward invariance [11], coincides with forward invariance provided that the solutions are unique. Specifically, Nagumo’s theorem states that a closed subset $K\subset\mathbb{R}^{n}$ is weakly forward invariant for the differential equation, with a continuous right-hand side $f$,

[TABLE]

if and only if

[TABLE]

where $\partial K$ refers to the boundary of $K$ and $T_{K}(x)$ denotes the contingent cone to $K$ at $x$, whose precise definition is recalled in the forthcoming Section 2. Nagumo’s theorem allows forward invariance to be assessed solely at the boundary of the set, without requiring explicit knowledge of the trajectories.

Subsequent works generalized Nagumo’s condition to stronger invariance notions and extended the result to differential inclusions and impulse differential inclusions. For instance, J.P. Aubin showed in [5, Theorem 5.3.4] that for a locally-Lipschitz set-valued map $F:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ with convex and compact images, the closed set $K$ is forward invariant for the differential inclusion

[TABLE]

if and only if

[TABLE]

This result was later extended to impulse differential inclusions [3], and Lipschitz continuity of $F$ is relaxed to directional Lipschitz continuity in [28].

More recently, the development of hybrid-systems theory [18] has motivated analogous studies. Indeed, such a general framework captures the system’s continuous-time behavior by a constrained differential inclusion of the form

[TABLE]

The fact that the differential inclusion in (3) is defined only on a closed set $C$ renders condition (2) overly conservative. For example, when $x$ is within $\partial C\cap\partial K$, vectors within $F(x)$ ‘pointing’ outside $C$ are, roughly speaking, not able to generate solutions. Hence, they should not be forced to be within $T_{K}(x)$. Less restrictive sufficient conditions for forward invariance in the context of hybrid systems are proposed in [8, 9]. Additional results for polyhedral sets [15], interconnected systems [35], and subsystem composition in the hybrid settings are proposed in [34]. To the best of our knowledge, the conditions in the aforementioned literature remain restrictive and do not rise to the standard of being both necessary and sufficient.

Finally, in many applications, the sets $C$ and $K$ correspond to the intersection of sublevel sets of differentiable scalar functions (i.e., sets defined by $g_{i}(x)\leq 0$ for all $i\in\{1,2,...\}$). We refer to these sets as practical sets, or polytopic sets when the defining functions are linear. Safety of practical sets is widely studied, through their defining functions called barrier functions, and they become a standard tool for enforcing safety and forward invariance [1, 17, 26, 14, 20]. These tools are especially powerful in the context of input-affine systems since safety conditions are enforced via solving quadratic programs [1, 39]. However, the challenging question of seeking necessary-and-sufficient conditions in the context of constrained systems remains open even when the sets $K$ and $C$ are practical or polytopic.

1.2 Contributions

In this paper, we establish a framework under which the relaxed Nagumo-inspired criterion

[TABLE]

is equivalent to forward invariance of the set $K$ for the constrained differential inclusion (3). When the boundary of $K$ lies strictly within $\mathrm{int}(C)$, condition (4) reduces to the classical Nagumo criterion (2) for unconstrained systems. However, as we show via a simple counterexample, condition (4) is only necessary in general. We therefore introduce three assumptions to conclude sufficiency of (4) when $\partial K\not\subset\mathrm{int}(C)$. A particular focus is made on a specific subset of critical points, denoted by $\mathcal{P}\subset\partial K\cap\partial C$, from where trajectories that leave the set $K$ “immediately” may exit $K$ even when (4) holds. In particular, we propagate the inclusion in (4), by assuming some regularities, so that it holds at points in $\mathcal{P}$. Furthermore, we impose a “transversal” intersection between the sets $K$ and $C$. The transversality condition we use applies to general sets and it is weaker than those commonly employed in the literature [5] and [12]. Finally, we restrict $F(x)$, when $x\in\mathcal{P}$, so that none of its vectors is tangent to $\partial K\cap\partial C$. Thereby, the first-order approximation of every solution starting from $\mathcal{P}$ is decisive and can indicate the side of $\partial K$ the solution is more leaning towards, allowing (4) to exclude the possibility of immediately leaving $K$. The importance of these assumptions is illustrated via counterexamples. In the second part of this paper, we apply our general result to the case where both sets $K$ and $C$ are practical. Within this setup, we show that the propagation assumption holds for free. Moreover, the transversality condition can be simplified to a more-restrictive but tractable condition involving the gradients of the defining functions. The latter condition is not needed when the sets are polytopic.

The remainder of this paper is organized as follows. Preliminaries related to differential inclusions and tangent cones are in Section 2. The problem formulation, its motivation, and the rationale of the proposed approach are in Section 3. Our main result for general sets is stated and proved in Section 4, and the considered assumptions are discussed, through counterexamples, in Section 5. The case of practical sets is treated in Section 6. For clarity, some of the intermediate proofs and technical lemmas are relegated to the Appendix.

A preliminary version of this work is in the conference paper [29], where intermediate results and detailed proofs were not included. Moreover, the context of practical and polytopic sets was not considered. Finally, the importance of having $F(x)$ non-tangent to $\partial K\cap\partial C$, for $x\in\mathcal{P}$, was not argued.

Notation. We denote by $\mathbb{N}$ the set of positive integers and by $\mathbb{N}_{\geq 0}$ the set of nonnegative integers. For each $n\in\mathbb{N}$, we let $[n]:=\{1,2,...,n\}$. For $K\subset\mathbb{R}^{n}$, we use $\text{cl}(K)$ to denote the closure of $K$ and $\text{int}(K)$ to denote the interior of $K$. By $\mathrm{Proj}_{K}(x):=\left\{y\in\mathbb{R}^{n}:|x-y|=\inf_{z\in K}|z-y|\right\}$, we denote the projection of a point $x$ onto the set $K$, where $|\cdot|$ is the Euclidean norm. The closed ball of radius $\epsilon$ centered at $x$ is denoted by $\mathbb{B}_{\epsilon}(x)$, and $\mathbb{B}:=\mathbb{B}_{1}(0)$ represents the unit ball centered at the origin. By $F:\rm{dom}\,F\rightrightarrows\text{Im}(F)$ or $x\leadsto F(x)$, we denote a set-valued map with $\rm{dom}\,F$ the domain of $F$ and $\text{Im}(F)$ the image of $F$. For $A\subset\rm{dom}\,F$, $F(A):=\{F(x):x\in A\}$ denotes the set of images of the elements in $A$ under $F$. Finally, for $\rm{dom}\,F,\text{Im}(F)\subset\mathbb{R}^{n}$, $F$ is said to be one-sided locally Lipschitz if, for each non-empty compact set $\mathcal{N}\subset\rm{dom}\,F$, there exists $k>0$ such that

[TABLE]

It is said to be locally Lipschitz if, for each non-empty compact set $\mathcal{N}\subset\rm{dom}\,F$, there exists $k>0$ such that

[TABLE]

2 Preliminaries

Consider the constrained differential inclusion

[TABLE]

where the set $C$ and the set-valued map $F:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ verify the following standing assumption.

• (SA)

The set $C$ is closed, $F(x)$ is non-empty, closed, and convex for all $x\in C$, and $F$ is continuous and verifies the one-sided local Lipschitzness property.

We refer the reader to [5, Definition 2.1.2] for the definitions of lower and upper semicontinuity as well as continuity in the context of set-valued maps.

We next specify the concept of solutions for constrained systems, recall the definition of forward invariance, and the tangent cones used throughout this work [4].

Definition 2.1** (Solution).**

The function $\phi:\rm{dom}\,\phi\rightarrow\text{Im}(\phi)$ is a solution to $\Sigma$ if $\text{Im}(\phi)\subset C$,

• •

Either $\rm{dom}\,\phi=[0,T]$ for some $T\geq 0$ or $\rm{dom}\,\phi=[0,T)$ for some $T\in\mathbb{R}_{\geq 0}\cup\{+\infty\}$, and

• •

The function $\phi$ is locally absolutely continuous and $\dot{\phi}(t)\in F(\phi(t))$ for almost all $t\in\rm{dom}\,\phi$.

Given a closed set $K\subset\mathbb{R}^{n}$, a solution $\phi$ starting from some $x\in C$ is said to leave $K$ immediately if there exists $T>0$ such that $\phi((0,T])\subset C\backslash K$.

Definition 2.2** (Forward invariance).**

The set $K\subset\mathbb{R}^{n}$ is forward invariant for $\Sigma$ if every solution $\phi$ to $\Sigma$ starting from $K$ satisfies $\phi(t)\in K$ for all $t\in\rm{dom}\,\phi$.

• The contingent cone to the set $K\subset\mathbb{R}^{n}$ at $x\in\text{cl}(K)$, denoted by $T_{K}(x)$, is given by

[TABLE]

Note that $T_{K}(x)$ is always closed. Furthermore, when the map $K\ni y\leadsto T_{K}(y)$ is lower semicontinuous at $x\in\text{cl}(K)$, then it is said to be sleek at $x$.

• The Clarke tangent cone to the set $K\subset\mathbb{R}^{n}$ at $x\in\text{cl}(K)$, denoted by $C_{K}(x)$, is given by

[TABLE]

where $x_{i}{\rightarrow}_{K}x$ means that the sequence $\{x_{i}\}_{i}\subset K$ and $x_{i}{\rightarrow}x$. Note that the map $x\leadsto C_{K}(x)$ is lower semicontinuous, and $C_{K}(x)$ is a closed and convex subset of $T_{K}(x)$. Furthermore, when $K$ is sleek at $x\in\text{cl}(K)$, then $C_{K}(x)=T_{K}(x)$.

• The adjacent contingent cone to the set $K\subset\mathbb{R}^{n}$ at $x\in\text{cl}(K)$, denoted by $T^{a}_{K}(x)$, is given by

[TABLE]

Note that $T^{a}_{K}(x)$ is always closed and

[TABLE]

Moreover, the set $K$ is said to be derivable at $x\in\text{cl}(K)$ if $T_{K}(x)=T^{a}_{K}(x)$.

• The Dubovitskiy cone to the set $K\subset\mathbb{R}^{n}$ at $x\in\text{cl}(K)$, denoted by $D_{K}(x)$, is given by

[TABLE]

It is also true that $v\in D_{K}(x)$ if and only if there exist $\alpha,\epsilon>0$ such that

[TABLE]

Although the aforementioned cones are sometimes introduced differently, the definitions we recall here are particularly useful for our proofs.

3 Problem Formulation

Given the constrained differential inclusion $\Sigma$ and a closed subset $K\subset\mathbb{R}^{n}$, our goal is to establish a fairly general framework such that the condition

[TABLE]

is equivalent to forward invariance of the set $K$. In general, condition (8) is only necessary. It becomes sufficient in the following restrictive situations.

• When $K\subset\text{int}(C)$, so that condition (8) reduces to

[TABLE]

Sufficiency of (9) is a consequence of Nagumo’s invariance theorem for unconstrained systems established in [5, Theorem 5.3.4] and [28].

• When $\partial K\cap\text{int}(C)$ is dense in $\partial(K\cap C)$, i.e., every $x\in\partial(K\cap C)$ is the limit of a sequence $\{x_{i}\}_{i}\subset\partial K\cap\text{int}(C)$, and $T_{K\cap C}$ is upper semicontinuous on $\partial K\cap\partial C$. In this case, we use the fact that (8) implies

[TABLE]

to conclude, under continuity of $F$ and upper semicontinuity of $T_{K\cap C}$, that

[TABLE]

and thus, by the density property, we obtain

[TABLE]

The latter guarantees forward invariance of the set $K\cap C$ by Nagumo’s invariance theorem, which implies forward invariance of $K$.

For general closed subsets $K$ and $C$, however, sufficiency of (8) is not always true; see the counterexamples in the forthcoming Section 5.1.

3.1 Rationale

Our approach to establish sufficiency of (8) under general sets $(C,K)$ is based on showing that any solution $\phi$ starting from $\partial K\cap\partial C$ that leaves the set $K$ immediately must fulfill the following key property.

• ($\star$)

There exists $T>0$ such that

[TABLE]

Under ($\star$), we show that (8) excludes the possibility that the solution $\phi$ leaves $K$ immediately.

According to Definition 2.1, the solutions starting from $\partial K\cap\partial C$ that leave $K$ immediately are topologically restricted to start within the set

[TABLE]

with $\mathcal{N}(x)$ denoting an arbitrary neighborhood of $x$. Hence, we will verify ($\star$) for the solutions starting from $\mathcal{P}$ by proposing a set of assumptions that constrain the sets $K$ and $C$ as well as the dynamics $F$ at points in $\mathcal{P}$. In particular, the proposed assumptions allow us to show that the set of initial speeds of a solution $\phi$ starting from $\phi(0):=x\in\mathcal{P}$ that leaves $K$ immediately, defined by

[TABLE]

must verify

[TABLE]

We then show that (12) is enough to conclude ($\star$).

3.2 Assumptions

We start noting that condition (8) does not necessarily impose a restriction on $F(x)$ when $x\in\mathcal{P}\subset\partial K\cap\partial C$, which makes it hard (and, in general, impossible) to guarantee, as suggested in (12), that

[TABLE]

Hence, we assume the following propagation property.

Assumption 1** (Propagation).**

$F(x)\subset T_{K}(x)$* $\forall x\in\mathcal{P}$. $\bullet$*

We will show, under Assumption 1, that any solution $\phi$ starting from $x\nolinebreak\in\nolinebreak\mathcal{P}$ that leaves the set $K$ immediately must satisfy $F_{\phi}(x)\nolinebreak\subset\nolinebreak T_{\partial K}(x)$. Furthermore, since we can easily show that $F_{\phi}(x)\nolinebreak\subset\nolinebreak T_{C}(x)$ under the considered concept of solutions, a key step towards verifying (13) consists in guaranteeing

[TABLE]

Additionally, to conclude that (also as suggested in (12))

[TABLE]

we establish that

[TABLE]

and use the following decisiveness assumption.

Assumption 2** (Decisiveness).**

$F(x)\cap T_{\partial K\cap\partial C}(x)=\emptyset$* for all $x\in\mathcal{P}$. $\bullet$*

When Assumption 2 is not verified, namely, when $F_{\phi}(x)\nolinebreak\cap\nolinebreak T_{\partial K\cap\partial C}(x)\neq\emptyset$ at some $x\in\mathcal{P}$, we are not able to guarantee (15), which is crucial to verify ($\star$) for the solutions that leave $K$ immediately.

Finally, to verify (14) and (16), we propose the following transversality assumption.

Assumption 3** (Transversality).**

*- For each $x\in\mathcal{P}$, $T_{\partial K}(x)=T^{a}_{\partial K}(x)$.

• For each $x\in\mathcal{P}$, $T_{C}(x)=T^{a}_{C}(x)$, and

($\dagger$) For each $x\in\mathcal{P}$, there exists a neighborhood of $x$, denoted by $\mathcal{N}(x)$, $c>0$, and $\alpha\in[0,1)$ such that*

[TABLE]

for some $(v_{1},v_{2})\in T_{\partial K}(x_{1})\times T_{C}(x_{2})$ such that $|\left[v_{1}^{\top}~v_{2}^{\top}\right]|\leq c|x_{1}-x_{2}|$. $\bullet$

Remark 3.1**.**

The first item in Assumption 3 is not needed to verify (14) nor (16). However, its role is to ensure that $T^{a}_{\partial K\cap C}(x)=T_{\partial K\cap C}(x)$ for all $x\in\mathcal{P}$, a useful property to prove that (12) implies ($\star$).

Remark 3.2**.**

We show in Lemma 7.5 in the Appendix that Assumption 1 and the first bullet in Assumption 3 hold for free under the following regularity property.

• (Pr1)

The map $T_{K}:\partial K\rightrightarrows\mathbb{R}^{n}$ is continuous on $\mathcal{P}$, and $\mathcal{N}(x)\cap\nolinebreak\partial K\cap\text{int}(C)\neq\emptyset$ for any $x\in\mathcal{P}$ and for any neighborhood $\mathcal{N}(x)$ around $x$.

4 Result for general sets

We are now ready to state our version of Nagumo’s invariance theorem for constrained systems.

Theorem 4.1**.**

Consider a closed subset $K\subset\mathbb{R}^{n}$ and a constrained differential inclusion $\Sigma$. Then,

• •

Forward invariance of $K$ $\Rightarrow$ (8).

• •

Under Assumptions 1-3, the equivalence holds.

Some intermediate results are key to the proof.

4.1 Intermediate Results

Proposition 4.2**.**

Consider system $\Sigma$ such that (8) holds. Consider a solution $\phi$ starting from $x\in\partial K$ that satisfies ($\star$). Then, $\phi$ cannot leave the set $K$ immediately, i.e., $\phi([0,T])\subset K$ for the same $T>0$ introduced in ($\star$).

Proof 4.3**.**

The proof applies the arguments in [28]; see also [23, Theorem 2]. Given $t\in[0,T)$, we use $y(t)$ to denote the projection of $\phi(t)$ on the set $K$ and, we let $\delta(t):=|\phi(t)-y(t)|$, to conclude that $\delta(t)=0$ for all $t\in(0,T)$.

We recall the difference of squares identity

[TABLE]

Using the previous identity, we conclude that, for any $h>0$ such that $t+h$ in $[0,T)$,

[TABLE]

Furthermore, for almost all $t\in(0,T)$, we replace $\phi(t+h)$ by

[TABLE]

with $o(h)$ the remainder of the first order Taylor expansion of $h\mapsto\phi(t+h)$ around $h=0$, which satisfies $\lim_{h\rightarrow 0}o(h)/h=0$. Using the previous limit, (18) and the inequality

[TABLE]

we obtain that

[TABLE]

From the previous inequality, and using property (17), we conclude that for almost all $t\in(0,T)$,

[TABLE]

Therefore,

[TABLE]

Next, we claim that

[TABLE]

Using (21), for each $\eta_{y}\in F(y(t))$, the term

[TABLE]

can be added in (20). As a result, for almost all $t\in(0,T)$

[TABLE]

Since $\dot{\phi}(t)\in F(\phi(t))$ for almost all $t\in(0,T)$, using the directional Lipschitzness property, we conclude that it is always possible to find $\eta^{*}_{y}\in F(y(t))$ such that

[TABLE]

Applying this inequality to (22) and replacing $\eta_{y}$ therein by $\eta^{*}_{y}$, we obtain that for almost all $t\in(0,T)$

[TABLE]

Hence, $\delta(t)=0$ for all $t\in(0,T)$ and the solution $\phi$ cannot leave the set $K$ immediately.

To prove the inequality in (21), we pick $t\in[0,T]$ and assume, without loss of generality, that $\phi(t)\notin K$. Hence,

[TABLE]

We start considering the inequality

[TABLE]

To obtain the previous inequality, we used the fact that $|\phi(t)-y(t)|=|\phi(t)|_{K}$ and that $|\cdot|_{K}$ is globally Lipschitz with a Lipschitz constant equal to $1$. Next, by taking the square in both sides of the last inequality and dividing by $h$, we obtain, for each $\eta_{y}\in F(y(t))$,

[TABLE]

which implies that

[TABLE]

Finally, letting $h\rightarrow 0^{+}$ through a suitable sequence, (21) is proved using the fact that $y(t)\in K$ (from (23)) and $\eta_{y}\in T_{K}(y(t))$, leading to $\liminf_{h\rightarrow 0^{+}}\frac{|y(t)+h\eta_{y}|_{K}}{h}=0$, see [4, Definition 1.1].

Proposition 4.4**.**

Consider system $\Sigma$ and a closed subset $K\subset\mathbb{R}^{n}$ such that Assumptions 1-3 hold. Let $\phi$ be a solution starting from $x\in\mathcal{P}$ that leaves the set $K$ immediately. Then,

[TABLE]

where

[TABLE]

Proof 4.5**.**

We first prove, under (SA), that

[TABLE]

Indeed, since $F(x)$ and $T_{C\backslash K}(x)$ are closed, we show that $v\in F(x)$ and $v\in T_{C\backslash K}(x)$ for all $v\in F_{\phi}(x)$.

To do so, we let a sequence $\{t_{i}\}^{\infty}_{i=1}\subset\mathbb{R}_{\geq 0}$ satisfying $\lim_{i\rightarrow\infty}t_{i}=0$ such that

[TABLE]

To show that $v\in F(x)$, thanks to the concept of solutions, we introduce a sequence $\{\delta^{j}_{i}\}^{N_{i}}_{j=1}\subset[0,t_{i}]$, with $N_{i}\in\{1,2,\ldots,\infty\}$, such that

• •

$t\mapsto\phi(t)$* is differentiable on each interval $(\delta^{j}_{i},\delta^{j+1}_{i})$.*

• •

$\dot{\phi}(s)\in F(\phi(s))$* for all $s\in(\delta^{j}_{i},\delta^{j+1}_{i})$.*

• •

$\delta^{1}_{i}=0$* and $\delta^{N_{i}}_{i}=t_{i}$.*

Now, using the Mean-Value Theorem, we conclude the existence of $c^{j}_{i}\in(\delta^{j}_{i},\delta^{j+1}_{i})$ such that

[TABLE]

The latter allows us to write

[TABLE]

where $v_{i}=\left[\frac{\phi(t_{i})-\phi(0)}{t_{i}}\right]$.

Now, since $F$ is continuous and has closed images, it follows that the graph of $F$ is closed. This allows us to conclude that $\lim_{t_{i}\rightarrow 0}\bigcup_{s\in[0,t_{i}]}\left\{F(\phi(s))\right\}\subset F(x).$ Hence, $v=\lim_{i\rightarrow\infty}\left[\frac{\phi(t_{i})-\phi(0)}{t_{i}}\right]\in F(x)$.

To show that $v\in T_{C\backslash K}(x)$, we use the fact that $\phi$ leaves the set $K$ immediately to conclude that for some $N\in\mathbb{N}$ big enough,

[TABLE]

where $v_{i}:=(\phi(t_{i})-\phi(0))/t_{i}$. (28) is enough to conclude that $v\in T_{C\backslash K}(x)\subset T_{C}(x)$.

Now, using (26)-(27), we prove that

[TABLE]

On one hand, we use Assumption 1 to obtain

[TABLE]

On the other hand, we apply [5, Theorem 4.3.3] to obtain

[TABLE]

Hence, combining (27) and (29)-(30), we obtain

[TABLE]

Next, using Lemma 7.1 with $(K_{1},K_{2})$ therein replaced by $(C,\partial K)$, under Assumption 3, we conclude that

[TABLE]

This allows us to conclude from (31) and (27) that $\rm{cl}\,(F_{\phi}(x))\subset T_{\partial K\cap C}(x).$ Now, using Assumption 2, we conclude that $F(x)\cap T_{\partial K\cap\partial C}(x)=\emptyset$, leading to

[TABLE]

At this point, (25) would follow if we show that

[TABLE]

To prove the latter equality, it is enough to show that

[TABLE]

For this, we use Lemma 7.1, while replacing $(K_{1},K_{2})$ therein by $(C,\partial K\backslash\text{int}(C))$, to conclude that (35) holds provided that $T_{C}(x)=T^{a}_{C}(x)$, and

• •

there exist $c>0$, $\alpha\in[0,1)$, and a neighborhood $\mathcal{N}(x)$ around $x$ such that

[TABLE]

for some $(v_{1},v_{2})\in T_{C}(x_{1})\times T_{\partial K\backslash\text{int}(C)}(x_{2})$ with $|\left[v_{1}^{\top},v_{2}^{\top}\right]|\leq c|x_{1}-x_{2}|$.

The latter item is verified under ($\dagger$) since

[TABLE]

Proposition 4.6**.**

Consider closed subsets $C$, $K\subset\mathbb{R}^{n}$ such that Assumption 3 holds. Let $x\in\partial K\cap\partial C$ and $\phi:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}^{n}$ be a function verifying

[TABLE]

with $\lim_{t\rightarrow 0^{+}}o(t)=0$ and the compact set $\Pi$ verifying

[TABLE]

Then ($\star$) is verified.

Proof 4.7**.**

To find contradiction, we assume the existence of

[TABLE]

and a sequence $\{y_{i}\}_{i=1}^{\infty}$ such that

[TABLE]

Using (36), we conclude the existence of $\{\pi_{i}\}_{i}\subset\Pi$ and $\{z_{i}\}_{i}\subset\mathbb{B}$ such that

[TABLE]

Subtracting $y_{i}$ from both sides of the previous inclusion and taking the absolute value, we obtain

[TABLE]

Through an appropriate sub-sequence, we let

[TABLE]

We will show that $w\neq\pi$. In fact, we assume that $w$ is finite, since otherwise $w\neq\pi$ would hold trivially. Now, since $x+t_{i}w_{i}=y_{i}\in\partial K\backslash\text{int}(C)$ for all $i\in\{1,2,\ldots\}$, we conclude that $w\in T_{\partial K\backslash\text{int}(C)}(x)$. In parallel, from (37), we conclude that $\pi\subset T_{\partial K\cap C}(x)\backslash T_{\partial K\backslash\text{int}(C)}(x)$.

As a result, for $\epsilon:=|\pi-w|>0$, we can write

[TABLE]

where

[TABLE]

Clearly, the sequence $\{r_{i}\}_{i}$ converges to [math].

Now, we use Assumption 3 and Lemma 7.1 to obtain

[TABLE]

Hence, since $\pi\in T_{\partial K\cap C}(x)$, we conclude that $\pi\in T^{a}_{\partial K\cap C}(x)$. Thus, there exists $\{\hat{\pi}_{i}\}_{i}\subset\mathbb{R}^{n}$ such that, for the same sequence $\{t_{i}\}_{i}$, we have

[TABLE]

That is, we established the existence of $\{x_{i}\}_{i}\subset\partial K\cap C$ such that

[TABLE]

and, at the same time, for the sequence $\{y_{i}\}_{i}\subset\partial K\backslash\text{int}(C)$ verifying (38), we have

[TABLE]

As a consequence, for $i$ large enough, it holds that

[TABLE]

Hence, for $i$ large enough, it follows that

[TABLE]

This contradicts (38), since

[TABLE]

Proposition 4.8**.**

Consider system $\Sigma$ and a closed subset $K\subset\mathbb{R}^{n}$ such that Assumptions 1-3 hold. Let $\phi$ be a solution starting from $x\in\mathcal{P}$. Then, (25) $\Rightarrow$ ($\star$).

Proof 4.9**.**

We show that

[TABLE]

The latter would entail that

[TABLE]

with $\lim_{t\rightarrow 0^{+}}o(t)=0$. From here, Proposition 4.6 can be used to conclude that $(\star)$ holds.

To show (41), we use the fact that $F$ is locally bounded an we show that every sequence $\{t_{i}\}^{\infty}_{i=1}$ as in (41) leads to a sequence $\left\{\frac{\phi(t_{i})-\phi(0)}{t_{i}}\right\}^{\infty}_{i=1}$ that is bounded. Indeed, by absolute continuity of solutions, we have

[TABLE]

which leads to

[TABLE]

Hence, by the concept of solutions, for all $t_{i}\geq 0$,

[TABLE]

Furthermore, by definition of $F_{\phi}(x)$, all the converging subsequences of $\left\{\frac{\phi(t_{i})-\phi(0)}{t_{i}}\right\}^{\infty}_{i=1}$ have their limit in $F_{\phi}(x)$. Hence, there exists a sequence $\{r_{i}\}_{i}\subset\mathbb{R}_{\geq 0}$ converging to [math] such that

[TABLE]

4.2 Proof of Theorem 4.1

To prove necessity of (8), we let $v\in F(x)$ for some $x\in\partial K\cap\text{int}(C)$. Using the continuity of $F$ plus the property of its images (the one-sided Lipschitzness property is not needed here), we show the existence of $T>0$ and a solution $\phi$ starting from $x$ such that

[TABLE]

for some continuous function $t\mapsto r(t)$ satisfying $\lim_{t\rightarrow 0}r(t)=0$. Indeed, using Michael’s selection theorem [24], we deduce the existence of a continuous map $y\mapsto u(y)\in F(y)$, with $\rm{dom}\,u$ containing a neighborhood of $x$, such that $u(x)=v$. Hence, there exists $T>0$ and a $\mathcal{C}^{1}$ function $\phi:[0,T]\rightarrow\mathbb{R}^{n}$ verifying

[TABLE]

Since $x\in\text{int}(C)$, we conclude that, for $T>0$ small enough, we have $\phi([0,T])\subset\text{int}(C)$ and $\phi$ is a solution to $\Sigma$. Furthermore, being continuously differentiable, the first-order approximation of $\phi$ verifies

[TABLE]

for a continuous function $t\mapsto r(t)$ satisfying $\lim_{t\rightarrow 0}r(t)=0$. Finally, since $K$ is forward invariant, we conclude that

[TABLE]

The latter is enough to conclude that $v\in T_{K}(x)$ by definition of the contingent cone.

The proof of sufficiency of (8) follows by contradiction. Indeed, assume the existence of a solution $\phi$ starting $x\in\mathcal{P}$ that leaves the set $K$ immediately. According to Proposition 4.4, such a solution must satisfy (25). Furthermore, under (25), Proposition 4.8 guarantees that the solution $\phi$ must satisfy ($\star$). At the same time, using Proposition 4.2, a solution satisfying ($\star$) cannot leave the set $K$ immediately. This is enough to conclude that such a solution $\phi$ cannot leave $K$ immediately. Furthermore, by continuity of solutions, we conclude that any solution $\phi$ starting from $\partial K\cap\mathrm{int}(C)$ must also satisfy ($\star$). Hence, by Proposition 4.2, it cannot leave the set $K$ immediately neither. Thus, every solution $\phi$ starting from $\partial K$ cannot leave the set $K$ immediately, which entails forward invariance of $K$.

5 Discussions

We give counterexamples demonstrating the importance of the proposed assumptions, followed by a discussion of alternative versions of Assumption 3 from the literature.

5.1 Counterexamples

In the following example, Assumption 1 is not verified.

Example 5.1**.**

Let $F(x):=\{[1,0]^{\top}\}$ and

[TABLE]

Having $\mathcal{P}=\{0\}$ and $T_{K}(0)=\{v\in\mathbb{R}^{2}:v_{1}\leq 0,~v_{2}\geq 0\}$, we conclude that Assumption 1 does not hold because $F(0)\not\in T_{K}(0)$. All the other assumptions do hold. Indeed,

[TABLE]

fulfilling Assumption 2. Moreover, all the items in Assumption (3) hold. Indeed, the first item holds by Proposition 6.5 in the forthcoming Section 6. The second item holds by sleekness of the set $C$. To show that $(\dagger)$ holds, we note first that

[TABLE]

Now, for any neighborhood $\mathcal{N}(0)$, we let

[TABLE]

so that

[TABLE]

Furthermore, for

[TABLE]

it holds that $y-z=w_{z}-w_{y}$. Moreover,

[TABLE]

Therefore, $|[w_{z}^{\top}~w_{y}^{\top}]|\leq\gamma|y-z|$ if $\gamma^{2}\geq 1+\frac{4(y_{1}^{2}+y_{1}^{2}z_{2})}{|y-z|^{2}}.$ Besides, for any $k>0$, we have $4(y_{1}^{2}+y_{1}^{2}z_{2})/(|y-z|^{2})<k$ whenever $z_{2}$ and $y_{1}$ are small enough. This is enough to conclude that ($\dagger$) holds at $x=0$, for $\alpha=0$ and $c=\sqrt{1+k}$, by choosing a small enough $\mathcal{N}(0)$. The phase portrait of this system including the solution $\phi$ is represented in Figure 1.

At the same time, (8) holds since for all $x\in\partial K\cap\text{int}(C)$,

[TABLE]

However, the solution $\phi(t):=[t,0]^{\top}$ leaves the set $K$.

In the next example, Assumption 2 is not verified.

Example 5.2**.**

Let $F(x):=\{[1,-x_{1}]^{\top}\}$ and

[TABLE]

Consider also the set $K:=\{x\in C:x_{2}\geq 0\}$. As in Example 5.1, $\mathcal{P}:=\{0\}$. We further note that

[TABLE]

[TABLE]

Since $F(0)=\{[1,0]^{\top}\}\subset T_{\partial K\cap\partial C}(0)$, we conclude that Assumption 2 does not hold. However, Assumption 1 holds, under (42). All the items in Assumption 3 hold as well: The first two items follow directly form (43) and (44), and $(\dagger)$ holds trivially because $\partial K\backslash C=\emptyset$. At the same time, (8) also holds trivially because $\partial K\cap\mathrm{int}(C)=\emptyset$. However, the solution $\phi(t):=[t,-t^{2}/2]^{\top}$ leaves the set $K$. The phase portrait of this system including the solution $\phi$ is represented in Figure 2.

In the last example, Assumption 3 is not verified.

Example 5.3**.**

Consider the system $\Sigma$ with

[TABLE]

Note that $\mathcal{P}=\{0\}$ and Assumption 1 holds by Lemma 7.5 in the Appendix. Moreover, Assumption 2 holds because

[TABLE]

However, we can show that the third item in Assumption 3 is not verified. To do so, we pick any neighborhood $\mathcal{N}(0)$, $c>0$, and $\alpha\in[0,1)$, and show the existence of $z\in(\partial K\backslash C)\cap\mathcal{N}(0)$ and $y\in(\partial C\backslash\partial K)\cap\mathcal{N}(0)$ such that, for any $(v,w)\in T_{\partial K}(z)\times T_{C}(y)$,

[TABLE]

First, $z\in(\partial K\backslash C)\cap\mathcal{N}(0)$ and $y\in(\partial C\backslash\partial K)\cap\mathcal{N}(0)$ imply that, for some $\epsilon>0$, we have

[TABLE]

Furthermore, we note that

[TABLE]

Next, we choose $z$ and $y$ such that $y_{1}-z_{1}=0$, which implies that $y-z=\frac{1}{2}[0,~-y_{1}^{2}]^{\top}$.

The left-hand side in (45) entails that

[TABLE]

for some $a_{1},a_{2}\in(-1,1)$. This is possible only when

[TABLE]

Hence,

[TABLE]

and, at the same time, $c^{2}|z-y|^{2}=c^{2}\frac{z_{1}^{4}}{4}$. By choosing $y_{1}$ sufficiently small such that $(1+a_{2})^{2}(1+1/y_{1}^{2})>c^{2}$, it follows that $|[v^{\top}~w^{\top}]|^{2}>c^{2}|z-y|^{2}$.

This being said, it is easy to see that (8) also holds since, for all $x\in\partial K\cap\mathrm{int}(C)$,

[TABLE]

Nevertheless, $K=\{x\in\mathbb{R}^{2}:x_{1}\geq 0,~x_{2}\geq 0\}$ is not forward invariant since the solution $\phi(t):=[t,-t^{2}/2]^{\top}$ leaves the set $K$. The phase portrait of this system including the solution $\phi$ is represented in Figure 3.

5.2 Alternatives to Assumption 3

We recall that Assumption 3 is used to ensure (14) and (16). According to Lemma 7.3 in the Appendix, ($\dagger$) in Assumption 3 is verified for a given $x\in\partial K\cap\partial C$ if

• ($\ddagger$)

$\exists\mathcal{N}(x)$ a neighborhood around $x$ such that

[TABLE]

Although ($\ddagger$) is in general more restrictive than ($\dagger$) in Assumption 3, see Example 5.4 below, it will allow us to verify ($\dagger$) using a simpler and more tractable assumption when the sets $(C,K)$ are practical.

Example 5.4**.**

Let

[TABLE]

At $0\in\mathcal{P}=\partial K\cap\partial C=K\cap C$, we can see that

[TABLE]

As a result,

[TABLE]

At the same time, for any $\mathcal{N}(0)$, we will find $(x,z)\in(\partial K\backslash C)\cap\mathcal{N}(0)\times(\partial C\backslash\partial K)\cap\mathcal{N}(0)$ such that

[TABLE]

Indeed, let $z:=[0,\epsilon,0]^{\top}\in\partial C\backslash\partial K$, for $\epsilon>0$, and let $x:=[5\epsilon^{2},2\epsilon,0]^{\top}\in\partial K\backslash C$. Now, for $\epsilon>0$ is sufficiently small, it conclude that which also verifies $(x,z)\in\mathcal{N}(0)\times\mathcal{N}(0)$. At the same, we can note that

[TABLE]

As a result, for any neighborhood $\mathcal{N}(0)$,

[TABLE]

and $(\ddagger)$ does not hold.

At the same time, ($\dagger$) trivially holds at $x=0$, for $\alpha=0$, $c=1$, $v_{2}=0$, and $v_{1}=x_{2}-x_{1}$, since $T_{\partial K}(0)=\{x\in\mathbb{R}^{3}:x_{3}=0\}.$

Remark 5.5**.**

While we succeeded to show that ($\dagger$) in Assumption 3 is enough to conclude $T_{\partial K\cap C}(x)=T_{\partial K}(x)\cap T_{C}(x)$ for all $x\in\mathcal{P}$, the latter equality is usually established in existing literature, at a given $x\in\mathcal{P}$, under one of the following (more-restrictive) conditions.

The sets $\partial K$ and $C$ are sleek and $T_{\partial K}(x)\cap\text{int}(T_{C}(x))\neq\emptyset$; see **[31]**. 2. 2.

There exists $v_{o}\in C_{\partial K}(x)$ such that

[TABLE]

see **[2, Proposition 4.3.7]**. 3. 3.

$T_{C}(x)=T^{a}_{C}(x)$* and $C_{\partial K}(x)-C_{C}(x)=\mathbb{R}^{n}$; see [2, Corollary 4.3.5].* 4. 4.

The sets $\partial K$ and $C$ are sleek and $N^{L}_{\partial K}(x)\cap N^{L}_{C}(x)=\{0\};$ see **[30, Theorem 6.42]**, where $N^{L}_{K}(x):=\limsup_{y\rightarrow x}N^{P}_{K}(y)$ and $N^{P}_{K}$ is the proximal normal cone.

6 Result for Practical Sets

In this section, we show that the propagation property in Assumption 1 is not needed when the sets $K$ and $C$ are practical. We also derive simpler versions of Assumptions 2 and 3.

We start recalling the notion of practical sets; see [7, Definition 4.9].

Definition 6.1** (Practical set).**

$K\subset\mathbb{R}^{n}$* is practical if there exists $d_{K}\in\{1,2,...\}$ and continuously differentiable functions $g_{K}^{i}:\mathbb{R}^{n}\to\mathbb{R}$ for all $i\in[d_{K}]$ such that*

[TABLE]

where $[d_{K}]:=\{1,2,\ldots,d_{K}\}$. Furthermore, for all $x\in\partial K$, there exists $v_{o}\in\mathbb{R}^{n}$ such that

[TABLE]

where

[TABLE]

is the set of active equality constraints at $x$.

Intuitively, condition (48) ensures that the active constraints intersect “cleanly” without forming cusps or degenerate tangencies. This guarantees that the local geometry of the set is faithfully captured by the linearization of the function $g_{K}(x)$. Consequently, the tangent cone $T_{K}(x)$ at every point $x\in K$ admits the algebraic representation

[TABLE]

Condition (48) is known as transversality in [2, Proposition 4.3.7] and [23, Assumption 1], and, as proven in [36], it is equivalent to the constraint-qualification condition in [30, Theorem 6.14].

In addition to the set $K$, we also consider the set $C\subset\mathbb{R}^{n}$ to be practical. In particular,

[TABLE]

where $d_{C}\in\mathbb{N}$ and $g_{C}^{i}:\mathbb{R}^{n}\to\mathbb{R}$ is continuously differentiable for all $i\in[d_{C}]$.

We will show later that the decisiveness property in Assumption 2 is verified under the following assumption.

Assumption 4** (Decisiveness for practical sets).**

For each $x\in\mathcal{P}$ and for each $\eta\in F(x)$ such that

[TABLE]

we have

[TABLE]

$\bullet$**

Furthermore, the transversality property in Assumption 3 can also be simplified to the following (more restrictive) one.

Assumption 5** (Transversality for practical sets).**

For each $x\in\mathcal{P}\subset\partial K\cap\partial C$ and for each $j\in\mathrm{Act}_{K}(x)$, there exists $v_{j}\in\mathbb{R}^{n}$ such that

[TABLE]

$\bullet$**

Assumption 5 guarantees, for a given $x\in\mathcal{P}$, the existence of a vector $v_{j}$, for all $j\in\text{Act}_{K}(x)$, that is tangent to the active boundary $j$ via (49). Simultaneously, $v_{j}$ points strictly inward of all the other active constraints of $K$ and $C$ at $x$ by, respectively, satisfying (50) and (51). This preserves tangent cone regularity property $T_{K\cap C}(x)=T_{K}(x)\cap T_{C}(x)$, and allows us to verify the first bullet in Assumption 3, i.e., $T_{\partial K}(x)=T^{a}_{\partial K}(x)$.

Remark 6.2** (Polytopic sets).**

We show in Lemma 7.11 in the Appendix that Assumption 3 holds for free when the sets $K$ and $C$ are polytopic, i.e., when

[TABLE]

However, Assumption 5 does not necessarily hold.

We are now ready to deduce a characterization of forward invariance of $K$ for $\Sigma$, when $(C,K)$ are practical.

Theorem 6.3**.**

Consider a practical set $K\subset\mathbb{R}^{n}$ and $\Sigma$ with the set $C$ therein also practical. Forward invariance of $K$ is equivalent to

[TABLE]

*provided that one of the following is verified.

-Assumptions 4 and 5 hold.

-Assumption 4 holds and $(K,C)$ are polytopic.

• Assumption 5 holds and $\partial K\cap\text{int}(C)$ is dense in $\partial(K\cap C)$.*

Remark 6.4**.**

In view of the aforementioned result, we conjecture that third bullet in Theorem 6.3 can be relaxed to $K$ being the closure of its interior.

6.1 Intermediate Results

Proposition 6.5**.**

Let $K\subset\mathbb{R}^{n}$ be practical and let $x\in\partial K$. Then,

$T_{K}(x)=\left\{v\in\mathbb{R}^{n}:\;\nabla g_{K}^{j}(x)^{\top}v\leq 0\quad\forall j\in\mathrm{Act}_{K}(x)\right\}$. 2. 2.

The map $K\ni x\leadsto T_{K}(x)$ is lower semicontinuous at $x$ and $T_{K}(x)=T^{a}_{K}(x)=C_{K}(x).$ 3. 3.

$D_{K}(x)=\left\{v\in\mathbb{R}^{n}:\nabla g_{K}^{j}(x)^{\top}v<0~~\forall j\in\mathrm{Act}_{K}(x)\right\}$. 4. 4.

$T_{\partial K}(x)=\partial T_{K}(x)=$* *

$\left\{v\in T_{K}(x):\exists j\in\mathrm{Act}_{K}(x):\nabla g_{K}^{j}(x)^{\top}v=0\right\}$.

Additionally, when the following condition holds,

• (Cd)

$\forall j\in\mathrm{Act}_{K}(x)$, there exists $v_{j}\in\mathbb{R}^{n}$ such that

[TABLE]

then,

For each $j\in\text{Act}_{K}(x)$, $\partial K_{j}\ni x\leadsto T_{\partial K_{j}}(x)$ is lower semicontinuous at $x$ and $T_{\partial K_{j}}(x)=C_{\partial K_{j}}(x)$, where

[TABLE]

• 6)

$T_{\partial K}(x)=T^{a}_{\partial K}(x)$.

Proof 6.6**.**

The first item follows from [2, Proposition 4.3.7].

To prove the second item, we consider $x\in\partial K$ and a neighborhood $\mathcal{N}(x)$ around $x$. We already know that, for any $y\in\mathcal{N}(x)\cap\partial K$,

[TABLE]

Furthermore, using a continuity argument, we conclude for $\mathcal{N}(x)$ sufficient small that

[TABLE]

Furthermore, note that for any $i\in[d_{K}]\backslash\mathrm{Act}_{K}(x)$, it holds that $g_{K}^{i}(x)<0$. Hence, by continuity of $g_{K}^{i}$, there exists a neighborhood, $\mathcal{N}_{i}(x)$ such that $g_{K}^{i}(y)<0$ for all $y\in\mathcal{N}_{i}(x)$. By taking $\mathcal{N}(x):=\bigcap_{i\in[d_{K}]\backslash\mathrm{Act}_{K}(x)}\mathcal{N}_{i}(x)$, it follows that

[TABLE]

Combining the latter to continuity of $\nabla g_{K}^{i}$ for all $i\in\mathrm{Act}_{K}(x)$, we conclude that, for any sequence $\{y_{i}\}^{\infty}_{i=1}\subset\mathcal{N}(x)\cap\partial K$ that converges to $x$, we have

[TABLE]

The latter plus the fact that $\nabla g_{K}^{k}(x)\neq 0$ for all $k\in\mathrm{Act}_{K}(x)$ are enough to conclude lower semicontinuity of $K\ni y\leadsto T_{K}(y)$ at $x$.

To prove the third item, we use sleekness of the set $K$ to conclude that $C_{K}(x)=T_{K}(x)$ for all $x\in\partial K$. Moreover, from [31, Theorem 2], we know that $\mathrm{int}(T_{K}(x))=\mathrm{int}(C_{K}(x))\subset D_{K}(x)$. On the other hand, taking into account that $D_{K}(x)\subset T_{K}(x)$ and that $D_{K}(x)$ is open, we conclude that $D_{K}(x)\subset\mathrm{int}(T_{K}(x))$. Hence, $D_{K}(x)=\mathrm{int}(T_{K}(x))$. Finally, since $\nabla g_{K}^{j}(x)\neq 0$ for all $j\in\mathrm{Act}_{K}(x)$, we conclude that

[TABLE]

To prove the fourth item, we combine the third item to [5, Theorem 4.3.3], to obtain

[TABLE]

To prove the fifth item, we pick $j\in\text{Act}_{K}(x)$ and show the existence of a neighborhood $\mathcal{N}(x)$ such that, for each $y\in\mathcal{N}(x)\cap\partial K_{j}$, there exists $\bar{v}_{j}(y)\in\mathbb{R}^{n}$ such that

[TABLE]

First, for $\mathcal{N}(x)$ sufficiently small, we use a continuity argument to conclude

[TABLE]

Note that $v_{j}\in\text{Ker}\left(\nabla g_{K}^{j}(x)^{\top}\right)$. Since $\nabla{g_{K}^{j}}(\cdot)^{\top}$ is continuous and $\nabla g_{K}^{j}(x)^{\top}\neq 0$, we conclude that for $\mathcal{N}(x)$ sufficiently small, there exists a continuous map $y\mapsto\bar{v}_{j}(y)\in\text{Ker}\left(\nabla g_{K}^{j}(y)^{\top}\right)$ defined on $\mathcal{N}(x)\cap\partial K_{j}$ that converges to $v_{j}$ as $y$ converges to $x$. At the same time, by continuity, when $\mathcal{N}(x)$ small enough we conclude that

[TABLE]

which is enough to conclude (54) under (55).

After that, we use [2, Proposition 4.3.7] under (54) to conclude that, for each $y\in\partial K_{j}\cap\mathcal{N}(x)$, we have

[TABLE]

Furthermore, since the set $K$ is practical, it follows by the second item that $T_{K}(x)=C_{K}(x)$ for all $x\in\partial K$, and $\partial K\ni y\leadsto T_{K}(y)$ is lower semicontinuous. Now, using continuity of $\nabla g_{K}^{j}$ and since $\nabla g_{K}^{j}(y)\neq 0$ for all $y\in\partial K_{j}\cap\mathcal{N}(x)$, we conclude that $\partial K_{j}\ni y\leadsto T_{\partial K_{j}}(y)$ is lower semicontinuous at $x$ and thus $T_{\partial K_{j}}(x)=C_{\partial K_{j}}(x)$.

Finally, to prove the sixth item, we let $v\in T_{\partial K}(x)$ for some $x\in\partial K$ and we show that $v\in T^{a}_{\partial K}(x)$. According to the fourth item, there exists $j\in\text{Act}_{K}(x)$ such that $\nabla g^{j}_{K}(x)v=0$. Let us define the vector function

[TABLE]

Note that, for all $\beta\in(0,1]$, we have

[TABLE]

Furthermore, since $T^{a}_{\partial K}(x)$ is closed, then $v\in T^{a}_{\partial K}(x)$ provided that $w_{j}(\beta)\in T^{a}_{\partial K}(x)$ for all $\beta\in(0,1]$. Hence, to ease the notation, we pick $w\in T_{\partial K}(x)$ that satisfies the same properties as $w_{j}(\beta)$ when $\beta\in(0,1]$, i.e.,

[TABLE]

and we show that $w\in T^{a}_{\partial K}(x)$. We do so by considering an arbitrary sequence $\{h_{k}\}\subset\mathbb{R}_{>0}$ such that $h_{k}\rightarrow 0$, and we let $y_{k}:=\text{Proj}_{\tilde{K}_{j}}(x+h_{k}w)$ be the projection of $x_{k}:=x+h_{k}w$ onto the set $\tilde{K}_{j}:=\{x\in\mathbb{R}^{n}:g^{j}_{K}(x)=0\}.$

We will verify the following two items, which are enough to conclude that $w\in T^{a}_{\partial K}(x)$.

• •

$y_{k}\in\partial K$* when $k$ is sufficiently large.*

• •

$w_{k}:=(y_{k}-x)/h_{k}\rightarrow w$.

To show the second item, we start showing that the sequence $\{w_{k}\}_{k}$ is uniformly bounded. Indeed, given $k\in\mathbb{N}$, we have

[TABLE]

Hence,

[TABLE]

and, at the same time,

[TABLE]

Leading to

[TABLE]

Next, since $y_{k}$ is the projection of $x_{k}$ onto $\tilde{K}_{j}$, we conclude for $k$ large enough that

[TABLE]

As a result, we obtain

[TABLE]

where the function $o(\cdot)$ is such that $\lim_{s\rightarrow 0}o(s)/|s|=0$.

Furthermore, we express $y_{k}$ as

[TABLE]

and we use the fact that $\nabla g^{j}_{K}(x)^{\top}w=0$, which leads to

[TABLE]

Using continuity of $\nabla g^{j}_{K}$, we conclude that

[TABLE]

Moreover, by boundedness of the sequence $\{w_{k}\}_{k}$, we conclude that

[TABLE]

Hence, $(w_{k}-w)\rightarrow 0$.

Finally, it remains to prove the first item, i.e., to show that $y_{k}=x+h_{k}w_{k}\in\partial K$ for large values of $k$. Indeed, it holds by definition that $g^{j}_{K}(y_{k})=0$ for all $k$. Furthermore, given $i\in\text{Act}_{K}(x)\backslash\{j\}$, we have

[TABLE]

By the properties of $w$ in (56), we get $\nabla g^{i}_{K}(x)^{\top}w<0$, which implies that $g^{i}_{K}(y_{k})\leq 0$ when $k$ is large, under boundedness of $\{w_{k}\}_{k}$ and since $(w_{k}-w)\rightarrow 0$. Hence, $y_{k}\in\partial K$ when $k$ is sufficiently large.

Proposition 6.7**.**

For practical sets $K,C\subset\mathbb{R}^{n}$, and $\mathcal{P}\subset\partial K\cap\partial C$, Assumption 5 $\Rightarrow$ Assumption 3.

Proof 6.8**.**

Given $x\in\mathcal{P}$, the first and the second bullets in Assumption 3 are verified using the second and the sixth items in Proposition 6.5.

Next, to verify ($\dagger$), we propose to verify ($\ddagger$) for the sets $(\partial K_{j},C)$, instead of the sets $(\partial K,C)$, for all $j\in\text{Act}_{K}(x)$, where $K_{j}$ was defined in (53).

Then, we apply Lemma 7.3 in the Appendix to conclude that ($\dagger$) holds for $(\partial K_{j},C)$, instead of $(\partial K,C)$, for all $j\in\text{Act}_{K}(x)$. In particular, for all $j\in\text{Act}_{K}(x)$, we conclude the existence of a neighborhood $\mathcal{N}(x)$, $c_{j}>0$, and $\alpha_{j}\in[0,1)$ such that,

[TABLE]

there exists $(v_{j},w_{j})\in T_{\partial K_{j}}(x_{1j})\times T_{C}(x_{2})$ such that

[TABLE]

Now, to verify ($\dagger$) for $(\partial K,C)$, we note that when $\mathcal{N}(x)$ is sufficiently small, then, for all $x_{1}\in((\partial K\backslash C)\cap\mathcal{N}(x))$, there exists $j_{1}\in\text{Act}_{K}(x)$ such that $x_{1}\in(\partial K_{j_{1}}\backslash C)\cap\mathcal{N}(x)$. Hence, by letting

[TABLE]

we conclude the existence of

[TABLE]

such that $|[v^{\top},w^{\top}]|\leq c|x_{1}-x_{2}|$ and $x_{2}-x_{1}\in v-w+\alpha(|x_{1}-x_{2}|)\mathbb{B}$.

Let us now verify ($\ddagger$) for the sets $(\partial K_{j},C)$, instead of the sets $(\partial K,C)$, for all $j\in\text{Act}_{K}(x)$. To do so, we note that the set $\partial K$ can be expressed as $\partial K=\bigcup_{j=1}^{d_{K}}\partial K_{j}$. Next, given $x\in\partial K$, we note that $T_{\partial K}(x)=\bigcup_{j=1}^{d_{K}}T_{\partial K_{j}}(x)=\bigcup_{j\in\text{Act}_{K}(x)}T_{\partial K_{j}}(x)$. Furthermore, given $x\in\mathcal{P}$ and $j\in\text{Act}_{K}(x)$, under  (49)-(50), we apply the fifth item in Proposition 6.5 to conclude that

[TABLE]

Now, we consider $v_{j}$ from Assumption 5 and a sufficiently small neighborhood, $\mathcal{N}(x)$, around $x$ such that

[TABLE]

[TABLE]

Such a neighborhood $\mathcal{N}(x)$ exists by continuity and since $\nabla g_{C}^{k}(x)^{\top}v_{j}<0$ for all $k\in\mathrm{Act}_{C}(x)$. Furthermore, using (57), we conclude that $v_{j}\in T_{\partial K_{j}}(x)=C_{\partial K_{j}}(x)$ and $w_{j}:=v_{j}-(x_{2}-x_{1j})\in C_{C}(x)$. Hence,

[TABLE]

we are able to find $(w_{j},v_{j})\in C_{C}(x)\times C_{\partial K_{j}}(x)$ such that $x_{2}-x_{1j}=v_{j}-w_{j}$.

Note that the equivalence in the following proposition is established in Lemma 7.9 in the Appendix, provided Condition (Cd) from Proposition 6.5 holds with the set $K$ therein replaced by $C$.

Proposition 6.9**.**

For practical sets $K,C\subset\mathbb{R}^{n}$ and $\mathcal{P}\subset\partial K\cap\partial C$, Assumption 4 $\Rightarrow$ Assumption 2.

Proof 6.10**.**

The proof is based on characterizing the maps $\partial K\ni x\leadsto T_{\partial K}(x)$ and $\partial C\ni x\leadsto T_{\partial C}(x)$ in terms of $\{g^{i}_{K}\}^{d_{K}}_{i=1}$ and $\{g^{k}_{C}\}^{d_{C}}_{k=1}$, respectively. Indeed, we already know that for all $x\in\mathcal{P}$,

[TABLE]

Plus Assumption 4, we verify Assumption 2.

6.2 Proof of Theorem 6.3

The necessity of (8) follows from Theorem 4.1 and the first item in Proposition 6.5.

To prove sufficiency of (8), we start using Lemma 7.7 in the Appendix to obtain from Assumption 5 that the propagation property in Assumption 1 holds, i.e.,

[TABLE]

Now, we verify Assumptions 2 and 3 under either one of the first two items in Theorem 6.3, which is enough to conclude sufficiency of (8), thanks to Theorem 4.1. Indeed, using Proposition 6.9, we conclude that the decisiveness property in Assumption 2 is verified under Assumption 4. Furthermore, Proposition 6.7 can be used to conclude that the transversality property in Assumption 3 is verified when Assumption 5 holds. Finally, Lemma 7.11 in the Appendix can be used to conclude that Assumption 3 holds naturally when $(K,C)$ are polytopic.

Under the third item in Theorem 6.3, we note that

[TABLE]

Furthermore, by the density property, we conclude that $\text{int}(K)\cap\partial C=\emptyset$ and that $\mathcal{P}=\partial K\cap\partial C$. Hence, (58) can be expressed as $F(x)\subset T_{K}(x)$ for all $x\in(\partial K\cap\partial C)$. In addition to (8), we conclude that

[TABLE]

Note that

[TABLE]

Hence, if we show that

[TABLE]

Then, we would conclude that

[TABLE]

which proves forward invariance of the set $K$.

To verify (59), it is enough to show that

[TABLE]

Indeed, under Assumption 5, we have shown in Proposition 6.7 that ($\dagger$) is verified. At the same time, ($\dagger$) entails the existence of a neighborhood of $x$, $\mathcal{N}(x)$, $c>0$, and $\alpha\in[0,1)$ such that

[TABLE]

for some $(v_{1},v_{2})\in T_{K}(x_{1})\times T_{C}(x_{2})$ such that $|\left[v_{1}^{\top},v_{2}^{\top}\right]|\leq c|x_{1}-x_{2}|$. Hence, according to Lemma 7.1 in the Appendix, we conclude that

[TABLE]

So, let us now verify (60) using contradiction. That is, we pick $x\in\partial K\cap\partial C$ and we assume the existence of $\eta\in F(x)$ such that $\eta\notin T_{C}(x)$. As a result, there exists $L_{x}\subset\text{Act}_{C}(x)$ such that

[TABLE]

Next, we introduce the vector function

[TABLE]

where $\{v_{j}\}_{j\in\text{Act}_{K}(x)}$ is introduced in Assumption 5.

Using continuity of $v$, we conclude the existence of $\lambda^{*}>0$ such that, for $v^{*}:=v(\lambda^{*})$, we have

[TABLE]

Additionally, since $\eta\in T_{K}(x)$, we conclude that

[TABLE]

Finally, we will show that a consequence of (61)-(62) is that $\partial C\cap\text{int}(K)\neq\emptyset$, which yields to a contradiction. Indeed, by Proposition 6.5, (62) implies that $v^{*}\in D_{K}(x)$ and (61) implies that $v^{*}\in T_{\partial C}(x)$. Hence, by definition of $T_{\partial C}$, we conclude the existence of $\{t_{i}\}^{\infty}_{i=1}\subset\mathbb{R}_{>0}$, $\{v_{i}\}^{\infty}_{i=1}\subset\mathbb{R}^{n}$ such that $t_{i}\rightarrow 0^{+}$, $v_{i}\rightarrow v^{*}$, and $y_{i}:=x+t_{i}v_{i}\in\partial C$ for all $i\in\{1,2,...\}$. At the same time, by definition of Dubovitskiy cone $D_{K}$, we conclude that $y_{i}\in\text{int}(K)\cap\partial C$ for all $i$ sufficiently large. $\blacksquare$

7 Conclusion

This paper provided a fine extension of Nagumo’s invariance criterion that is suitable for constrained systems. For the proposed condition to imply forward invariance, three assumptions are imposed and their importance is highlighted via counterexamples. Dropping the second assumption at the expense of hardening the transversality condition is the subject of further investigations. The main result is further specialized to practical sets, which allowed us to drop one of the assumptions and to provide a simpler and more tractable formulation of the remaining ones.

Appendix

The following lemma is inspired by [2], and its proof mimics the arguments used in the proof of the constrained inverse function theorem; see [2, Section 3.4.2 and the proof of Theorem 3.4.5].

Lemma 7.1**.**

Consider closed subsets $K_{1}$, $K_{2}\subset\mathbb{R}^{n}$ with $K_{1}$ derivable on $\partial K_{1}\cap\partial K_{2}$, i.e.,

[TABLE]

Given $x\in\partial K_{1}\cap\partial K_{2}$, we assume the following property.

• ($\diamond$)

$\exists\mathcal{N}(x)$* a neighborhood of $x$, $c>0$, and $\alpha\in[0,1)$ such that*

[TABLE]

for some $(v_{1},v_{2})\in T_{K_{1}}(x_{1})\times T_{K_{2}}(x_{2})$ such that $|[v_{1}^{\top}~v_{2}^{\top}]|\leq c|x_{1}-x_{2}|$.

Then,

[TABLE]

Proof 7.2**.**

It is straightforward to show that

[TABLE]

Hence, to establish (64)-(65), it is enough to show that

[TABLE]

In the sequel, we only verify  (66)-(67) can be verified using the exact same arguments.

Consider a vector $v\in T_{K_{1}}(x)\cap T_{K_{2}}(x)$ for some $x\in\partial K_{1}\cap\partial K_{2}$. By definition of the contingent cone, there exist $\{t_{i}\}_{i}\rightarrow 0$ and $\{w_{i}\}_{i}\rightarrow v$ such that $x_{2i}:=x+t_{i}w_{i}\in K_{2}$. Now, using (63), we conclude that, for the same sequence $\{t_{i}\}_{i}$, there exists $\{z_{i}\}_{i}\rightarrow v$ such that $x_{1i}:=x+t_{i}z_{i}\in K_{1}$.

The next step consists in showing the existence of $\{(x^{*}_{1i},x^{*}_{2i})\}_{i}\subset K_{1}\times K_{2}$, $l>0$, and $\sigma:\{1,2,\ldots\}\rightarrow\{1,2\}$ such that, through an appropriate subsequence,

[TABLE]

[TABLE]

The latter would imply that

[TABLE]

and $x^{*}_{\sigma(i)i}=x+t_{i}v_{i}\in K_{1}\cap K_{2}$, which is enough to show that $v\in T_{K_{1}\cap K_{2}}(x)$.

Note that, when $x_{1i}\in K_{2}\cap K_{1}$ (resp., $x_{2i}\in K_{1}\cap K_{2}$), then we just take $x^{*}_{1i}=x^{*}_{2i}=x_{1i}$ (resp., $x^{*}_{1i}=x^{*}_{2i}=x_{2i}$) to conclude that the inequality in (69) holds for $l=1$ and $\sigma(i)=1$ (resp., $\sigma(i)=2$).

In the sequel, we use the transversality condition ($\diamond$) to prove the following claim.

Claim 1**.**

Given $\{x_{i}:=(x_{1i},x_{2i})\}_{i}\subset(K_{1}\backslash K_{2})\times(K_{2}\backslash K_{1})$ such that $x_{i}\rightarrow(x,x)$, there exists $\{(x^{*}_{1i},x^{*}_{2i})\}_{i}\subset K_{1}\times K_{2}$, $l>0$, and $\sigma:\{1,2,\ldots\}\rightarrow\{1,2\}$ such that (68) and (69) hold at least through a certain subsequence.

Proof of Claim 1:* We introduce the function*

[TABLE]

Then, applying the Ekeland Variational Principle to the function $|f(\cdot)|$, we conclude that, for each $\varepsilon>0$, there exists a sequence $\{\bar{x}_{i}:=(\bar{x}_{1i},\bar{x}_{2i})\}\subset K_{1}\times K_{2}$ such that, for all $i\in\{1,2,\ldots\}$, we have

[TABLE]

Using the first inequality, we conclude that for $l:=2/\varepsilon$,

[TABLE]

Hence, we conclude that $\bar{x}_{i}\rightarrow(x,x)$.

The proof is completed if we show , for a certain choice of $\varepsilon>0$, the existence of $\sigma:\{1,2,\ldots\}\rightarrow\{1,2\}$ such that, after passing to an appropriate subsequence, we find $\{\bar{x}_{\sigma(i)i}\}_{i}\subset K_{1}\cap K_{2}$.

To find a contradiction, we assume that, for any choice of $\varepsilon>0$, we have

[TABLE]

Next, we show the existence of $\bar{c}>0$ and $\bar{\alpha}\in[0,1)$ such that, for $\bar{x}_{i}\in(K_{1}\backslash K_{2})\times(K_{2}\backslash K_{1})$, there is $v_{i}:=(v_{1i},v_{2i})\in T_{K_{1}}(\bar{x}_{1i})\times T_{K_{2}}(\bar{x}_{2i})$ and $w_{i}\in\mathbb{R}^{n}$ such that

[TABLE]

[TABLE]

Indeed, the transversality assumption ($\diamond$) allows us to conclude, when $\bar{x}_{i}\in(\partial K_{1}\backslash K_{2})\times(\partial K_{2}\backslash K_{1}),$ the existence of $v_{i}:=(v_{1i},v_{2i})\in T_{K_{1}}(\bar{x}_{1i})\times T_{K_{2}}(\bar{x}_{2i})$ and $w_{i}\in\mathbb{R}^{n}$ such that $-f(\bar{x}_{i})=\nabla f(\bar{x}_{i})^{\top}v_{i}+w_{i}$ with $|[v_{1i}^{\top}~v_{2i}^{\top}]|\leq c|f(\bar{x}_{i})|$ and $|w_{i}|\leq\alpha|f(\bar{x}_{i})|.$

Otherwise, when $\bar{x}_{1i}\in\text{int}(K_{1})\backslash K_{2}$ (resp. $\bar{x}_{2i}\in\text{int}(K_{2})\backslash K_{1}$), since $T_{K_{1}}(\bar{x}_{1i})=\mathbb{R}^{n}$ (resp., $T_{K_{2}}(\bar{x}_{2i})=\mathbb{R}^{n}$), we can just take $v_{i}:=(\bar{x}_{2i}-\bar{x}_{1i},0)$ (resp. $v_{i}:=(0,\bar{x}_{1i}-\bar{x}_{2i})$) so that (72) holds with $|[v_{1i}^{\top}~v_{2i}^{\top}]|=|f(\bar{x}_{i})|$ and $|w_{i}|=0$. Hence, whenever $\bar{x}_{i}\in(K_{1}\backslash K_{2})\times(K_{2}\backslash K_{1})$, there exists $v_{i}:=(v_{1i},v_{2i})\in T_{K_{1}}(\bar{x}_{1i})\times T_{K_{2}}(\bar{x}_{2i})$ and $w_{i}\in\mathbb{R}^{n}$ such that  (72)-(73) hold with $\bar{c}:=\max\{c,1\}$ and $\bar{\alpha}:=\max\{0,\alpha\}$.

By definition of the contingent cone, we conclude the existence of $h_{p}\rightarrow 0$ and $e_{p}\rightarrow 0$ such that

[TABLE]

and $f(y_{i})=f(\bar{x}_{i})+h_{p}\nabla f(\bar{x}_{i})^{\top}(v_{i}+e_{p})+h_{p}o(h_{p}).$

Using (72), we obtain

[TABLE]

Hence,

[TABLE]

Now, using the second inequality in (70), we obtain $|f(\bar{x}_{i})|-\varepsilon|\bar{x}_{i}-y_{i}|<|f(y_{i})|$. Hence,

[TABLE]

The latter allows us to conclude that

[TABLE]

Dividing by $h_{p}$ and letting $p\rightarrow\infty$, we obtain

[TABLE]

Now, if $\varepsilon>0$ is such that $\bar{\alpha}+\varepsilon\bar{c}<1$, we conclude that $|f(\bar{x}_{i})|=0$ and thus $\bar{x}_{i1}=\bar{x}_{i2}$, contradicting (71).

In the following lemma, we replace ($\diamond$) by a simpler, and yet a more conservative, condition.

Lemma 7.3**.**

Consider closed subsets $K_{1}$, $K_{2}\subset\mathbb{R}^{n}$, and let $x\in\partial K_{1}\cap\partial K_{2}$. The property ($\diamond$) in Lemma 7.1 is verified provided that

• ($\diamond\diamond$)

$\exists$* a neighborhood $\mathcal{N}(x)$ around $x$ such that*

[TABLE]

Proof 7.4**.**

A key argument to prove the lemma consists of using the Open Mapping Theorem [2, Theorem 2.2.1] to conclude the existence of $l>0$ such that, for any $(x_{1},x_{2})\in((\partial K_{1}\backslash K_{2})\cap\mathcal{N}(x))\times((\partial K_{2}\backslash K_{1})\cap\mathcal{N}(x))$, there exists $(c_{1},c_{2})\in C_{K_{1}}(x)\times C_{K_{2}}(x)$ such that

[TABLE]

Indeed, we note that the map $F:C_{K_{1}}(x)\times C_{K_{2}}(x)\rightarrow C_{K_{1}}(x)-C_{K_{2}}(x)$ given by $F(c_{1},c_{2}):=c_{1}-c_{2}$ admits a closed and convex graph which is a cone. Hence, by the Open Mapping Theorem, we conclude that $F^{-1}$ is locally Lipschitz.

Next, we introduce the set

[TABLE]

We can see, under property ($\diamond\diamond$), that

[TABLE]

where $\text{int}_{K}(\cdot)$ stands for the interior of $(\cdot)$ relative to $K$. In other words, there exists $\epsilon>0$ such that $(y_{o}+\epsilon\mathbb{B})\cap K\subset C_{K_{1}}(x)-C_{K_{2}}(x)$. Next, we note that $(c_{1o},c_{2o}):=(0,0)\in F^{-1}(y_{o}).$ The Lipschitzness property of $F^{-1}$ allows us to conclude that, there exists $\epsilon>0$ sufficiently small and $l>0$ such that, $\forall y\in(y_{o}+\epsilon\mathbb{B})\cap K\subset C_{K_{1}}(x)-C_{K_{2}}(x)$, we can find $(c_{1},c_{2})\in C_{K_{1}}(x)\times C_{K_{2}}(x)$ such that $F((c_{1},c_{2}))=c_{1}-c_{2}=y$ and

[TABLE]

Next, since the maps $x\leadsto C_{K_{1}}(x)$ and $x\leadsto C_{K_{2}}(x)$ are lower semicontinuous cones, we conclude that, for each $\varepsilon>0$, there exists a neighborhood $\mathcal{N}(x)$ such that, for each $(c_{1},c_{2})\in C_{K_{1}}(x)\times C_{K_{2}}(x)$ and for each $(x_{1},x_{2})\in((\partial K_{1}\backslash K_{2})\cap\mathcal{N}(x))\times((\partial K_{2}\backslash K_{1})\cap\mathcal{N}(x))$, there exists $(v_{1},v_{2})\in T_{K_{1}}(x_{1})\times T_{K_{2}}(x_{2})$ such that $|[v_{1}^{\top}~v_{2}^{\top}]-[c_{1}^{\top}~c_{2}^{\top}]|\leq\varepsilon|[c_{1}^{\top}~c_{2}^{\top}]|$. As a result,

[TABLE]

where $|w|\leq\varepsilon l|x_{2}-x_{1}|$ and $|[v_{1}^{\top}~v_{2}^{\top}]|\leq(1+\varepsilon)l|x_{2}-x_{1}|$. Finally, we choose $\varepsilon$ small so that $\varepsilon l<1$.

The following lemma allows us to verify Assumption 1 and the first item in Assumption 3 by imposing further regularities on the sets $K$ and $C$.

Lemma 7.5**.**

Consider closed subsets $K$, $C\subset\mathbb{R}^{n}$, and let $\mathcal{P}\subset\partial K\cap\partial C$ such that the following property holds.

• (Pr1)

The map $T_{K}:\partial K\rightrightarrows\mathbb{R}^{n}$ is continuous on $\mathcal{P}$, and $\mathcal{N}(x)\cap\nolinebreak\partial K\cap\text{int}(C)\neq\emptyset$ for any $x\in\mathcal{P}$ and for any neighborhood $\mathcal{N}(x)$ around $x$.

Additionally, we let $F:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ be continuous such that (8) holds. Then, for all $x\in\mathcal{P}$, $F(x)\subset T_{K}(x)$ and $T^{a}_{\partial K}(x)=T_{\partial K}(x)$.

Proof 7.6**.**

Given $x\in\mathcal{P}$, (Pr1) entails the existence of $\{x_{i}\}^{\infty}_{i=1}\subset\partial K\cap\text{int}(C)$ such that $x=\lim_{i}x_{i}$. Moreover, based on (8), we know that $F(x_{i})\subset T_{K}(x_{i})$ for all i $\in\{1,2,\ldots\}$. Hence, since both $T_{K}$ and $F$ are continuous at $x$, it follows that $F(x)\subset T_{K}(x)$.

To prove $T^{a}_{\partial K}(x)=T_{\partial K}(x)$, it is enough to show that the map $T_{\partial K}:\partial K\rightrightarrows\mathbb{R}^{n}$ is continuous at every $x\in\mathcal{P}$. Indeed, we note that the set $K$ is sleek; hence, $C_{K}(x)=T_{K}(x)$. Moreover, from [31, Theorem 2], we know that $\mathrm{int}(C_{K}(x))\subset D_{K}(x)$. On the other hand, taking into account that $D_{K}(x)\subset T_{K}(x)$, and that $D_{K}(x)$ is open, it holds that $D_{K}(x)\subset\mathrm{int}(T_{K}(x))$. Hence, $D_{K}(x)=\mathrm{int}(T_{K}(x))$. Combining the latter to [5, Theorem 4.3.3], we obtain that

[TABLE]

Finally, since $T_{K}:\partial K\rightrightarrows\mathbb{R}^{n}$ is continuous, then $\partial T_{K}\nolinebreak(\nolinebreak=\nolinebreak T_{\partial K}):\partial K\rightrightarrows\mathbb{R}^{n}$ is also continuous.

In the following lemma, we show that Assumption 1 holds for free when $(K,C)$ are practical sets verifying Assumption 5.

Lemma 7.7**.**

Consider practical sets $K,C\in\mathbb{R}^{n}$, let $P\subset\partial K\cap\partial C$ and $F:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ be continuous such that (8) holds. Then Assumption 5 implies Assumption 1.

Proof 7.8**.**

Given $x\in\mathcal{P}$ and $i\in\text{Act}_{K}(x)$, we show that

[TABLE]

where $\partial K_{i}:=\{x\in K:g_{K}^{i}(x)=0\}$.

Under (74), we conclude the existence of a sequence $\{y^{i}_{k}\}^{\infty}_{k=1}\subset\partial K_{i}\cap\text{int}(C)$ such that $y^{i}_{k}\rightarrow x$ as $k\rightarrow\infty$. Furthermore, under (8), we conclude that

[TABLE]

A continuity argument allows us to conclude that

[TABLE]

The latter is enough to verify Assumption 1 according to the first bullet in Proposition 6.5.

Now, to verify (74) for a given $i\in\text{Act}_{K}(x)$, we start using the fourth bullet in Proposition 6.5 to conclude that

[TABLE]

Furthermore, from Assumption 5, there exists a vector $v_{i}\in T_{\partial K_{i}}(x)$ such that

[TABLE]

As a result, there exists $\{w^{i}_{k}\}_{k}\subset\mathbb{R}^{n}$ such that $w^{i}_{k}\rightarrow v_{i}$ as $k\rightarrow\infty$ and $\{h_{k}\}_{k}\subset\mathbb{R}_{>0}$ such that $h_{k}\rightarrow 0$ as $k\rightarrow\infty$ such that $y_{k}:=x+h_{k}w^{i}_{k}\in\partial K_{i}$ for all $k$. Also,

[TABLE]

for all $j\in\mathrm{Act}_{C}(x)$. As a result, for $k$ sufficiently large, we conclude that $g^{j}_{C}(y_{k})<0$ for all $j\in\mathrm{Act}_{C}(x)$, and thus $y_{k}\in\text{int}(C)$ for all $j\in\mathrm{Act}_{C}(x)$, proving (74).

In the following lemma, when $(K,C)$ are practical and (Cd) holds with $K$ therein replaced by $C$, we establish the equivalence between Assumptions 2 and 4.

Lemma 7.9**.**

Consider practical sets $K,C\subset\mathbb{R}^{n}$ and $\mathcal{P}\subset\partial K\cap\partial C$ such that Assumption 5 holds. Assume that for each $x\in\mathcal{P}$ and $j\in\mathrm{Act}_{C}(x)$, there exists $v_{j}\in\mathbb{R}^{n}$ such that

[TABLE]

Then Assumption 2 is equivalent to Assumption 4.

Proof 7.10**.**

The proof is based on showing that

[TABLE]

To verify (75), we show that ($\dagger$) is verified while replacing $C$ therein is by $\partial C$. To do so, we show the following property.

• •

($\ddagger$) holds while replacing $(\partial K,C)$ therein by $(\partial K_{j},\partial C_{i})$ for all $(j,i)\in\text{Act}_{K}(x)\times\text{Act}_{C}(x)$.

Then, applying Lemma 7.3, we conclude the existence of a neighborhood $\mathcal{N}(x)$, $c_{ji}>0$, and $\alpha_{ji}\in[0,1)$ such that

[TABLE]

there exists $(\bar{v}_{j},\bar{w}_{i})\in T_{\partial K_{j}}(x_{1j})\times T_{\partial C_{i}}(x_{2i})$ such that $|[\bar{v}_{j}^{\top}~\bar{w}_{i}^{\top}]|\leq c_{ji}|x_{1j}-x_{2i}|$ and

[TABLE]

Next, for $\mathcal{N}(x)$ sufficiently small, we conclude that

[TABLE]

there exists $j\in\text{Act}_{K}(x)$ and $i\in\text{Act}_{C}(x)$ such that

[TABLE]

Hence, by letting

[TABLE]

we conclude the existence of

[TABLE]

such that $|[v^{\top}~w^{\top}]|\leq c|x_{1}-x_{2}|$ and

[TABLE]

which shows ($\dagger$).

Let us now verify the property in the latter bullet. Given $x\in\mathcal{P}$ and $j\in\text{Act}_{K}(x)$, by considering the vector $v_{j}$ in Assumption 5 and a sufficiently small neighborhood $\mathcal{N}(x)$ around $x$, we have

[TABLE]

Furthermore, by the fifth item in Proposition 6.5, we obtain

[TABLE]

and that

[TABLE]

Next, we introduce the vector

[TABLE]

and we show the existence of $\beta_{j}\in[0,1]$ such that

[TABLE]

Claim 2**.**

There exists $\mathcal{N}(x)$ sufficiently small such that,

[TABLE]

there exist $\beta_{j}\in[0,1]$ and $l_{j}\in\text{Act}_{C}(x)$ such that $w_{j}(\beta_{j})\in T_{C}(x)$ and $\nabla g_{C}^{l_{j}}(x)^{\top}w_{j}(\beta_{j})=0$.

Under the latter claim, since

[TABLE]

we conclude the existence of $i\in\text{Act}_{C}(x)$ such that $w_{j}(\beta_{j})\in T_{\partial C_{i}}(x)$. Furthermore, under the first bullet in Assumption 4, we can apply the fifth bullet in Proposition 6.5 (while replacing $K$ therein by $C$) to obtain

[TABLE]

As a consequence,

[TABLE]

*we are able to find $(w^{*}_{i},v^{*}_{j}):=(w_{j}(\beta_{j}),\beta_{j}v_{j})\in C_{\partial C_{i}}(x)\times C_{\partial K_{j}}(x)$ such that $x_{2i}-x_{1j}=v^{*}_{j}-w^{*}_{i}$.

Proof of Claim 2: To find a contradiction, we assume that, for each $\mathcal{N}(x)$ arbitrary small, we can find*

[TABLE]

such that

[TABLE]

In particular, we have

[TABLE]

At the same time, using the fact that $x_{1j}\notin C$, we conclude the existence of $k\in\text{Act}_{C}(x)$ such that

[TABLE]

where the vector function $O(\cdot)$ satisfies $\lim_{\cdot\rightarrow 0}O(\cdot)=0$.

Now, since

[TABLE]

and

[TABLE]

can be made arbitrarily small when the neighborhood $\mathcal{N}(x)$ is so, we conclude that

[TABLE]

Hence, we just showed that $w_{j}(\beta=0)\notin T_{C}(x)$.

On the other hand, we know from (76) that

[TABLE]

As a result, using closeness of $T_{C}(x)$ and continuity of $\beta\mapsto w_{j}(\beta)$, we conclude the existence of $\beta_{j}\in(0,1]$ such that

[TABLE]

This contradicts (77).

In the next lemma, we show that Assumption 3 holds for free when the sets $(K,C)$ are polytopic.

Lemma 7.11**.**

Let $K:=\{x\in\mathbb{R}^{n}:A_{K}x\leq b_{K}\}\subset\mathbb{R}^{n}$, where $A_{K}\in\mathbb{R}^{m_{K}\times n}$, $b_{K}\in\mathbb{R}^{m_{K}}$, and the latter inequality is considered element-wise. Then, for all $x\in\partial K$,

[TABLE]

Let additionally $C:=\{x\in\mathbb{R}^{n}:A_{C}x\leq b_{C}\}\subset\mathbb{R}^{n}$, where $A_{C}\in\mathbb{R}^{m_{C}\times n}$ and $b_{C}\in\mathbb{R}^{m_{C}}$. Then, $(\dagger)$ in Assumption 3 holds for $\mathcal{P}=\partial K\cap\partial C$.

Proof 7.12**.**

For simplicity, we will denote the $j$-th row of $A_{K}$ as $a_{K}^{j}\in\mathbb{R}^{1\times n}$, and the corresponding $j$-th element of $b_{K}$ as $b_{K}^{j}\in\mathbb{R}$. First, using Lemma 7.13 hereafter, it is known that

[TABLE]

Furthermore, we note that, for any $j\in[m_{k}]\backslash\mathrm{Act}_{K}(x)$, it holds that $a_{K}^{j}x<b_{K}^{j}$. Hence, there exits a neighborhood $\mathcal{N}_{j}(x)$ such that $a_{K}^{j}y<b_{K}^{j}$ for all $y\in\mathcal{N}_{j}(x)$. By taking $\mathcal{N}(x):=\bigcap_{j\in[m_{k}]\backslash\mathrm{Act}_{K}(x)}\mathcal{N}_{j}(x)$, it follows that

[TABLE]

This combined with (78) can be used to obtain that $T_{K}(x)\subset T_{K}(y)$ for all $y\in\mathcal{N}(x)\cap{K}$. This proves lower semicontinuity of $x\leadsto T_{K}(x)$ on $K$.

To prove that $T_{\partial K}(x)=T_{\partial K}^{a}(x)$, we let $v\in T_{\partial K}(x)$ and show that $v\in T_{\partial K}^{a}(x)$. For this, we choose a continuous function $\beta:\mathbb{R}_{>0}\to[1,+\infty)$ such that $\lim_{t\to 0^{+}}\beta(t)=1$ and the ratio $t/\beta(t)$ is small enough to verify $x+vt/\beta(t)\in\mathcal{N}(x)$ for all $t\geq 0$. Since $v\in T_{K}(x)$, using (78) and (80), we conclude that $a_{K}^{j}\left(x+tv/\beta(t)\right)\leq b_{K}^{j}$ for all $j\in[m_{K}]$, for all $t\geq 0$. Hence, $x+vt/\beta(t)\in K$ for all $t\geq 0$. Furthermore, since $v\in T_{\partial K}(x)$, then there exists $L_{x}\subset\text{Act}_{K}(x)$ such that

[TABLE]

Hence, $x+vt/\beta(t)\in\partial K$ for all $t\geq 0$. Keeping this in mind, for any sequence $\{t_{i}\}_{i=0}^{\infty}$ that converges to [math], the sequence $\{v_{i}\}_{i=0}^{\infty}$, defined as $v_{i}:=v/\beta(t_{i})$, converges to $v$ and satisfies $x+v_{i}t_{i}\in\partial K$ for all $i\in\{0,1\ldots\}$. This is enough to conclude that $v\in T_{\partial K}^{a}(x)$. Hence, the first item holds.

To prove $(\dagger)$, we pick a point $x\in\partial K\cap\partial C$ and a small enough neighborhood $\mathcal{N}(x)$ such that (80) holds. Hence,

[TABLE]

Given $x_{1}\in(\partial K\backslash C)\cap\mathcal{N}(x)$ and $x_{2}\in(\partial C\backslash\partial K)\cap\mathcal{N}(x)$, we claim the existence of $v_{2}\in\mathbb{R}^{n}$ such that

[TABLE]

Note that $v_{2}\in T_{C}(x_{2})$ in this case and, by taking $v_{1}:=(x_{2}-x_{1})+v_{2}$, we conclude that $a_{K}^{j}v_{1}=0$ for all $j\in\mathrm{Act}_{K}(x_{1})$; hence, $v_{1}\in T_{\partial K}(x_{1})\subset T_{K}(x_{1})$. Also, since $(x_{2}-x_{1})=v_{1}-v_{2}$. We conclude that $(\dagger)$ holds.

We now prove the existence of $v_{2}$ solution to (82). Indeed, let assume the existence of the scalar sequences $\left(\{\lambda_{j}\}_{j\in\text{Act}_{K}(x_{1})},\{\sigma_{l}\}_{l\in\text{Act}_{C}(x_{2})}\right)$ such that

[TABLE]

($v_{2}$ would exist trivially otherwise). Hence, we show that

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Indeed, from (81), we note that

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Finally, using (83), we obtain

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Hence, (84) is verified.

The following lemma provides a characterization of the contingent cone to polytopic sets.

Lemma 7.13**.**

Let $K\subset\mathbb{R}^{n}$ be defined as

[TABLE]

Then,

[TABLE]

where

[TABLE]

and $a^{j}\in\mathbb{R}^{1\times n}$ and $b^{j}\in\mathbb{R}$ denote the $j$-th row of $A$ and $b$, respectively.

Proof 7.14**.**

We will only prove (86). The proof of (87) follows similar arguments, taking into account that

[TABLE]

To show (86), we will prove the two inclusions:

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To prove (88), we pick $w\in T_{K}(x)$ and show that

[TABLE]

To do so, we use [5, Definition 1.1.3] to obtain that

[TABLE]

where $d_{K}(x+hw)=\inf_{z\in K}|x+hw-z|$ is the distance from $x+hw$ to the set $K$. From (91) and the polytopic nature of the set $K$, we conclude that there exists $h>0$ such that $d_{K}(x+hw)=0$, i.e., $x+hw\in K$. Hence,

[TABLE]

which implies that

[TABLE]

This proves (88).

To prove (89), we pick

[TABLE]

and show that $w\in T_{K}(x)$. Since $x\in\partial K$, we have that

[TABLE]

Expressions (92) and (93) can be combined to conclude by continuity that, for every $h>0$ sufficiently small, we have

[TABLE]

Hence, $(x+hv)\in K$ for small enough $h>0$, and thus $v\in T_{K}(x)$.