Forward Euler Solutions and Weakly Invariant Time-Delayed Systems

Abstract

This paper presents a necessary and sufficient condition for the weak invariance property of a time-delayed system parametrized by a differential inclusion. The aforementioned condition generalizes the well-known Hamilton-Jacobi inequality that characterizes weakly invariant systems in the nondelay setting. The forward Euler approximation scheme used in the theory of discontinuous differential equations is extended to the time-delayed context by incorporating the delay and tail functions featuring the dynamics. Accordingly, an existence theorem of weakly invariant trajectories is established under the extended forward Euler approach.

1. Introduction

The main goal of this paper is to establish a characterization of the so-called weakly invariant property of a closed set 𝑆∈ℝ^𝑛 with respect to the set of solutions of the time-delay differential inclusion 𝑥̇(𝑡)∈𝐹(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿(𝑡))),𝑡∈𝐼. (1.1) Throughout, the prescribed delay function 𝛿(⋅) is nonnegative and continuous on the interval 𝐼∶=[0,+∞). The multifunction 𝐹∶𝐼×ℝ^𝑛×ℝ^𝑛↦ℝ^𝑛 is endowed with the following structural hypotheses (SH). (i)𝐹(𝑡,𝑥,𝑦) is a nonempty, convex, and compact set for every (𝑡,𝑥,𝑦). (ii)There are constants 𝛾,𝑐>0 such that for each (𝑡,𝑥,𝑦)sup‖𝑣‖∶𝑣∈𝐹𝑡,𝑥,𝑦≤𝛾‖𝑥‖+𝑦+𝑐. (1.2)(iii)𝐹(⋅,⋅,⋅) is almost upper semicontinuous, which in presence of the previous assumptions, is equivalent to the following condition: for every compact interval 𝐼̃⊂𝐼 and each 𝜀>0, there exists a closed set 𝒩_𝜀⊆𝐼̃ with Lebesgue measure 𝜇(𝐼̃⧵𝒩_𝜀)<𝜀, so that the graph 𝒢(𝐹)∶=𝑡,𝑥,𝑦,𝑣∶𝑣∈𝐹𝑡,𝑥,𝑦 (1.3) is closed in 𝒩_𝜀×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛. A solution (or trajectory) to (1.1) is an absolutely continuous function 𝑥 satisfying (1.1) for almost all 𝑡∈[𝑡₀,∞), for some 𝑡₀≥0. A more precise description and assumptions on the elements defining the data are provided in the next section.

Time delay has been shown to appear in many deterministic dynamical systems (see for instance [1–4]). When this feature is present, its study is essential to understand important qualitative and quantitative aspects of the dynamics. Furthermore, ignoring an existing time delay may result in an ill-posed model [5]. Based upon the nature of the state space, time-delay systems can be investigated by means of either infinite ([6–8]) or finite dimensional models ([1, 9, 10]). In the present work, the considered dynamics evolve according to the differential inclusion (1.1), which fits in the category of finite dimensional models, since the delay component of its right-hand side is being identified with the pointwise history of its trajectories. This setting, we believe, will prove to be satisfactory to further study strong invariance properties and their possible interconnections with Hamilton-Jacobi theory for problems involving time-delayed differential inclusions on finite dimensional spaces.

A complete exposition on the invariance concept used in this paper for the nondelayed setting can be consulted in [6, 11]. We also refer the reader to the comprehensive work [12] where notions of approximate invariance are introduced and a complete set of criteria for the weak and strong invariance properties are obtained in infinite dimensional Hilbert spaces. Here we briefly recall the weak invariance terminology in terms of the time-delayed differential inclusion (1.1). Given Δ>0, a closed set 𝑆⊆ℝ^𝑛 and a multifunction 𝐹, the pair (𝑆,𝐹) is said to be weakly invariant if for every initial condition (𝑡₀,𝑥₀)∈𝐼×𝑆 and every tail function 𝜙∶[−Δ,𝑡₀]→ℝ^𝑛 with 𝜙(𝑡₀)=𝑥₀, there is a solution 𝑥 to (1.1) satisfying 𝑥(𝑡)=𝜙(𝑡) on [−Δ,𝑡₀] and 𝑥(𝑡)∈𝑆 for all 𝑡≥𝑡₀.

The original motivation to study invariance properties rises from the qualitative theory of dynamical systems under ordinary differential equations. Nevertheless, their applicability branched out to other areas, including partial differential equations, dynamic optimization, and control theory (see [6, 11, 13–16]).

The literature centered upon the invariance topic itself is abundant. We highlight the primary works [17–20] given in the context of vector field differential equations and the references [6–8, 13, 21–24] in the differential inclusion setting. All of the aforementioned works provide either sufficient or necessary conditions for the weak invariance property. However, it is noteworthy to mention that to the best of the authors' knowledge, weak invariance for dynamics exhibiting a time-delay behavior have only been discussed in the references [6–8], with functional differential inclusions being employed to model the underlined dynamics. The conditions for weak invariance provided in [6–8] are of tangential type only and roughly speaking they assert that weak invariance is equivalent to having, at each 𝑥∈𝑆, the corresponding set of velocities overlapping the Bouligand cone of 𝑆 at 𝑥: 𝑇^𝐵_𝑆(𝑥)∶=lim_𝑖→∞𝑥_𝑖−𝑥𝑡_𝑖∶𝑥→−_𝑖^𝑆𝑥,𝑡_𝑖↓0, (1.4) where 𝑥→−_𝑖^𝑆𝑥 means that 𝑥_𝑖∈𝑆 converges to the point 𝑥, which is identified as the initial condition of the weakly invariant trajectory of (1.1). The closed cone (1.4) coincides with the tangent space if 𝑆 is a smooth manifold, but of course 𝑇^𝐵_𝑆(𝑥) encloses a more general tangency concept.

A condition that is dual with the previous and that has been revisited by several authors (see [6, 11, 12, 15–17, 22, 24, 25]) to characterize the weak invariance property in the nondelayed framework is the following (we recall here the autonomous version). The pair (𝑆,𝐹) is weakly invariant if and only if ℎ_𝐹(𝑥,𝜉)≤0,∀𝑥∈𝑆,∀𝜉∈𝑁^𝑃_𝑆(𝑥), (1.5) where ℎ_𝐹(𝑥,𝜉)∶=inf𝑣,𝜉∶𝑣∈𝐹(𝑥), (1.6) is the Hamiltonian of the multifunction 𝐹∶ℝ^𝑛↦ℝ^𝑛 and 𝑁^𝑃_𝑆(𝑥)∶=𝜉∶∃𝜎>0suchthat𝑥^′−𝑥,𝜉≤𝜎𝑥^′−𝑥²∀𝑥^′∈𝑆 (1.7) is the Proximal Normal Cone to 𝑆 at 𝑥.

The Hamiltonian inequality (1.5) has proven to be an important link between some of the invariance properties of nondelayed systems (𝑆,𝐹) and the theory of generalized solutions of Hamilton-Jacobi equations (see [11, 16, 18, 22, 26, 27]). As it was pointed out previously, this Hamiltonian interpretation is not available for time-delayed systems. To our knowledge, the only results deriving a Hamilton-Jacobi equation for time-delayed systems involve a game-theoretic approach in which the dynamics are prescribed by a continuous differential equation [28]. However, it is well understood that a great deal of applications stemming from optimal control theory, including discontinuous differential equations models, can be reformulated in terms of the more general paradigm (1.1). Therefore, the investigation about the form and the vality of (1.5) under the context of the time-delay differential inclusion (1.1) is of natural interest and represents in particular the main goal of this article.

The remaining material of this paper has been organized in three additional sections. Section 2 introduces the extension of the forward Euler solution concept given in [29] to Cauchy problems with time-delay components and establishes the corresponding existence theorem. Section 3 reveals the connection between the new solution concept and the trajectories of the time-delayed differential inclusion (1.1). The final section (Section 4) contains our main result, which is a Hamiltonian characterization of the weak invariance property for time-delayed systems modeled by (1.1).

2. Delayed Forward Euler Solutions

Motivated by the problem of asymptotic stabilization, Bressan introduced the forward Euler approximation scheme in [29] to provide a suitable class of generalized solutions for nondelayed discontinuous differential equations. The new category of solutions subsumes the one generated by the classic Euler polygonal approach and provides some topological advantages when compared with the classes of Krasovskii and Filippov solutions that are used in the theory of discontinuous feedback controls [30, 31]. The aim of this section is to extend Bressan's solution concept to a more general setting where the dynamics exhibit a time delay behavior. This will serve as the initial step to show that weakly invariant trajectories of (1.1) can be identified within a wider class of solutions of certain initial value problems given in terms of time-delayed feedback selections. In this context, we consider for 𝑇>0 and every 0≤𝑡₀<𝑇 the following time-delayed Cauchy problem: 𝑥̇(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿(𝑡))),𝑡∈𝑡₀𝑥(𝑡)(𝑡),𝑇=𝜙,𝑡∈−Δ,𝑡₀, (2.1) where the tail function 𝜙∶[−Δ,𝑡₀]→ℝ^𝑛 is Lebesgue measurable and bounded, the delay function 𝛿∶[0,𝑇]→[0,Δ] is continuous and for some positive constants 𝛾 and 𝑐, the vector field 𝑓∶[0,𝑇]×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 satisfies the following linear growth condition: 𝑓𝑡,𝑥,𝑦≤𝛾‖𝑥‖+𝑦+𝑐,∀𝑡,𝑥,𝑦∈[0,𝑇]×ℝ^𝑛×ℝ^𝑛. (2.2) We start with a partition of the interval [𝑡₀,𝑇]𝜋=𝑡₀,𝑡₁,…,𝑡_𝑘, (2.3) where 𝑡_𝑘=𝑇. As usual, the diameter of 𝜋 is defined by 𝜇_𝜋=max{𝑡_𝑖−𝑡_𝑖−1∶1≤𝑖≤𝑘}. Given an initial point 𝑥₀ and a set of outer perturbations 𝑞₀,𝑞₁,…,𝑞_𝑘, all of them in ℝ^𝑛, we construct a delayed forward Euler polygonal as follows: 𝜓_𝜋(𝑡)=𝜙(𝑡)if−Δ≤𝑡<𝑡₀,𝜓_𝜋𝑡₀=𝑥₀𝜓_𝜋(𝑡)=𝜓_𝜋𝑡_𝑖+𝑓𝑡_𝑖,𝜓_𝜋𝑡_𝑖,𝜓_𝜋𝑡_𝑖−𝛿𝑡_𝑖+𝑞_𝑖𝑡−𝑡_𝑖for0≤𝑖≤𝑘−1,𝑡∈𝑡_𝑖,𝑡_𝑖+1. (2.4)(2.5) A function 𝑥∶[−Δ,𝑇]→ℝ^𝑛 is a delayed forward Euler solution (DFES) to the initial value problem (2.1) if there are sequences 𝑥^𝑚₀→𝜙(𝑡₀), 𝑞^𝑚_𝑖→0, for all 𝑖=0,1,…,𝑘, such that the corresponding sequence of delayed forward Euler polygonals 𝜓_𝜋𝑚 converges uniformly to 𝑥, as 𝜇_𝜋𝑚→0.

The existence of a DFES is established by proving the stabilization of the previous scheme for certain subsequences of delayed forward Euler polygonals. This is done with the help of the following two discrete versions of Gronwall's inequality.

Lemma 2.1. Assume 𝑥_𝑖, 𝑦_𝑖 are sequences of nonnegative real numbers having the property that for some 𝛼≥0𝑦_𝑖≤𝛼+_{0≤𝑗<𝑖}𝑦_𝑗𝑥_𝑗,∀𝑖≥0. (2.6) Then the following estimate holds: 𝑦_𝑖≤𝛼𝑒^{(∑0≤𝑗<𝑖𝑥𝑗)},∀𝑖≥0. (2.7)

Proof. To prove the lemma we first show by induction that if (2.6) holds, then 𝑦_𝑖≤𝛼1+_{0≤𝑗<𝑖}𝑥_{𝑗𝑗<𝑘<𝑖}1+𝑥_𝑘,∀𝑖≥0. (2.8) In fact, since a sum with zero terms is defined as 0 and the product of zero factors is defined as 1, inequality (2.6) implies 𝑦₀≤𝛼=𝛼(1+0), which is (2.8) for 𝑖=0. Let 𝑖>0 and assume that (2.8) holds for 𝑗<𝑖. Using hypothesis (2.6) we can write 𝑦_𝑖≤𝛼+_{0≤𝑗<𝑖}𝑥_𝑗𝑦_𝑗≤𝛼+_{0≤𝑗<𝑖}𝑥_𝑗𝛼1+_{0≤𝑘<𝑗}𝑥_{𝑘𝑘<𝑟<𝑗}1+𝑥_𝑟=𝛼1+_{0≤𝑗<𝑖}𝑥_𝑗+_{0≤𝑗<𝑖}𝑥_{𝑗0≤𝑘<𝑗}𝑥_{𝑘𝑘<𝑟<𝑗}1+𝑥_𝑟. (2.9) On the other hand, notice that (using induction again on 𝑖) _{0≤𝑗<𝑖}𝑥_{𝑗0≤𝑘<𝑗}𝑥_{𝑘𝑘<𝑟<𝑗}1+𝑥_𝑟=_{0≤𝑘<𝑖}𝑥_{𝑘𝑘<𝑗<𝑖}𝑥_{𝑗𝑘<𝑟<𝑗}1+𝑥_𝑟. (2.10) The expansion of the last sum and product on the right hand side of (2.10) yields 1+_{𝑘<𝑗<𝑖}𝑥_{𝑗𝑘<𝑟<𝑗}1+𝑥_𝑟=1+𝑥_𝑘+1+𝑥_𝑘+21+𝑥_𝑘+1+⋯+𝑥_𝑖−11+𝑥_𝑘+1⋯1+𝑥_𝑖−2=1+𝑥_𝑘+11+𝑥_𝑘+2+⋯+𝑥_𝑖−11+𝑥_𝑘+2⋯1+𝑥_𝑖−2⋮=1+𝑥_𝑘+11+𝑥_𝑘+2⋯1+𝑥_𝑖−1=_{𝑘<𝑟<𝑖}1+𝑥_𝑟. (2.11) The following estimate follows from the tautology ∑_{0≤𝑗<𝑖}𝑥_𝑗=∑_{0≤𝑘<𝑖}𝑥_𝑘 and from taking (2.10) into (2.9) and then using (2.11): 𝑦_𝑖≤𝛼1+_{0≤𝑘<𝑖}𝑥_𝑘+_{0≤𝑘<𝑖}𝑥_{𝑘𝑘<𝑗<𝑖}𝑥_{𝑗𝑘<𝑟<𝑗}1+𝑥_𝑟=𝛼1+_{0≤𝑘<𝑖}𝑥_𝑘1+_{𝑘<𝑗<𝑖}𝑥_{𝑗𝑘<𝑟<𝑗}1+𝑥_𝑟=𝛼1+_{0≤𝑘<𝑖}𝑥_{𝑘𝑘<𝑟<𝑖}1+𝑥_𝑟. (2.12) The last inequality completes the induction. Finally, we use (2.8) to obtain the exponential estimate (2.7). For every 𝑟 it is clear that 1+𝑥_𝑟≤𝑒^𝑥𝑟. Therefore, 𝑦_𝑖≤𝛼1+_{0≤𝑘<𝑖}1+𝑥_𝑘−1_{𝑘<𝑟<𝑖}1+𝑥_𝑟=𝛼1+_{0≤𝑘<𝑖𝑘≤𝑟<𝑖}1+𝑥_𝑟−_{𝑘<𝑟<𝑖}1+𝑥_𝑟=𝛼1+_{0≤𝑟<𝑖}1+𝑥_𝑟−_{𝑖−1<𝑟<𝑖}1+𝑥_𝑟=𝛼1+_{0≤𝑟<𝑖}1+𝑥_𝑟−1=𝛼_{0≤𝑟<𝑖}1+𝑥_𝑟≤𝛼𝑒^{(∑0≤𝑟<𝑖𝑥𝑟)}, (2.13) which is precisely the sought-after inequality.

Lemma 2.2. Let 𝑟₀,𝑟₁,…,𝑟_𝑁 be nonnegative numbers satisfying 𝑟_𝑖+1≤1+𝑑_𝑖𝑟_𝑖+𝐷_𝑖,𝑖=0,1,…,𝑁−1, (2.14) where 𝑑_𝑖≥0, 𝐷_𝑖≥0 and 𝑟₀=0. Then 𝑟_𝑁≤𝑒^{(∑𝑁−1𝑖=0𝑑𝑖)𝑁−1}_𝑖=0𝐷_𝑖. (2.15)

Proof. This lemma is an exercise proposed in page 183 of [11].

It is noteworthy to mention that the existence of forward Euler solutions has been clarified by Bressan in the context of nondelay patchy vector fields by comparing the new set of solutions with the set of Carathéodory solutions (see Theorem 3, part (iv) in [29]). The following existence result contrasts with Bressan's approach by following the ideas from Theorem 1.7 in ([11], Chapter 4) and extending it to the time-delayed framework under the more general class of vector fields satisfying (2.2).

Theorem 2.3. Suppose that 𝑓 satisfies the linear growth condition (2.2) and is otherwise arbitrary. Then, at least one DFES exists for the time-delayed Cauchy problem (2.1) and the restriction of any DFES on [𝑡₀,𝑇] is Lipschitz.

Proof. To relax the notation, we assume without loss of generality that 𝑡₀=0. Let 𝑁 be a natural number. For the uniform partition of [0,𝑇]𝜋∶=𝑡_𝑖=𝑇𝑁𝑖∶𝑖=0,1,…,𝑁, (2.16) let 𝜓_𝜋 be the corresponding delayed forward Euler polygonal with outer perturbations 𝑞_𝑖, nodes 𝑥_𝑖=𝜓_𝜋(𝑡_𝑖), and delayed nodes 𝑦_𝑖=𝜓_𝜋(𝑡_𝑖−𝛿(𝑡_𝑖)), for 𝑖=0,1,…,𝑁. We assume max{‖𝑞_𝑖‖∶0≤𝑖≤𝑁}≤1/𝑁 and ‖𝑥₀−𝜙(0)‖<1/𝑁. Let 𝑖∈{0,1,…,𝑁} be fixed. Due to the hypothesis on 𝑓, for every 𝑡∈(𝑡_𝑖,𝑡_𝑖+1) we have 𝜓̇_𝜋(𝑡)=𝑓𝑡_𝑖,𝑥_𝑖,𝑦_𝑖+𝑞_𝑖≤𝛾𝑥_𝑖+𝑦_𝑖+(1+𝑐). (2.17) Let 𝛽=max{‖𝜙(𝑡)‖∶𝑡∈[−Δ,0]} and choose an index 𝑝_𝑖∈{0,1,…,𝑖−1} such that ‖𝑥_𝑝𝑖‖=max{‖𝑥_𝑝‖∶0≤𝑝≤𝑖−1}. The definition of 𝑦_𝑖 implies ‖𝑦_𝑖‖≤‖𝑥_𝑝𝑖‖+𝛽. Using this observation, (2.4), and (2.17) we can write 𝑥_𝑖+1−𝜙(0)≤𝑥_𝑖+1−𝑥_𝑖+𝑥_𝑖(0≤𝛾𝑥−𝜙)_𝑖+𝑦_𝑖𝑇+𝑐+1𝑁+𝑥_𝑖(0≤𝛾𝑥−𝜙)_𝑖−𝜙(0)+𝜙(0)+𝑦_𝑖𝑇+𝑐+1𝑁+𝑥_𝑖(0≤𝛾𝑇−𝜙)𝑁+1𝑥_𝑖−𝜙(0)+𝛾𝛽+𝑦_𝑖𝑇+𝑐+1𝑁≤𝛾𝑇𝑁+1𝑥_𝑖−𝜙(0)+3𝛾𝛽+𝛾𝑥_𝑝𝑖(0𝑇−𝜙)+𝑐+1𝑁. (2.18) Let 𝑀∶=3𝛾𝛽+𝑐+1. Using the identifications 𝑟_𝑖=𝑥_𝑖−𝜙(0),𝑑_𝑖𝑇=𝛾𝑁,𝐷_𝑖=𝑀+𝛾𝑥_𝑝𝑖(0𝑇−𝜙)𝑁 (2.19) and applying Lemma 2.2 to the last inequality, we have 𝑥_𝑖+1−𝜙(0)≤𝑒^𝑖𝛾𝑇_𝑗=0𝑀+𝛾𝑥_𝑝𝑗(0𝑇−𝜙)𝑁=𝑒^𝛾𝑇𝑀^𝑖_𝑗=0𝑇𝑁+𝛾𝑒^𝑖𝛾𝑇_𝑗=0𝑥_𝑝𝑗(0𝑇−𝜙)𝑁≤𝑒^𝛾𝑇𝑀𝑇+𝛾𝑒^𝛾𝑇𝑇𝑁^𝑖_𝑗=0𝑥_𝑗−𝜙(0). (2.20) If we set 𝛼=𝑒^𝛾𝑇𝑀𝑇 and 𝜌=𝛾𝑒^𝛾𝑇𝑇, the application of Lemma 2.1 on (2.20) leads to 𝑥_𝑖−𝜙(0)≤𝛼𝑒^𝜌, (2.21) and thus, all nodes 𝑥_𝑖 lie in 𝐵(𝜙(0),𝛼𝑒^𝜌). By convexity the delayed forward Euler polygonal, including its tail 𝜙, must lie completely within the ball 𝐵(𝜙(0),𝛼𝑒^𝜌+2𝛽). For 𝑡∈(𝑡_𝑖,𝑡_𝑖+𝑖) inequality (2.17) then implies 𝜓̇_𝜋(𝑡)≤𝛾𝑥_𝑖+𝑦_𝑖+𝑞_𝑖+𝑐≤𝛾2𝛼𝑒^𝜌+3𝛽+𝑐+1=∶𝑘. (2.22) Therefore, the restriction of 𝜓_𝜋 on [0,𝑇] is Lipschitz of rank 𝑘. If 𝜋_𝑚 is a sequence of uniform partitions such that 𝜇_𝜋𝑚→0 as 𝑚→∞, we can apply the theorem of Arzela-Ascoli to the family {𝜓_𝜋𝑚} restricted on [0,𝑇], which is equicontinuous and uniformly bounded, to obtain a subsequence that converges uniformly to a continuous function 𝑥 on [0,𝑇]. Notice that not only does 𝑥 inherit the Lipschitz constant 𝑘 on [0,𝑇], but also 𝑥(0)=lim_𝑚→∞𝜓_𝜋𝑚(0). Finally, a DFES of (2.1) is obtained by appending 𝑥 and 𝜙.

3. Trajectories of Time-Delay Differential Inclusions

This section identifies a particular set of trajectories of the time-delay differential inclusion problem (TDDI) 𝑥̇(𝑡)∈𝐹(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿(𝑡)))a.e.𝑡∈𝑡₀,𝑥(𝑠)(𝑠),∞=𝜙for𝑠∈−Δ,𝑡₀, (3.1) where the data above consists of the interval 𝐼=[0,+∞), a multifunction 𝐹∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 satisfying SH, a positive constant Δ, an initial time 𝑡₀∈𝐼, a continuous delay function 𝛿∶𝐼→[0,Δ], and a tail function 𝜙∶[−Δ,𝑡₀]→ℝ^𝑛 that is assumed to be Lebesgue measurable and bounded. As in the nondelay case [11], we show that any DFES associated to selections of the multifunction 𝐹 are indeed trajectories of TDDI. The following lemma plays a key role in this regard. The proof is a slight modification of Lemma 1.1 in [32] and therefore it is omitted.

Lemma 3.1. Suppose 𝑥 is a continuous function on [𝑡₀,𝑇] and satisfies ‖𝑥(𝑡)‖≤𝑎+𝑏^𝑡_𝑡0‖𝑥(𝑠)‖𝑑𝑠+𝑐^𝑡_𝑡0‖𝑥(𝑠−𝛿(𝑠))‖𝑑𝑠,𝑡∈𝑡₀,𝑥(𝑡)(𝑡),𝑇=𝜙,𝑡∈−Δ,𝑡₀, (3.2) where 𝑎,𝑏,𝑐≥0 and 𝜙(⋅) is measurable and bounded by 𝜆>0 on [−Δ,𝑡₀]. Then the following inequality holds: ‖𝑥(𝑡)‖≤𝑎+𝑐𝑡−𝑡₀𝜆𝑒^{(𝑏+𝑐)(𝑡−𝑡0)},𝑡∈𝑡₀,𝑇. (3.3)

The next theorem is the main result of this section. It generalizes Corollary 1.12 of ([11], Chapter 4) to the context of a time-delayed differential inclusion with trajectories defined on a semi-infinite interval. For this purpose, it is necessary to extend the concept of a DFES to semi-infinite intervals by considering selections of 𝐹. Recall that a selection is a function 𝑓∶𝐼×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 satisfying 𝑓(𝑡,𝑥,𝑦)∈𝐹(𝑡,𝑥,𝑦) for almost all 𝑡∈𝐼 and all (𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑛. Under this extended domain, we agree that a function 𝑥 is a delayed forward Euler solution of the corresponding Cauchy problem for 𝑓 if for every 𝑇>0 the restriction of 𝑥 on [−Δ,𝑇] is a DFES for 𝑓 in the regular sense. This definition becomes coherent under the same argument that is used to extend nondelayed Euler solutions: an appropriate sequence of delayed forward Euler polygonals on [−Δ,∞) is constructed, then a convergent subsequence on [−Δ,𝑇] is obtained for a fixed 𝑇>0, a further convergent subsequence on [−Δ,2𝑇], and so on (see [11]).

Theorem 3.2. Let 𝑓 be a selection of 𝐹 as above. If 𝑥 is a DFES for the Cauchy problem (2.1) on [−Δ,∞), then 𝑥 is a trajectory of 𝐹.

Proof. Let 𝜓_𝑚 be a sequence of delayed forward Euler polygonals that converge uniformly to 𝑥 and that coincide with 𝜙 on [−Δ,𝑡₀]. Choose 𝑇>𝑡₀. Let 𝜋_𝑚={𝑡^𝑚₀,𝑡^𝑚₁,…,𝑡^𝑚_𝑁} be the partition of [𝑡₀,𝑇] associated with 𝜓_𝑚, and for 𝑡∈(𝑡^𝑚_𝑖,𝑡^𝑚_𝑖+1) we define the following functions: 𝜏_𝑚(𝑡)=𝑡^𝑚_𝑖,𝑤_𝑚(𝑡)=𝜓_𝑚𝜏_𝑚(𝑡)−𝜓_𝑚(𝑡),𝑧_𝑚(𝑡)=𝜓_𝑚𝜏_𝑚(𝑡)−𝛿𝜏_𝑚(𝑡)−𝜓_𝑚(𝑡−𝛿(𝑡)). (3.4) For any 𝛼>0, it is readily seen that 𝜏_𝑚⁻¹(𝛼,∞)∶=^𝑁⋃_𝑖=0𝑡^𝑚_𝑖,𝑡^𝑚_𝑖+1,if𝛼<𝑡^𝑚₀,_𝑁𝑖=⋃_𝑗+1𝑡^𝑚_𝑖,𝑡^𝑚_𝑖+1,if𝑡^𝑚_𝑗≤𝛼<𝑡^𝑚_𝑗+1,𝑗∈{0,1,…,𝑁−1},∅,if𝑡^𝑚_𝑁−1≤𝛼, (3.5) which guarantees the measurability of 𝜏_𝑚. Moreover, 𝜓_𝑚 and 𝛿 are continuous functions, whence the composites 𝜓_𝑚∘𝜏_𝑚 and 𝛿∘𝜏_𝑚 turn out to be measurable, and so also do the linear combinations 𝑤_𝑚 and 𝑧_𝑚.
If 𝑘 is the Lipschitz constant for 𝜓_𝑚 on [𝑡₀,𝑇], then the following estimates hold: 𝑤_𝑚(𝑡)≤𝑘𝜇_𝜋𝑚,𝜏_𝑚(𝑡)−𝑡≤𝜇_𝜋𝑚,𝑧_𝑚(𝑡)≤𝑘𝜇_𝜋𝑚+𝛿(𝑡)−𝛿𝜏_𝑚(𝑡). (3.6) Thus 𝑤_𝑚,𝑧_𝑚 both converge to zero in 𝐿²[𝑡₀,𝑇] and 𝜏_𝑚(𝑡) converges to 𝑡 a.e. Since 𝑓 is a selection of 𝐹, at every nonpartition point 𝑡 we have 𝜓̇_𝑚(𝑡)∈𝐹𝜏_𝑚(𝑡),𝜓_𝑚(𝑡)+𝑤_𝑚(𝑡),𝜓_𝑚(𝑡−𝛿(𝑡))+𝑧_𝑚(𝑡). (3.7) Therefore, for almost all 𝑡∈[𝑡₀,𝑇] the linear growth condition on 𝐹 implies 𝜓̇_𝑚(𝑡)≤𝛾𝜓_𝑚(𝑡)+𝜓_𝑚(𝑡−𝛿(𝑡))+𝛾𝑤_𝑚(𝑡)+𝑧_𝑚(𝑡)+𝑐. (3.8) We now apply Gronwall's inequality ([11], page 179) on (3.8), followed by Lemma 3.1 with 𝑎=𝑒^{𝛾(𝑇−𝑡0)}‖𝜓_𝑚(0)‖+𝛾𝑒^{𝛾(𝑇−𝑡0)}(‖𝑤_𝑚‖₁+‖𝑧_𝑚‖₁+𝑐(𝑇−𝑡₀))), 𝑏=0, and 𝑐=𝛾𝑒^{𝛾(𝑇−𝑡0)} to obtain 𝜓_𝑚(𝑡)≤𝑎+𝛾𝑒^{𝛾(𝑡−𝑡0)}𝑡−𝑡₀𝜆𝑒^{𝛾(𝑡−𝑡0)𝑒0)𝛾(𝑡−𝑡},𝑡∈𝑡₀,𝑇. (3.9) The last inequality leads to a uniform bound for ‖𝜓_𝑚‖ and hence for ‖𝜓̇_𝑚‖₂ due to (3.8). Given the theorem of Arzela-Ascoli and the weak compactness in 𝐿²_𝑛[𝑡₀,𝑇], we may choose converging subsequences of {𝜓_𝑚} and {𝜓̇_𝑚}, respectively (without relabeling), so that 𝜓_𝑚(𝑡)=𝜓_𝑚𝑡₀+^𝑡_𝑡0𝜓̇_𝑚(𝑠)𝑑𝑠→𝜙𝑡₀+^𝑡_𝑡0𝑥̇(𝑠)𝑑𝑠=𝑥(𝑡),as𝑚→∞. (3.10) A simple variant of the sequential compactness result given in Chapter 3 of [11] that holds under SH can be applied to obtain that 𝑥 is a trajectory of 𝐹 on [𝑡₀,𝑇]. Repeating the argument described above on [𝑡₀,2𝑇] with the subsequence {𝜓_𝑚} shows that 𝑥 is a trajectory of 𝐹 on [𝑡₀,2𝑇]. We may continue in this fashion to establish the desired property for 𝑥 on 𝐼.

4. The Main Result

The following observation provides a sufficient condition for delayed forward Euler solutions of (2.1) to approach a closed set 𝑆⊂ℝ^𝑛. It plays in particular a crucial role in the main contribution of this article (Theorem 4.2 below) by linking the existence result given in Theorem 3.2 and the weak invariance property of TDDI.

Theorem 4.1. Suppose 𝑓∶[0,𝑇]×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 satisfies (2.2) and let 𝑥 be a DFES for 𝑓 on [−Δ,𝑇]. Let Ω be an open set containing 𝑥(𝑡) for all 𝑡∈[−Δ,𝑇] and suppose that for every (𝑡,𝑧,𝑦)∈[𝑡₀,𝑇]×Ω×Ω there exists 𝑠∈𝑝𝑟𝑜𝑗_𝑆(𝑧) such that ⟨𝑓(𝑡,𝑧,𝑦),𝑧−𝑠⟩≤0. Then one has 𝑑_𝑆(𝑥(𝑡))≤𝑑_𝑆𝑥𝑡₀∀𝑡∈𝑡₀,𝑇. (4.1)

Proof. Let 𝜓_𝜋 be one element from the sequence of delayed forward Euler polygonals converging uniformly to 𝑥 and let 𝜋={𝑡₀,𝑡₁,…,𝑡_𝑁} be its associated partition. Let 𝑞₀,𝑞₁,…,𝑞_𝑁 be the external perturbations defining 𝜓_𝜋 and as in the previous result, we denote by 𝑥_𝑖 and 𝑦_𝑖 their associated nodes and delayed nodes. Given 𝜀>0, due to the uniform convergence we may assume that 𝜓_𝜋 lies completely in Ω and ‖𝑞_𝑖‖<𝜀/2 for 𝑖=0,1,…,𝑁. The hypothesis asserts that for each 𝑖 there is a point 𝑠_𝑖∈proj_𝑆(𝑥_𝑖) such that ⟨𝑓(𝑡_𝑖,𝑥_𝑖,𝑦_𝑖),𝑥_𝑖−𝑠_𝑖⟩≤0. If 𝑘 is as in (2.22), then the following estimates hold: 𝑑²_𝑆𝑥_𝑖+1≤𝑥_𝑖+1−𝑠_𝑖²since𝑠_𝑖=𝑥∈𝑆_𝑖+1−𝑥_𝑖²+𝑥_𝑖−𝑠_𝑖²+2𝑥_𝑖+1−𝑥_𝑖,𝑥_𝑖−𝑠_𝑖≤𝑘²𝑡_𝑖+1−𝑡_𝑖²+𝑑²_𝑆𝑥_𝑖+2^{𝑡𝑖+𝑖}_𝑡𝑖𝑓𝑡_𝑖,𝑥_𝑖,𝑦_𝑖+𝑞_𝑖,𝑥_𝑖−𝑠_𝑖𝑑𝑡≤𝑘²𝑡_𝑖+1−𝑡_𝑖²+𝑑²_𝑆𝑥_𝑖+2^{𝑡𝑖+𝑖}_𝑡𝑖𝑞_𝑖𝑥_𝑖−𝑠_𝑖𝑑𝑡≤𝑘²𝑡_𝑖+1−𝑡_𝑖²+𝑑²_𝑆𝑥_𝑖+𝜀𝑡_𝑖+1−𝑡_𝑖𝑑_𝑆𝑥_𝑖. (4.2) The recursive application of last inequality yields 𝑑²_𝑆𝑥_𝑖+1≤𝑑²_𝑆𝑥₀+^𝑖_𝑗=0𝑘²(𝑡_𝑗+1−𝑡_𝑗)²+^𝑖_𝑗=0𝜀𝑡_𝑗+1−𝑡_𝑗𝑑_𝑆𝑥_𝑗≤𝑑²_𝑆𝑥₀+𝑘²𝜇_𝜋^𝑁−1_𝑗=0𝑡_𝑗+1−𝑡_𝑗+^𝑖_𝑗=0𝜀𝑡_𝑗+1−𝑡_𝑗𝑑_𝑆𝑥_𝑗≤𝑑²_𝑆𝑥₀+𝑘²𝜇_𝜋𝑇−𝑡₀+^𝑖_𝑗=0𝜀𝑡_𝑗+1−𝑡_𝑗𝑑_𝑆𝑥_𝑗. (4.3) Let Γ∶={𝑗∶𝑑_𝑆(𝑥_𝑗)<1}. If 𝑗∉Γ, then 𝑑_𝑆(𝑥_𝑗)≤𝑑²_𝑆(𝑥_𝑗). From this last and (4.3) we have 𝑑²_𝑆𝑥_𝑖+1≤𝑑²_𝑆𝑥₀+𝑘²𝜇_𝜋𝑇−𝑡₀+_𝑗∈Γ𝜀𝑡_𝑗+1−𝑡_𝑗+_𝑗∉Γ𝜀𝑡_𝑗+1−𝑡_𝑗𝑑²_𝑆𝑥_𝑗≤𝑑²_𝑆𝑥₀+𝑘²𝜇_𝜋+𝜀𝑇−𝑡₀+^𝑖_𝑗=0𝜀𝑡_𝑗+1−𝑡_𝑗𝑑²_𝑆𝑥_𝑗. (4.4) We now apply Lemma 2.1 with 𝛼=𝑑²_𝑆(𝑥₀)+(𝑘²𝜇_𝜋+𝜀)(𝑇−𝑡₀) to obtain 𝑑²_𝑆𝑥_𝑖≤𝑑²_𝑆𝑥₀+𝑘²𝜇_𝜋+𝜀𝑇−𝑡₀𝑒^{∑𝑖𝑗=0𝜀(𝑡𝑗+1−𝑡𝑗)}≤𝑑²_𝑆𝑥₀+𝑘²𝜇_𝜋+𝜀𝑇−𝑡₀𝑒^{𝜀(𝑇−𝑡0)}. (4.5) The previous estimates holds at any node and then after passing to the limit when 𝜇_𝜋→0 we obtain 𝑑²_𝑆(𝑥(𝑡))≤𝑑²_𝑆𝑥₀+𝜀𝑇−𝑡₀𝑒^{𝜀(𝑇−𝑡0)},∀𝑡∈[0,𝑇], (4.6) which in turn implies 𝑑_𝑆(𝑥(𝑡))≤𝑑_𝑆𝑥₀,∀𝑡∈𝑡₀,𝑇, (4.7) since 𝜀 is arbitrary.

In view of the previous result, weakly invariant trajectories for TDDI are naturally identified as delayed forward Euler solutions of particular selections of 𝐹. Accordingly, the Hamiltonian (1.5) plays an important role in the design of these selections and features the following weakly invariant criterion, which is the main contribution of this paper.

Theorem 4.2. Let 𝑆⊂ℝ^𝑛 be closed. Under the assumptions of TDDI, the pair (𝑆,𝐹) is weakly invariant if and only if ℎ𝑡,𝑥,𝑦,𝜁≤0,a.e.,𝑡∈𝐼,∀𝑥,𝑦∈𝑆×ℝ^𝑛,∀𝜁∈𝑁^𝑃_𝑆(𝑥). (4.8) In this case, (4.8) holds at all points of density of a certain countable family of pairwise disjoint closed sets 𝐼_𝑘⊂𝐼 (𝑘=0,1,…), for which 𝐹(⋅,⋅,⋅) is upper semicontinuous on each 𝐼_𝑘×ℝ^𝑛×ℝ^𝑛.

Remarks 1. Recall that a point 𝑡∈ℝ is a point of density of a measurable set ℳ⊂ℝ if lim_𝛿→0𝜇([𝑡−𝛿,𝑡+𝛿]∩ℳ)2𝛿=1, (4.9) where 𝜇 denotes the Lebesgue measure in ℝ. The set of all points of density of ℳ is denoted by ℳ and has full measure in ℳ. Furthermore, if 𝒱⊂ℳ, then the equality cl(𝒱)=cl(𝒱⧵𝒩) holds for any null set 𝒩⊂ℝ. It is clear that if ℳ is closed, then ℳ⊂ℳ (see [33]).

Proof. Let 𝐴⊂𝐼 be the set of full measure such that (4.8) holds for all 𝑡∈𝐴. For every (𝑡,𝑥,𝑦)∈𝐴×ℝ^𝑛×ℝ^𝑛 we choose an element 𝑠_𝑥∈proj_𝑆(𝑥) and then select 𝑓(𝑡,𝑥,𝑦)∈𝐹(𝑡,𝑠_𝑥,𝑦) so that ℎ𝑡,𝑠_𝑥,𝑦,𝑥−𝑠_𝑥=⟨𝑓𝑡,𝑥,𝑦,𝑥−𝑠_𝑥⟩. (4.10) We extend 𝑓, if necessary, by setting 𝑓(𝑡,𝑥,𝑦)∶=0 for all (𝑡,𝑥,𝑦)∈(𝐼⧵𝐴)×ℝ^𝑛×ℝ^𝑛. It follows that the multifunction 𝐹 induces the linear growth property (2.2) on 𝑓 and also ⟨𝑓(𝑡,𝑥,𝑦),𝑥−𝑠_𝑥⟩≤0 for all (𝑡,𝑥,𝑦)∈𝐼×ℝ^𝑛×ℝ^𝑛. Consequently, if 𝑥 is a DFES of the problem 𝑥̇(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿(𝑡))) a.e., 𝑡∈[𝑡₀,𝑇], with 𝜙(𝑡₀)∈𝑆, Theorem 4.1 implies 𝑥(𝑡)∈𝑆 for all 𝑡∈[𝑡₀,𝑇]. We may extend 𝑥 on 𝐼=[𝑡₀,∞) by considering the previous Cauchy problem on the interval [𝑡₀,2𝑇] and so on. It turns out that the resulting arc is a solution to TDDI. In fact, define 𝐹_𝑠𝑡,𝑥,𝑦∶=co𝐹𝑡,𝑠,𝑦∶𝑠∈proj_𝑆(𝑥),∀𝑡,𝑥,𝑦∈𝐼×ℝ^𝑛×ℝ^𝑛. (4.11) Then 𝐹_𝑠 satisfies the standing hypotheses and we have 𝐹_𝑠(𝑡,𝑥,𝑦)=𝐹(𝑡,𝑥,𝑦) when 𝑥∈𝑆. It is clear that 𝑓(𝑡,𝑥,𝑦)∈𝐹_𝑠(𝑡,𝑥,𝑦) for almost all 𝑡∈𝐼 and all (𝑥,𝑦)∈ℝ^𝑛×ℝ^𝑛. Therefore, Theorem 3.2 implies that 𝑥 is a trajectory of 𝐹_𝑠. Since 𝑥(𝑡)∈𝑆 for all 𝑡≥𝑡₀, it follows that 𝐹_𝑠(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿(𝑡)))=𝐹(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿(𝑡))) and hence 𝑥 is also a trajectory of 𝐹.
To prove the converse, let us consider 𝐼̃=[0,1] and recall that the almost upper semicontinuity of 𝐹 implies the existence of a countable family of pairwise disjoint closed sets 𝐼_𝑘⊂𝐼̃ such that 𝐹(⋅,⋅,⋅) is upper semicontinuous on 𝐼_𝑘×ℝ^𝑛×ℝ^𝑛 for all 𝑘 and the union ⋃_𝑘≥1𝐼_𝑘 has full measure in 𝐼̃ (see page 29 in [34] or page 24 in [35]). Let 𝐼_𝑘̃ denote the points of density of 𝐼_𝑘, and define 𝐴∶=𝐼̃⧵⋃_𝑘≥1𝐼_𝑘̃, which has null measure, and take (𝑡₀,𝑥₀,𝑦₀)∈𝐼_𝑘̃×𝑆×ℝ^𝑛 for some 𝑘. Let {𝑡_𝑗} be a sequence in [𝑡₀,∞)⋂𝐼_𝑘 satisfying 𝑡_𝑗→𝑡₀ and set ℎ_𝑗=𝑡_𝑗−𝑡₀. By hypothesis, there is a weakly invariant solution 𝑥 to TDDI such that 𝑥(𝑡₀)=𝑥₀. The absolute continuity of 𝑥, the compact-convex valuedness of 𝐹 and the upper semicontinuity of 𝐹 on 𝐼_𝑘×ℝ^𝑛×ℝ^𝑛, lead to the existence of a subsequence {𝑡_𝑗𝑘}, which we relabel as {𝑡_𝑗}, such that 𝑣∶=lim_𝑗→∞𝑥𝑡_𝑗−𝑥𝑡₀ℎ_𝑗=lim_𝑗→∞1ℎ_𝑗^𝑡𝑗_𝑡0𝑥̇(𝑠)𝑑𝑠∈𝐹𝑡₀,𝑥₀,𝑦₀. (4.12) On the other hand, recall that 𝑥(𝑡_𝑗)∈𝑆 for all 𝑗. Therefore, for any 𝜉∈𝑁^𝑃_𝑆(𝑥₀) the definition (1.7) asserts the existence of 𝜎>0 such that 𝑣,𝜉=lim_𝑗→∞𝑥𝑡_𝑗−𝑥𝑡₀ℎ_𝑗,𝜉≤𝜎lim_𝑗→∞𝑥𝑡_𝑗−𝑥𝑡₀ℎ_𝑗𝑥𝑡_𝑗−𝑥𝑡₀=0, (4.13) which implies the Hamiltonian inequality in the statement at the point (𝑡₀,𝑥₀,𝑦₀). Finally, we repeat the previous argument on each of the intervals [1,2],[2,3],… to obtain that (4.8) is satisfied almost everywhere on the whole interval 𝐼.