{'fraction_non_alphanumeric': 0.07526428546342148, 'fraction_numerical': 0.025111509149018368, 'mean_word_length': 4.112028725314183, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Equivalence of minimax and viscosity solutions of path-dependent Hamilton-Jacobi equations

In the paper, we consider a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton-Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of inequalities for coinvariant sub- and super-gradients. We prove that these two notions are equivalent, which is the main result of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton-Jacobi equation and a right-end boundary condition. The proof is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein-Preiss variational principle with an appropriate guage-type functional.


Introduction
In the theory of Hamilton-Jacobi (HJ) equations (nonlinear first-order partial differential equations), several approaches to a notion of a generalized solution are known. Among others, the minimax and viscosity approaches are the most developed.
The minimax approach originates in the positional differential games theory (see, e.g., [22,24]) and can be seen as the further development of the unification constructions of differential games [23]. According to this approach [42,45], a generalized (minimax) Fréchet derivatives as in, e.g., [12,13], they involve some special pathwise derivatives of functionals. In this paper, we follow [20,27] (see also [34]) and use so-called coinvariant (ci-) derivatives, while one can also use so-called horizontal and vertical derivatives [15]. For a discussion of the relationship between these notions, the reader is referred to [18,Section 5.2]. Minimax solutions of path-dependent HJ equations were introduced in [27,29] and comprehensively studied in, e.g., [2,18,33,34]. On the other hand, in the development of the viscosity approach, the absence of the local compactness property of the continuous function space becomes a more essential problem. To overcome this, based on the idea of [40], it was proposed in [32] to consider a modified notion of a viscosity solution. This modification consists in introducing into the definition a special expanding sequence of compact sets which union is everywhere dense. Note also that the role of test functionals in this definition is played by functionals that are smooth in the sense of ci-differentiability (rather than Fréchet differentiability). This technique and some of its variants and generalizations, which also allow one to use compactness arguments in the proofs, were further developed in, e.g., [16,17,19,33,47,50].
However, recently, several papers have appeared [8,9,37,38,48,49,[51][52][53] in which it is proposed to consider another notions of viscosity solutions of path-dependent HJ equations that are more consistent with the classical case. The present paper is also motivated by the same reasons and can be assigned to this area of research. Let us shortly describe some of the basic ideas of the cited papers that, in particular, allowed one to overcome the problem with the local compactness property. (i) In [37,38], it was suggested to move from the space of continuous functions to the wider space of piecewise continuous functions and to deal with a certain class of locally Lipschitz continuous solutions only. In application to optimal control problems in timedelay systems, this means the separation of the current value of the state vector and the history, while the Lipschitz condition is expressed in terms of the Euclidean norm of the current state and the integral norm of the history. In this case, it was shown that the proof of the uniqueness result for viscosity solutions can be carried out relying on the finite-dimensional optimization technique only.
(ii) Similar constructions were developed in [48,49,52]. The key tool used in the proof of the uniqueness result is a so-called left maximization principle in the space of piecewise continuous functions, which is established for functionals satisfying certain growth estimate and uniform continuity property (close to that from item (i)). The proof of this principle also uses only finite-dimensional optimization arguments.
(iii) A different approach was proposed in [8,9,51,53], where features of optimization problems for functionals over the space of continuous functions were taken into account in a direct way. The progress is related to the construction of some special smooth (in the sense of ci-differentiability or horizontal and vertical differentiability) functionals, which allows one to built smooth gauge-type functionals and apply Borwein-Preiss variational principle from [25,Theorem 1] (see also, e.g., [4,Theorem 2.5.2]). In particular, in [50], a smooth functional was found that in some sense equivalent to the square of the uniform norm, and this functional formed the basis of the results of [51,53] (see also [18]).
In this paper, we deal with viscosity solutions of the considered path-dependent HJ equation that are defined in terms of a pair of inequalities for so-called coinvariant (ci-) sub-and supergradients, which are naturally consistent with the notion of ci-differentiability. This definition was given in [34,Section 14], but no uniqueness results were 3 established for such a viscosity solution. Our main result is that, under certain assumptions, this definition of a viscosity solution is equivalent to that of a minimax solution from, e.g., [18]. In particular, as a direct corollary, we obtain comparison and uniqueness theorems for viscosity solutions of a Cauchy problem for the considered Hamilton-Jacobi equation and a right-end boundary condition. In general, our proof follows the scheme from [45,Theorem 4.3], which was already partially adapted to the path-dependent setting in [37]. Namely, we establish a property of the ci-subdifferential, which is similar to Property of Subdifferential [44]. To this end, we apply the technique going back to the use of Multidirectional Mean-Value Inequality [6]. In the spirit of [9], the absence of the local compactness is overcome by constructing a guage-type functional, which is carried out on the basis of the functional from [50], and by applying Borwein-Preiss variational principle from [25,Theorem 1]. It should be emphasized that the result is proved under an additional assumption concerning continuity properties of the considered solutions, which are close to the Lipschitz condition described above in item (i). Note also that the comparison and uniqueness theorems for viscosity solutions of Cauchy problems for path-dependent HJ equations obtained in the present paper are not covered by the results of [8,37,38,48,51,52], since, on the one hand, we do not require any growth estimates as well as any uniform continuity or boundedness properties of the solutions we are dealing with, and, on the other hand, we assume a weaker set of conditions on the Hamiltonian and the boundary functional. A more detailed discussion of the presented results is given in the main body of the paper.
The paper is organized as follows. In Section 2, we describe the path-dependent HJ equation under consideration, introduce the notions of minimax and viscosity solutions of this equation, and formulate our main result about their equivalence. Section 3 is devoted to a discussion of this result and some of its corollaries. In particular, we derive comparison and uniqueness theorems for viscosity solutions, obtain several criteria for viscosity solutions, and present an application to optimal control problems in time-delay systems, which also implies an existence result for a viscosity solution in the case of path-dependent Bellman equations. In Section 4, we give the proof of the main result.

Basic notation
Let n ∈ N, h > 0, and T > 0 be fixed. Let R n be the n-dimensional Euclidean space with the inner product ·, · and the norm · . For every t ∈ [0, T ], let C([−h, t], R n ) be the Banach space of all continuous functions x : [−h, t] → R n equipped with the norm Let us consider the set In other words, G consists of all pairs (t, x(·)) with t ∈ [0, T ] and x(·) ∈ C([−h, t], R n ). The set G is equipped with the metric ρ ∞ : G × G → R given by (see, e.g., [18,Section 5.1] and the references therein) Here and below, we use the notation a ∧ b min{a, b} for all a, b ∈ R, and, respectively, for every (t, x(·)) ∈ G, we define the function x(· ∧ t) ∈ C([−h, T ], R n ) as follows: The metric space (G, ρ ∞ ) is complete due to completeness of the space C([−h, T ], R n ). We also introduce the sets Along with the metric ρ ∞ , we consider the auxiliary metric ρ 1 : G × G → R given by for all (t, x(·)), (τ, y(·)) ∈ G. Note that (2.6) For every (t, x(·)) ∈ G, let Lip(t, x(·)) be the set of all functions z(·) ∈ C([−h, T ], R n ) that satisfy the condition z t (·) = x(·) and are Lipschitz continuous on [t, T ].
Assumption 2.1. The Hamiltonian H satisfies the conditions below: (i) For every s ∈ R n , the functional G ∋ (t, x(·)) → H(t, x(·), s) ∈ R is continuous.
(ii) There exists c H > 0 such that, for all (t, x(·)) ∈ G and all s, r ∈ R n , Note that Assumption 2.1 implies continuity of the Hamiltonian H.
In the literature, for the HJ equation (2.8), a Cauchy problem is usually studied under the right-end boundary condition where σ : C([−h, T ], R n ) → R is a given boundary functional, which is assumed to be at least continuous. However, since the presence of a specific boundary condition is not essential for the main result of the paper, we will mainly focus on minimax and viscosity solutions of the HJ equation (2.8) rather than those of the Cauchy problem (2.8), (2.9).

Minimax solutions
In order to give a definition of a minimax solution of the HJ equation (2.8), we need to introduce some notation.
Let us denote by Φ + the set of all functionals ϕ : G → R that satisfy the following condition of lower semi-continuity: for every point (t, x(·)) ∈ G and every sequence is valid provided that ρ 1 ((t (k) , x (k) (·)), (t, x(·))) → 0 as k → ∞ and there exists α > 0 such that Note that, if a functional ϕ : G → R is lower semicontinuous with respect to the auxiliary metric ρ 1 , then ϕ ∈ Φ + . On the other hand, any functional ϕ ∈ Φ + is lower semi-continuous (with respect to the basic metric ρ ∞ ).
In accordance with [34, Section 14], we give Note that Definition 2.2 can be considered as a quite natural extension of the definition of viscosity solutions of HJ equations with partial derivatives in terms of inequalities for sub-and supergradients from, e.g., [10] (see also, e.g., [45,Sections 4.4 and 6.3]) to the path-dependent setting.

Main result
The main result of the paper is Theorem 2.1. Let Assumption 2.1 hold. Then, a functional ϕ : G → R is an upper minimax (resp., a lower minimax, resp. a minimax) solution of the HJ equation (2.8) if and only if ϕ is an upper viscosity (resp., a lower viscosity, resp. a viscosity) solution of this equation.
The proof of Theorem 2.1 is given in Section 4. It is carried out by the scheme from [45,Theorem 4.3], where a similar result is obtained for HJ equations with partial derivatives (more precisely, see the proof of the equivalence (U 5) ⇔ (U 6) in [45,Section 4.6]). Note that this scheme was already partially adapted to the path-dependent setting in [37].
The next section is devoted to a discussion of Theorem 2.1 and some of its corollaries.

Discussion of main result and its corollaries
It is convenient to divide the discussion into three parts.
(iii) We call ϕ a minimax solution (resp., a viscosity solution) of the Cauchy problem (2.8), (2.9) if ϕ is an upper as well as a lower minimax (resp., an upper as well as a lower viscosity) solution of this problem.
In addition to Assumption 2.1, we make for all (t, x(·)), (t, y(·)) ∈ X and all s ∈ R n . (ii) The functional σ is continuous.
The next theorem is a comparison result for upper and lower viscosity solutions.
Theorem 3.1. Suppose that Assumptions 2.1 and 3.1 hold. Let ϕ + be an upper and let ϕ − be a lower viscosity solution of the Cauchy problem (2.8), (2.9). Then, the inequality below is valid:  Remark 3.1. The comparison theorems from [8,37,38,48,51,52] require that the correspondingly defined upper and lower viscosity solutions satisfy certain additional conditions like growth estimates, uniform continuity, or boundedness. In contrast to these results, Theorem 3.1 assumes only that ϕ + ∈ Φ + and ϕ − ∈ Φ − , i.e., these functionals ϕ + and ϕ − are only semi-continuous (see Section 2.4). As a consequence, we conclude that Corollary 3.1 ensures the uniqueness of a viscosity solution in a quite wide set of functionals Φ + ∩ Φ − . In addition, we note that Assumptions 2.1 and 3.1 are weaker than those considered in the cited papers.
Concerning an existence result for a viscosity solution of the Cauchy problem (2.8), (2.9), the reader is referred to Section 3.3 below.

Some other criteria for minimax and viscosity solutions
In the theory of HJ equations with partial derivatives, various criteria for minimax and viscosity solutions are known (see, e.g., [45,Sections 4.4 and 6.3] and [10,11]). Theorem 2.1 provides an analog of one of these criteria for the path-dependent HJ equation (2.8).
In this section we show that, in addition, Theorem 2.1 allows us to immediately obtain analogues of some other criteria. For brevity, we deal with upper solutions only, while the corresponding results for lower solutions can be derived similarly. We first consider a general case when ϕ ∈ Φ + (as in Definitions 2.1 and 2.2), and then we focus on a particular case when ϕ satisfies an additional local Lipschitz condition.
Let ϕ : G → R, let (t, x(·)) ∈ G 0 , and let L ⊂ R n be a non-empty convex compact set. The lower right derivative of the functional ϕ at the point (t, x(·)) in the multi-valued direction L is defined by where the symbol [L] ε stands for the closed ε-neighborhood of L in R n , and ∂ − ϕ(t, x(·); ω(·)) is the lower right derivative along the function ω(·) (see (2.16)).
Proposition 3.1. Suppose that Assumption 2.1 holds. Then, for every lower semicontinuous functional ϕ : G → R, condition (2.14) is equivalent to the following one: In condition (3.3), the functional ϕ s is defined by ϕ and s according to (2.13), and the set B cH (t, x(·)) is introduced in (2.12). Now, let us provide a similar criterion for an upper viscosity solution of (2.8).
The proof is given in Appendix A.1. Note that (3.3) implies (3.4), while the reverse implication is not trivial. However, Theorem 2.1 allows us to conclude that, under Assumption 2.1, these conditions are actually equivalent for every functional ϕ ∈ Φ + .
In addition, it seems interesting to obtain also a criterion for condition (2.20) in terms of ci-smooth test functionals. Proposition 3.3. For every functional ϕ : G → R, condition (2.20) is equivalent to the following one: for every point (t, x(·)) ∈ G 0 and every ci-smooth functional ψ : G → R, if, for any function z(·) ∈ Lip(t, x(·)), there exists δ z ∈ (0, T − t] such that (3.5) then the inequality below is fulfilled: The proof is given in Appendix A.1. Note that the condition formulated in Proposition 3.3 requires the point (t, x(·)) to be a local minimum of the difference between the functional ϕ and the test functional ψ only along every function z(·) ∈ Lip(t, x(·)), which correlates with the definition of ci-differentiability (see (2.7)).
Summarizing, we arrive at Remark 3.2. Based on the idea of [40], another approach to the development of the theory of viscosity solutions of the Cauchy problem (2.8), (2.9) was proposed in [32].
In that paper, the definition of a viscosity solution is given in terms of ci-smooth test functionals and involves a certain expanding sequence of compact subsets of G which union is dense in G. Under mild assumptions, it is proved that a viscosity solution is a minimax solution (defined with the help of conditions (2.14) and (2.15)) and that a viscosity solution is unique. Relying on these facts and a uniqueness result for minimax solutions, it is obtained that the minimax and viscosity solutions coincide. However, apart from this connection via minimax solutions and uniqueness results, it is not clear how the definition of a viscosity solution from [32] and that introduced in the present paper relate to each other.

Case of Lipschitz continuous functionals
In this section, we concretize infinitesimal conditions (2.20) and (3.4) in a particular case when ϕ satisfies an additional local Lipschitz condition in the second variable with respect to the auxiliary metric ρ 1 from (2.4).
Let us denote by Φ Lip the set of all functionals ϕ : G → R that satisfy the following local Lipschitz condition: for any α > 0, there exists λ ϕ > 0 such that Let ϕ : G → R, (t, x(·)) ∈ G 0 , and l ∈ R n . The lower right derivative of the functional ϕ at the point (t, x(·)) in the (single-valued) direction l is defined by where the function z [l] (·) ∈ Lip(t, x(·)) is given by and ∂ − ϕ(t, x(·); z [l] (·)) is the lower right derivative along the function z [l] (·) (see (2.16)). Let us first recall that, according to [31,Section 5.4] (see also [35,Proposition 4]), for every functional ϕ ∈ Φ Lip , every point (t, x(·)) ∈ G 0 , and every non-empty convex compact set L ⊂ R n , the equality below holds: As a consequence, for every Further, let us note the following property.
Proposition 3.4. Let ϕ ∈ Φ Lip and (t, x(·)) ∈ G 0 . Then, for every z(·) ∈ Lip(t, x(·)), there exists l ∈ R n such that The proof is given in Appendix A.2. Applying Proposition 3.4, we derive that, for every ϕ ∈ Φ Lip and every (t, x(·)) ∈ G 0 , the equalities below take place: and Thus, by Corollary 3.2, we have Theorem 3.2. Let Assumption 2.1 hold and let ϕ ∈ Φ + ∩ Φ Lip . Then, condition (2.14) is equivalent to each of the following two conditions: (i) The inequality below is valid: In other words, if we want to verify infinitesimal conditions (2.20) and (3.4) for a specific functional ϕ ∈ Φ Lip , we do not need to consider all functions z(·) from Lip(t, x(·)), and we can restrict ourselves only to the (finite-dimensional) set of the functions z [l] (·) for all l ∈ R n (see (3.8)).
Note that condition (3.13) clearly follows from condition (3.10), but, in order to obtain the reverse implication, we use Corollary 3.2 of Theorem 2.1. An existence result for a viscosity solution of the Cauchy problem (2.8), (2.9) that belongs to Φ Lip is given in the next section.

Application to optimal control problems
Path-dependent HJ equations of form (2.8) arise from optimal control problems and differential games for time-delay systems (see, e.g., [2,19,20,27,30,33]). In this regard, Theorem 2.1 can be applied to characterize the value functionals in such problems under quite general assumptions. The goal of this section is to illustrate this fact and, in particular, to show that the requirements ϕ ∈ Φ − and ϕ ∈ Φ + in Definitions 2.1 and 2.2 are not too restrictive.
Let us consider the optimal control problem described by an initial data (t, x(·)) ∈ G, the dynamic equationż under the initial condition z t (·) = x(·) (3.15) 13 and the cost functional to be minimized. Here, τ is time, z(τ ) ∈ R n is the current state, u(τ ) ∈ U is the current control, U ⊂ R m is a compact set, m ∈ N, and z τ (·) is the history of the system motion z(·) realized up to τ (see (2.6)).
satisfy the following conditions: (i) The mappings f and χ are continuous.
Then, the value functional ϕ 0 : G → R is given by According to, e.g., [33], the Bellman equation associated to the optimal control problem (3.14)-(3.16) is the path-dependent HJ equation (2.8) with the Hamiltonian H(t, x(·), s) min u∈U s, f (t, x(·), u) + χ(t, x(·), u) , ∀(t, x(·)) ∈ G, ∀s ∈ R n . (3.18) In particular, it follows from [33, Theorem 2] that, under Assumption 3.2, the value functional ϕ 0 is continuous and satisfies conditions (2.14) and (2.15) with H from (3.18). By construction, ϕ 0 meets the boundary condition (2.9). Moreover, we have The proof is given in Appendix A.3. Thus, we conclude that ϕ 0 is a minimax solution of the associated Cauchy problem (2.8), (2.9). Therefore, and since Assumption 3.2 implies that the Hamiltonian H from (3.18) and the boundary functional σ satisfy Assumptions 2.1 and 3.1, we obtain that ϕ 0 is a viscosity solution of this problem by Theorem 2.1. In addition, such a viscosity solution is unique due to Corollary 3.1. As a result, we get Finally, let us consider the following condition on the functional σ, which is stronger than condition (iv) of Assumption 3.2.
In this case, we have  In particular, Theorem 3.3 gives us a sufficient condition for existence of a viscosity solution of the Cauchy problem (2.8), (2.9), while Corollary 3.3 provides a sufficient condition for existence of a viscosity solution ϕ : G → R of this problem satisfying the additional requirement ϕ ∈ Φ Lip .

Proof of Theorem 2.1
This section is devoted to the proof of Theorem 2.1. Note that we prove only the first part of the theorem concerning upper solutions. The scheme of the proof is the following. In Section 4.1, we establish a variational principle, which is based on Borwein-Preiss variational principle from [25,Theorem 1]. In Section 4.2, we apply the variational principle to prove a certain property of the ci-subdifferential (see (2.17)), which can be seen as an analog of Property of Subdifferential [44, Theorem 1.1] (see also [6,Theorem 3.1] and [45, Section A6]). In Section 4.3, we provide three lemmas that imply the first part of Theorem 2.1. The second part, which concerns lower solutions, can be proved similarly, while the third part is a direct consequence of the two previous ones. 15

Variational principle
For every α > 0, let us denote The goal of this section is to prove Lemma 4.1. Let α > 0, let X ⊂ G be a non-empty closed set such that X ⊂ G(α) ∩ G 0 , and let ϕ : X → R be a functional that is lower semi-continuous and bounded from below. Then, for every κ ∈ (0, 1], there exist a functional ψ : G → R and a point (t * , x * (·)) ∈ X such that the following statements hold: (i) The inequality below is valid: The functional ψ is ci-differentiable at the point (t * , x * (·)), and the corresponding ci-derivatives satisfy the estimates We prove this lemma on the basis of the variational principle from [25,Theorem 1]. In order to apply this principle, we first need to construct a guage-type functional.

Guage-type functional
Following [50], let us consider the functional V : G → R given by for all (t, x(·)) ∈ G. Recall that the notation x(·) [−h,t] is introduced in (2.1). By [50, Lemma 2.3] (see also [18,Section 4 and Appendix B]), the functional V is ci-smooth and its ci-derivatives are as follows: and for all (t, x(·)) ∈ G 0 . Moreover, the estimates below hold:

Comments
In this section, we make some comments concerning Lemma 4.1 and its proof. 3] (see also, e.g., formula (4.5) in that paper), we note that, according to (4.4), the point (t * , x * (·)) provides the minimum over the whole set X and not only over its subset {(t, x(·)) ∈ X | t ≥ t * }. This difference is important for the proof of Lemma 4.3 given below (see, e.g., relation (4.52)).
(ii) The fact that we define the functional µ α equal to c α for t < τ (see (4.9)) is used to obtain inequalities (4.18), which allow us to get ci-differentiability of the functional ψ at the point (t * , x * (·)) directly by ci-smoothness of the functional V .
Below, we make some comments on Lemma 4.3 and its proof.  .40) is not compact with respect to the auxiliary metric ρ 1 and, therefore, in view of (2.5), with respect to the metric ρ ∞ too. Indeed, let us fix e ∈ R n with e = 1 and ω(·) ∈ Ω and, for every k ∈ N, consider the functioñ x (k) (·) ∈ C([−h, t * + δ], R n ) given bỹ if ξ ∈ (t * + δ/k, t * + δ]. Then, we have {(t * + δ,x (k) (·))} ∞ k=1 ⊂ X. However, the sequence {(t * + δ,x (k) (·))} ∞ k=1 does not have a subsequence converging to some point (t * + δ,x(·)) ∈ G with respect to ρ 1 , since, roughly speaking, this sequence actually converges with respect to ρ 1 but to the point (t * + δ,x * (·)) with the discontinuous functionx * : [−h, t * + δ] → R defined bỹ As a result, we conclude that the existence of a minimum of the functional ϕ k from (4.46) on X does not follow from lower semi-continuity of ϕ k or even from the inclusion ϕ k ∈ Φ + , which is valid since functional (4.47) in fact belongs to Φ + . To overcome this difficulty, Lemma 4.1 is applied. (ii) The only place in the proof where we use the assumption ϕ ∈ Φ + and not just the fact that ϕ is lower semi-continuous is the proof of inequality (4.58), which in turn is based on relation (4.60). 28 (iii) The choice of the functional Ψ in form (4.43) leads us to the fact that we derive convergence (4.54) with respect to the auxiliary metric ρ 1 . In order to obtain such a convergence but with respect to the metric ρ ∞ , which would allow us to replace the assumption ϕ ∈ Φ + with the requirement that ϕ is lower semi-continuous, we could try to take the functionalV from (4.28) instead of Ψ and use the first inequality in (4.7). However, in this case, there would be difficulties with the ci-differentiability properties (4.68) and (4.70). Indeed, as shown in Example 4.1 (see also item (iii) of Remark 4.1), one of these properties may fail to be valid if t (k) = τ (k) .
In conclusion of this section, let us make a comment additional to the discussion from Section 3.2.2. Note that, similarly to (3.9), for any functional ϕ ∈ Φ Lip , any point (t, x(·)) ∈ G 0 , and any non-empty convex compact set L ⊂ R n , we have This equality can be proved by essentially repeating the arguments from [31, Section 5.4], since, in order to obtain a relation like (5.8) in that paper, we can use definition (4.32) of the value d − 0 ϕ(t, x(·); L). In this connection, see also [35,Proposition 4]. Hence, Lemma 4.3 implies Corollary 4.1. Let ϕ ∈ Φ + ∩ Φ Lip , let (t * , x * (·)) ∈ G 0 , and let L ⊂ R n be a non-empty convex compact set. Suppose that inf l∈L ∂ − * ϕ(t * , x * (·); l) > 0.
Thus, comparing Corollary 4.1 with [37, Lemma 4.7], we note that an important difference is that the latter result provides the existence of a point (t, x(·)) and a pair (p 0 , p) satisfying inequality (4.36) but only with a piecewise continuous function x(·), which is due to a different technique used in the proof. In turn, this circumstance leads to the need to consider the HJ equation (2.8) in the wider space of all piecewise continuous functions.  Proof. It follows from Proposition 3.1 that (2.14) implies (3.3). Hence, it suffices to show that (3.3) implies (2.20). Note that this fact was proved in [34,Proposition 14.1]. However, since the proof is available in Russian only, we present it below for the reader's convenience.
Proof. The proof can be carried out by repeating the arguments from the second part of the proof of [29, Theorem 8.1] (see also Proposition 3.1 above). Indeed, the only difference is that the inequality in (4.78) involves the value d − 0 ϕ s (t, x(·); B cH (t, x(·))) (see (4.32)) instead of the directional derivative d − ϕ s (t, x(·); B cH (t, x(·))) (see (3.1)). Nevertheless, it follows directly from (4.32) that condition (4.78) is sufficient in order to obtain relations (8.15) and (8.16) from [29]. The details are omitted.
Remark 4.4. Let us suppose that Assumption 2.1 hold and take a lower semi-continuous functional ϕ : G → R. Then, by Proposition 3.1, condition (2.14) implies condition (3.3), which, in turn, implies condition (4.78) due to inequality (4.33). Thus, as a corollary of Lemma 4.5, we obtain that all three conditions (2.14), (3.3), and (4.78) are equivalent. The proof of Lemma 4.6 uses the following lower semi-continuity property of the value d − 0 ϕ(t, x(·); L) from (4.32) with respect to the variation of the set L.