A Zero-Sum Deterministic Impulse Controls Game in Infinite Horizon with a New HJBI QVI
aa r X i v : . [ m a t h . O C ] J a n A Zero-Sum Deterministic Impulse Controls Game inInfinite Horizon with a New HJBI QVI
Brahim El Asri ∗ , Hafid Lalioui † and Sehail Mazid ‡ Abstract
In the present paper, we study a two-player zero-sum deterministic differential gamewith both players adopting impulse controls, in infinite time horizon, under rather weakassumptions on the cost functions. We prove by means of the dynamic programming prin-ciple (DPP) that the lower and upper value functions are continuous and viscosity solutionsto the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality(QVI). We define a new HJBI QVI for which, under a proportional property assumption onthe maximizer cost, the value functions are the unique viscosity solution. We then provethat the lower and upper value functions coincide.
Keywords:
Deterministic differential game, Impulse controls, Dynamic programming principle,Viscosity solutions, Quasi-variational inequality.
AMS classifications:
Differential games are concerned with the problem that multiple players make decisions,according to their own advantages and trade-off with other peers, in the context of dynamicsystems. The theory of two-player zero-sum differential games was initiated by R. Isaacs [23]and L.S. Pontryagin et al [8] at the beginning of 60’s, in the early 80’s the theory of viscositysolutions was pioneered by the seminal papers of M.G. Crandall and P.L. Lions [10] and M.G.Crandall et al [11]. The notion of strategies and the rigorous definitions of lower and uppervalue functions are due to R.J. Elliott and N.J. Kalton [20, 21], L.C. Evans and P.E. Souganidis[22] began to study differential games by means of the viscosity theory, proving that the twovalue functions are the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman-Issacs (HJBI) partial differential equations (PDEs) for finite horizon problem, M. Bardi and ∗ Ibn Zohr University, Equipe. Aide à la decision, ENSA, B.P. 1136, Agadir, Morocco. e-mail:[email protected]. † Ibn Zohr University, Equipe. Aide à la decision, ENSA, B.P. 1136, Agadir, Morocco. e-mail:hafi[email protected]. Financially supported by CNRST, Rabat, Morocco (Grant 17 UIZ 2019). ‡ Ibn Zohr University, Department of Mathematics, Faculty of Sciences, Morocco. e-mail: [email protected].
1. Capuzzo-Dolcetta [3] described the implementation of the viscosity solutions approach to anumber of significant model problems in optimal deterministic control and differential games.The deterministic differential games, apart from the mathematical interest in its own right,enjoy a wide range of applications in various fields of engineering, such as medicine, biology,economics and finance, see for more information A. Bensoussan and J.L. Lions [7]. The deter-ministic impulse control problems in finite horizon were studied by many authors, J.M. Yong [31]considered problems where one player takes continuous controls whereas the other uses impulsecontrol, G. Barles et al [4] treated a minimax problem driven by two controls, one is continuousand another impulsive. For the infinite horizon case as considered in the present paper, we citethe works of S. Dharmatti and A.J. Shaiju [13, 15], S. Dharmatti and M. Ramaswamy [14] andG. Barles [5].In this paper, we consider the state y . ( . ) of the two-player zero-sum deterministic differentialgame involving impulse controls in infinite time horizon described by the solution of the followingsystem: y x (0) = x ∈ R n ;˙ y x ( t ) = b (cid:16) y x ( t ) (cid:17) , t = τ m , t = ρ k ; y x ( τ + m ) = y x ( τ − m ) + ξ m Y k ≥ { τ m = ρ k } , τ m ≥ , ξ m = 0; y x ( ρ + k ) = y x ( ρ − k ) + η k , ρ k ≥ , η k = 0 , (1)where m ∈ N ∗ , k ∈ N ∗ and y x ( t ) is the state variable of the system at time t , R n -valued, withinitial state y x (0) = x . The state y . ( . ) is driven by two impulse controls, u control of player − ξ defined by a double sequence ( τ m , ξ m ) m ≥ and v control of player − η defined by a double sequence ( ρ k , η k ) k ≥ , the actions ξ m and η k belong to the spaces of control actions U ⊂ R n and V ⊂ R n ,respectively. The infinite product Q k ≥ { τ m = ρ k } signifies that when the two players act togetheron the system at the same time, we take into account only the action of player − η . The gain(resp. cost) functional J for player − ξ (resp. player − η ) is defined as follows: J ( x ; u, v ) = Z ∞ f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m ) Y k ≥ { τ m = ρ k } + X k ≥ χ ( η k ) exp ( − λρ k ) , where c and χ are the zero lower bound impulse cost functions for player − ξ and player − η ,respectively, and the discount factor λ is a positive real. We note that the cost of a player is thegain for the other (zero-sum), meaning that when a player performs an action he/she has to paya positive cost, resulting in a gain for the other player. The function f represents the runninggain.The terminology of a quasi-variational inequality (QVI) was introduced in [7] to deal withimpulse control problems. The definition of lower and upper value functions for a differentialgame as defined in [20, 21, 22] leads to characterize the value of the game as the unique viscositysolution of a corresponding QVI. Moreover the relationship between the theory of two-player zero-sum deterministic differential games and viscosity solutions was first shown in [22], N. Barron etal [2] and P.E. Souganidis [29, 30].Recently, A. Cosso [9] and B. El Asri and S. Mazid [17] studied a stochastic impulse controlsproblem in finite horizon. They have proved, based on the dynamic programming principle(DPP) and viscosity solutions theory, that the differential game admits a value, however in both2orks the authors impose a stronger constraint that involves both cost functions, which is givenby: ∃ h : [0 , T ] → (0 , + ∞ ) such that c ( t, ξ + η + ξ ) ≤ c ( t, ξ ) − χ ( t, η ) + c ( t, ξ ) − h ( t ) , where c and χ are the impulse cost functions from U and V , respectively, ξ , ξ ∈ U and η ∈ V ,as a consequence they had to require V ⊂ U ⊂ R n .Our aim in this work is to investigate the two-player zero-sum deterministic impulse controlsproblem in infinite horizon given by the system (1). In particular, we describe the problemby a classic HJBI QVI, which we replace by a new HJBI QVI in order to characterize, inviscosity solution sense under rather weak assumptions on the cost functions, the value functionof the differential game studied as the unique viscosity solution of the new HJBI QVI. In thiswork we only adopt, in addition to the classical assumptions of the impulse control problems, aproportional property assumption on the maximizer cost function c which is given by: ∀ k > , ∀ ξ ∈ U such that kξ ∈ U we have c ( kξ ) ≤ kc ( ξ ) , (2)note that assumption (2) is of great interest in the literature, as an application we cite the workdeveloped in recent years in the field of biology, see L. Mailleret and F.Grognard [28].For our game the associated QVI is given by the following double-obstacle HJBI equation,where the Hamiltonian involves only the first order partial derivatives: max n min h λv ( x ) − Dv ( x ) .b ( x ) − f ( x ) , v ( x ) − H csup v ( x ) i , v ( x ) − H χinf v ( x ) o = 0 , (3)where Dv ( . ) denotes the gradient of the function v : R n → R , and the first (resp. second)obstacle is defined through the use of the infimum (resp. maximum) cost operator H χinf (resp. H csup ), where H χinf v ( x ) = inf η ∈ V h v ( x + η ) + χ ( η ) i (cid:16) resp. H csup v ( x ) = sup ξ ∈ U h v ( x + ξ ) − c ( ξ ) i(cid:17) . We prove, using the dynamic programming principle, the existence of the value functions for ourdifferential game as viscosity solutions of the HJBI QVI (3), but the uniqueness of the viscositysolution is not guarantee under standing assumptions, which means that the value functioncannot enjoy anymore the property being the unique viscosity solution of the classic HJBI QVI(3).Furthermore, we consider a new HJBI QVI, where the term of impulsions v ( . ) − H csup v ( . ) (second obstacle) is replaced by the differential term F cinf (cid:16) Dv ( . ) (cid:17) defined for all x ∈ R n bymeans of the operator F cinf as follows: F cinf (cid:16) Dv ( x ) (cid:17) = inf ξ ∈ U h − Dv ( x ) .ξ + c ( ξ ) i . (4)Therefore, under assumption (2) and classical assumptions of the impulse control problems, weshow the existence and the uniqueness results in the viscosity solution sense for the new HJBIQVI. Indeed, for the existence result, we give an equivalence in the viscosity supersolution sensebetween the classic HJBI QVI (3) and the new HJBI QVI, then, for the uniqueness, we establisha comparison theorem. 3he paper is organized as follows: in section 2, we present the impulse controls problemstudied, we give its related definitions and assumptions and we introduce our new HJBI QVI.In section 3, we prove some classic results on the lower and upper value functions, we first showthat both satisfy the dynamic programming principle property, then we prove that they arecontinuous in R n . Section 4 is devoted, on one hand, to the viscosity characterization of theclassic HJBI QVI (3) by deducing that both value functions are its viscosity solutions, and, onthe other hand, we prove that the new HJBI QVI has the same bounded continuous viscositysupersolutions as the classic HJBI QVI (3), then we deduce the viscosity characterization of thenew HJBI QVI. In the last section, we look more carefully to the new HJBI QVI by proving thatthe value functions of our infinite horizon two-player zero-sum impulse controls problem are hisunique viscosity solution. Further, the lower and upper value functions coincide and the gameadmits a value. Throughout this paper, we let n be a fixed positive integer, the time variable T belongs to [0 , + ∞ ] , k and m are in N ∗ and we let the discount factor λ be a fixed positive real.Let us assume H1 : [ H b,f ] : The functions b : R n → R n and f : R n → R are bounded and Lipschitz continuous withconstant C b and C f , respectively. [ H c,χ ] : The cost functions c : U → R + and χ : V → R + are from two subsets of R n , U and V ,respectively, into R + , non negative and satisfy the zero lower bound property given by: inf ξ ∈ U c ( ξ ) > and inf η ∈ V χ ( η ) > . Also for all ξ , ξ ∈ U and η , η ∈ V , we let the impulse cost functions satisfy c ( ξ + ξ ) ≤ c ( ξ ) + c ( ξ ) and χ ( η + η ) < χ ( η ) + χ ( η ) . Moreover, we assume H2 that encompass the proportional impulse costs for player − ξ , thatis the function c satisfies ∀ k > , ∀ ξ ∈ U such that kξ ∈ U we have c ( kξ ) ≤ kc ( ξ ) . Regarding assumption H1 , [ H b,f ] implies that there exists a unique global solution y x ( . ) tothe above dynamical system (1), while the assumption [ H c,χ ] provides the classical frameworkfor the study of the impulse control problems. Assumption H2 , which is of great interest inthe literature, leads to the existence and the uniqueness results for the new HJBI QVI definedhereafter in (6).For the rest of the paper we denote by | . | and k . k the Euclidian vector norm in R and R n ,respectively, and for a bounded and continuous function F from R n to R (resp. R n ) we define k F k ∞ = sup x ∈ R n | F ( x ) | (cid:16) resp. k F k ∞ = sup x ∈ R n k F ( x ) k (cid:17) . .2 Impulse Controls Game Problem Here we shall be interested in the two-player zero-sum deterministic differential game de-scribed in the introduction set by the dynamical system (1). The horizon (the interval in whichtime varies) is infinite. The state of the system y . ( t ) at the instant t lies in R n , with initialvalue y x (0) = x . The mapping t → y x ( t ) describe the evolution of the system provided by adeterministic model ˙ y x ( t ) = b (cid:16) y x ( t ) (cid:17) , where b is a function from R n to R n satisfies assumption [ H b,f ] .At certain impulse instants τ m and ρ k , the state undergoes impulses (jumps) ξ m and η k , respec-tively, that is: y x ( τ + m ) = y x ( τ − m ) + ξ m Y k ≥ { τ m = ρ k } , τ m ≥ , ξ m = 0; y x ( ρ + k ) = y x ( ρ − k ) + η k , ρ k ≥ , η k = 0 , the impulse time sequences { τ m } m ≥ and { ρ k } k ≥ are two non-decreasing sequences of [0 , ∞ ] which satisfy τ m , ρ k → + ∞ when m, k → + ∞ , and the impulse value sequences { ξ m } m ≥ and { η k } k ≥ are two sequences of elements of U ⊂ R n and V ⊂ R n , respectively.The state y x ( . ) of the system is driven by two impulse controls, ( τ m , ξ m ) m ≥ control of player − ξ and ( ρ k , η k ) k ≥ control of player − η . The infinite product Q k ≥ { τ m = ρ k } signifies that when thetwo players act together on the system at the same time, only the action of player − η is tackinginto account.We call U (resp. V ) the space of impulse control u (resp. v ) for player − ξ (resp. player − η )and we denote u := ( τ m , ξ m ) m ≥ (cid:16) resp. v := ( ρ k , η k ) k ≥ (cid:17) . For any initial state x the controls u and v generate a trajectory y x ( . ) solution of the system (1). We are given a gain (resp. cost)functional J ( x ; u, v ) for player − ξ (resp. player − η ), which represents the criterion to maximize(resp. minimize) by applying the control u (resp. v ): J ( x ; u, v ) = Z ∞ f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m ) Y k ≥ { τ m = ρ k } + X k ≥ χ ( η k ) exp ( − λρ k ) , where c and χ are the impulse cost functions (jump costs) from U and V , respectively, R + -valuedand satisfy assumption [ H c,χ ], the running gain f : R n → R satisfies assumption [ H b,f ] and λ ≥ ,representing the discount factor.Typically, in a two-player game, the player who moves first would not choose a fixed action.Instead, he/she prefers to employ a strategy which can give different responses to different futureactions the other player will take. Hence, besides the admissible controls, we define, following[20, 21], the notion of non-anticipative strategies as follows: Definition 1. (Non-anticipative strategy) The non-anticipative strategy set A for player − ξ isthe collection of all non-anticipative maps α from V to U , i.e., for any v and v in V , if v ≡ v ,then α ( v ) ≡ α ( v ) .The non-anticipative strategy set B for player − η is the collection of all non-anticipative maps β from U to V , i.e., for any u and u in U , if u ≡ u , then β ( u ) ≡ β ( u ) . In the game, player − ξ aims to maximize the gain functional J and contrarily player − η aims to minimize. We may now give the definition of the lower and upper value functions for5ur two-player zero-sum deterministic differential game.We define the lower value function V − ( . ) and the upper value function V + ( . ) of the game,respectively, by the following expressions: V − ( x ) = inf β ∈B sup u ∈U J (cid:16) x ; u, β ( u ) (cid:17) ; V + ( x ) = sup α ∈A inf v ∈V J (cid:16) x ; α ( v ) , v (cid:17) . If V − = V + we say that the game admits a value and V := V − = V + is called the value function of the game. For the impulse controls problem studied in the present paper, the associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality turns out to be the same for the two valuefunctions, because of the two players can not act simultaneously on the system, and it is givenby the following double-obstacle equation: max n min h λv ( x ) − Dv ( x ) .b ( x ) − f ( x ) , v ( x ) − H csup v ( x ) i , v ( x ) − H χinf v ( x ) o = 0 , (5)where H χinf and H csup are the cost operators defined, respectively, as follows: H χinf v ( x ) = inf η ∈ V h v ( x + η ) + χ ( η ) i ; H csup v ( x ) = sup ξ ∈ U h v ( x + ξ ) − c ( ξ ) i , and Dv ( . ) denotes the gradient of the function v : R n → R .Therefore, we define the new Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality (6),where the term of impulsions v ( . ) − H csup v ( . ) is replaced by the differential term F cinf (cid:16) Dv ( . ) (cid:17) ,through the use of the operator F cinf as follows: max n min h λv ( x ) − Dv ( x ) .b ( x ) − f ( x ) , F cinf (cid:16) Dv ( x ) (cid:17)i , v ( x ) − H χinf v ( x ) o = 0 , (6)where the operator F cinf is defined as follows: F cinf (cid:16) Dv ( x ) (cid:17) = inf ξ ∈ U h − Dv ( x ) .ξ + c ( ξ ) i . Note that a differential term as in QVI (6) was introduced in G. Barles [6] and used, to deal withthe particular case of null infimum jump costs in the infinite horizon impulse control problem,in N. El Farouq [19].The main objectives of this paper are: (i)
Focusing on the existence of the solution in viscosity sense for both quasi-variational in-equalities (5) and (6). 6 ii)
Showing that the QVI (6) admits the lower and upper value functions as the unique solutionof viscosity.For the rest of the paper we call QVI (5) the classic HJBI QVI, we call QVI (6) the newHJBI QVI and we adopt the following definition of the viscosity solution:
Definition 2. (Viscosity Solution) Let V : R n → R be a continuous function. V is called: (i) A viscosity subsolution of the classic HJBI QVI (resp. new HJBI QVI) if for any x ∈ R n and any function φ ∈ C ( R n ) such that V ( x ) = φ ( x ) and x is a local maximum point of V − φ , we have: max n min h λV ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , V ( x ) − H csup V ( x ) i , V ( x ) − H χinf V ( x ) o ≤ , (cid:16) resp. max n min h λV ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , F cinf (cid:16) Dφ ( x ) (cid:17)i , V ( x ) − H χinf V ( x ) o ≤ (cid:17) . (ii) A viscosity supersolution of the classic HJBI QVI (resp. new HJBI QVI) if for any x ∈ R n and any function φ ∈ C ( R n ) such that V ( x ) = φ ( x ) and x is a local minimum point of V − φ , we have: max n min h λV ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , V ( x ) − H csup V ( x ) i , V ( x ) − H χinf V ( x ) o ≥ , (cid:16) resp. max n min h λV ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , F cinf (cid:16) Dφ ( x ) (cid:17)i , V ( x ) − H χinf V ( x ) o ≥ (cid:17) . (iii) A viscosity solution of the classic HJBI QVI (resp. new HJBI QVI) if it is both a viscositysubsolution and supersolution of the classic HJBI QVI (resp. new HJBI QVI).
Letting y x ( . ) and y x ′ ( . ) be the trajectory generated by u ∈ U and v := β ( u ) ∈ V from x and x ′ , respectively, where β ∈ B . We then have the following characterization of the trajectories y . ( . ) , for which the proof follows from Gronwall’s Lemma and can be found in P.L. Lions [24]: Lemma 1.
Under assumption H1 , for any x, x ′ ∈ R n and any t ≥ we have: k y x ( t ) − y x ′ ( t ) k≤ exp ( C b t ) k x − x ′ k . We show hereafter that the lower and upper value functions are bounded in R n . Proposition 1.
Under assumption H1 , the lower and upper value functions are bounded in R n .Proof. We make the proof only for the lower value function V − , the other case being analogous.By the definition of V − , for all x ∈ R n and all non-anticipative strategy β ∈ B we have V − ( x ) ≤ sup u ∈U J (cid:16) x ; u, β ( u ) (cid:17) . β ( u ) := ( ρ k , η k ) k ≥ for player − η wherethere is no impulse time, i.e., ρ = + ∞ , we get V − ( x ) ≤ sup u ∈U (cid:20)Z ∞ f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m ) (cid:21) . Next, for all ε > , there exists a strategy u ε := ( τ εm , ξ εm ) ∈ U such that V − ( x ) ≤ Z ∞ f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ εm ) exp ( − λτ εm ) + ε. Since c is a non negative function and f is bounded, we then get the existence of a constant C > such that V − ( x ) ≤ C. Similarly, for the set of strategies u ∈ U for player − ξ for which there is no impulse time, i.e., τ = + ∞ , we have V − ( x ) ≥ inf β ∈B (cid:20)Z ∞ f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt + X k ≥ χ ( η k ) exp ( − λρ k ) (cid:21) . Let ε > , then there exists a strategy β ε ( u ) := ( ρ εk , η εk ) ∈ V where β ε ∈ B , for which we have V − ( x ) + ε ≥ Z ∞ f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt + X k ≥ χ ( η εk ) exp ( − λρ εk ) . Since χ is a non negative function and f is bounded, we deduce the existence of a constant C > such that V − ( x ) ≥ − C . Hence we obtain the thesis.
We now present, for our two-player zero-sum deterministic differential game in infinite timehorizon, the dynamic programming principle in the following theorem. The DPP, as one of theprinciple and most commonly used approaches in solving optimal control problems, meaningthat an optimal control viewed from today will remain optimal when viewed from tomorrow andstands for a basic property in dealing with our problem.
Theorem 1. (Dynamic Programming Principle) Under assumption H1 , given x ∈ R n and T > , we have the dynamic programming principle: V − ( x ) = inf β ∈B sup u ∈U (cid:20)Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ k } + X k ≥ χ ( η k ) exp ( − λρ k )11 { ρ k ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT ) (cid:21) , (7)8 nd V + ( x ) = sup α ∈A inf v ∈V (cid:20)Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ k } + X k ≥ χ ( η k ) exp ( − λρ k )11 { ρ k ≤ T } + V + (cid:16) y x ( T ) (cid:17) exp ( − λT ) (cid:21) . Proof.
We give the proof only for the lower value function V − , similarly for V + . We first let ε > , u ∈ U and assume, for some x ∈ R n and some T > , that V − ( x ) < W T ( x ) , where W T ( x ) is the right-hand side of (7). We let the difference be W T ( x ) − V − ( x ) = 2 ε and we choose β ε anon-anticipative strategy that approximates V − ( x ) up to ε , and denote β ε ( u ) := ( ρ εk , η εk ) k ≥ thejumps it produces. We then have W T ( x ) − ε ≥ sup u ∈U (cid:20)Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + J (cid:16) y x ( T ); u, β ε ( u ) (cid:17)(cid:21) . The above inequality can be rewritten, for U and U T the restrictions of U to [0 , T ] × U and [ T, + ∞ ] × U , respectively, as follows: W T ( x ) − ε ≥ sup u ∈U (cid:20)Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + sup u ∈U T J (cid:16) y x ( T ); u, β ε ( u ) (cid:17)(cid:21) . (8)Observe that once y x ( T ) is known, the knowledge of u over [0 , T ] × U is useless to evaluate sup u ∈U T J . Therefore, since λ ≥ , the restriction of U to U T satisfies V − (cid:16) y x ( T ) (cid:17) exp ( − λT ) ≤ sup u ∈U T J (cid:16) y x ( T ); u, β ε ( u ) (cid:17) , replacing in inequality (8) leads to a contradiction. Finally, for all x ∈ R n and all T > , wededuce V − ( x ) ≥ W T ( x ) . Now let us assume to the contrary, for some x ∈ R n and some T > , that V − ( x ) > W T ( x ) andlet the difference be V − ( x ) − W T ( x ) = 3 ε for ε > . We denote by β ε ∈ B the non-anticipativestrategy that approximates W T ( x ) up to ε , where β ε ( u ) := ( ρ εk , η εk ) k ≥ are the jumps it producesfor u := ( τ m , ξ m ) m ≥ ∈ U . We then get V − ( x ) − ε ≥ sup u ∈U (cid:20)Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT ) (cid:21) , u ∈ U the restriction of U to [0 , T ] , we have V − ( x ) − ε ≥ Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT ) . Furthermore, we choose a non-anticipative strategy β ε ∈ B of the game over [ T, + ∞ ] thatapproximates V − (cid:16) y x ( T ) (cid:17) again up to ε . The concatenation β ε of β ε and β ε is a non-anticipativestrategy of the game over [0 , + ∞ ] , then we deduce for the non-anticipative strategy β ε ∈ B andall control u ∈ U the following: V − ( x ) − ε ≥ J (cid:16) x ; u, β ε ( u ) (cid:17) , a contradiction. Finally, for all x ∈ R n and all T > , we deduce V − ( x ) ≤ W T ( x ) . The proof is now complete.We next use the DPP to show that both the lower value function and the upper value functionare continuous in R n . Theorem 2.
Under assumption H1 , the lower and upper value functions are continuous in R n .Proof. We make the proof only for the lower value function V − , the other case being analogous.We first show that V − is upper semi-continuous. For any x, x ′ ∈ R n and T > , according tothe DPP for the lower value function, we have V − ( x ) = inf β ∈B sup u ∈U (cid:20)Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ k } + X k ≥ χ ( η k ) exp ( − λρ k )11 { ρ k ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT ) (cid:21) and V − ( x ′ ) = inf β ∈B sup u ∈U (cid:20)Z T f (cid:16) y x ′ ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ k } + X k ≥ χ ( η k ) exp ( − λρ k )11 { ρ k ≤ T } + V − (cid:16) y x ′ ( T ) (cid:17) exp ( − λT ) (cid:21) . Now fix an arbitrary ε > and pick, for β ε ∈ B , a strategy β ε ( u ) := ( ρ εk , η εk ) k ≥ which satisfiesthe following: V − ( x ′ ) + ε ≥ sup u ∈U (cid:20)Z T f (cid:16) y x ′ ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ m ) exp ( − λτ m )11 { τ m ≤ T } Y k ≥ { τ m = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + V − (cid:16) y x ′ ( T ) (cid:17) exp ( − λT ) (cid:21) . (9)10urther, we pick a control u ε := ( τ εm , ξ εm ) m ≥ ∈ U which satisfies the following: V − ( x ) − ε ≤ Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ εm ) exp ( − λτ εm )11 { τ εm ≤ T } Y k ≥ { τ εm = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT ) . Since the inequality (9) holds for any control u , then u ε satisfies V − ( x ′ ) + ε ≥ Z T f (cid:16) y x ′ ( t ) (cid:17) exp ( − λt ) dt − X m ≥ c ( ξ εm ) exp ( − λτ εm )11 { τ εm ≤ T } Y k ≥ { τ εm = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + V − (cid:16) y x ′ ( T ) (cid:17) exp ( − λT ) . It follows from the two last inequalities that V − ( x ) − V − ( x ′ ) ≤ Z T h f (cid:16) y x ( t ) (cid:17) − f (cid:16) y x ′ ( t ) (cid:17)i exp ( − λt ) dt − X m ≥ c ( ξ εm ) exp ( − λτ εm )11 { τ εm ≤ T } Y k ≥ { τ εm = ρ εk } + X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT )+ X m ≥ c ( ξ εm ) exp ( − λτ εm )11 { τ εm ≤ T } Y k ≥ { τ εm = ρ εk } − X k ≥ χ ( η εk ) exp ( − λρ εk )11 { ρ εk ≤ T } − V − (cid:16) y x ′ ( T ) (cid:17) exp ( − λT ) + 2 ε. Thus, we get V − ( x ) − V − ( x ′ ) ≤ Z T k f (cid:16) y x ( t ) (cid:17) − f (cid:16) y x ′ ( t ) (cid:17) k exp ( − λt ) dt + (cid:12)(cid:12)(cid:12) V − (cid:16) y x ( T ) (cid:17) − V − (cid:16) y x ′ ( T ) (cid:17)(cid:12)(cid:12)(cid:12) exp ( − λT ) + 2 ε. By Lemma 1, the Lipschitz continuity of f and the boundedness of V − , we deduce that thereexists a constant C > such that V − ( x ) − V − ( x ′ ) ≤ C f k x − x ′ k Z T exp (cid:16) ( C b − λ ) t (cid:17) dt + 2 Cexp ( − λT ) + 2 ε. (10)Therefore, if λ = C b , we obtain V − ( x ) − V − ( x ′ ) ≤ C f C b − λ k x − x ′ k h exp (cid:16) ( C b − λ ) T (cid:17) − i + 2 Cexp ( − λT ) + 2 ε. (11)Now we choose T such that exp ( − C b T ) = k x − x ′ k / with k x − x ′ k < . Hence, in theright-hand side of (11), the first term goes to when x → x ′ , i.e., T → ∞ , indeed, it is equal to C f C b − λ k x − x ′ k / (cid:16) exp ( − λT ) − k x − x ′ k / (cid:17) , where T → ∞ . We then deduce, by letting x → x ′ and ε → ,the upper semi-continuity of the lower value function: lim sup x → x ′ V − ( x ) ≤ V − ( x ′ ) . In the case where λ = C b , it suffice to let some ˆ λ < λ = C b , so we go back to (10) and weproceed, since exp (cid:16) ( C b − λ ) T (cid:17) < exp (cid:16) ( C b − ˆ λ ) T (cid:17) and exp ( − λT ) < exp ( − ˆ λT ) , as above withthe case ˆ λ = C b , we then conclude by letting x → x ′ and ε → .Analogously we get the lower semi-continuity: lim inf x → x ′ V − ( x ) ≥ V − ( x ′ ) . Then the lower value function is continuous in R n . The aim of the present section is to show that the lower and upper value functions are viscositysolutions to both classic HJBI QVI and new HJBI QVI. To do so we first give, in Lemma 2, someproperties of the value functions, hence we get the aim for the classic HJBI QVI. Furthermore,we show the equivalence in viscosity supersolution sense between the classic HJBI QVI and thenew HJBI QVI.We begin with the following technical lemma:
Lemma 2.
Under assumption H1 , the lower value function satisfies for all x ∈ R n the followingproperty: V − ( x ) ≤ H χinf V − ( x ) . Let x ∈ R n be such that V − ( x ) < H χinf V − ( x ) , then V − ( x ) ≥ H csup V − ( x ) .The same results hold true for the upper value function V + .Proof. We give the proof for the lower value function V − , similarly for V + . Letting x ∈ R n andconsidering, for player − η , the strategy β ( u ) := ( ρ k , η k ) k ≥ ∈ V where β ∈ B . Next, choose β ′ ∈ B such that β ′ ( u ) := (0 , η ; ρ , η ; ρ , η ; ..... ) , we then obtain V − ( x ) ≤ sup u ∈U J (cid:16) x ; u, β ′ ( u ) (cid:17) = sup u ∈U J (cid:16) x + η ; u, β ( u ) (cid:17) + χ ( η ) , from which we deduce the following inequality: V − ( x ) ≤ inf η ∈ R n h V − ( x + η ) + χ ( η ) i . Now let us assume that V − ( x ) < H χinf V − ( x ) for some x ∈ R n . From the DPP for V − , by taking T = 0 , we get V − ( x ) = inf ρ ∈{ , + ∞} , η ∈ V sup τ ∈{ , + ∞} , ξ ∈ U (cid:20) − c ( ξ )11 { τ =0 } { ρ =+ ∞} + χ ( η )11 { ρ =0 } + V − ( x + ξ { τ =0 } { ρ =+ ∞} + η { ρ =0 } ) (cid:21) , V − ( x ) = inf ρ ∈{ , + ∞} " inf η ∈ V h χ ( η ) + V − ( x + η ) i { ρ =0 } + sup τ ∈{ , + ∞} , ξ ∈ U h − c ( ξ )11 { τ =0 } + V − ( x + ξ { τ =0 } ) i { ρ =+ ∞} . Since V − ( x ) < H χinf V − ( x ) , we get V − ( x ) = sup τ ∈{ , + ∞} , ξ ∈ U h − c ( ξ )11 { τ =0 } + V − ( x + ξ { τ =0 } ) i . Therefore V − ( x ) ≥ sup ξ ∈ U h V − ( x + ξ ) − c ( ξ ) i . Now we are ready to show the relation between our deterministic impulse controls problemand the classic HJBI QVI (5), indeed, we prove the following theorem.
Theorem 3.
Under assumption H1 , the lower and upper value functions are viscosity solutionsto the classic Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.Proof. The proof is based on the DPP and it is inspired from [5], we give the prove only for V − , the other case being analogous. We first prove the subsolution property. Suppose V − − φ achieves its local maximum in B δ ( x ) , where B δ ( x ) is the open ball of center x and radius δ > ,with V − ( x ) = φ ( x ) , where φ is a function in C ( R n ) and x ∈ R n . From Lemma 2 we always have V − ( x ) − H χinf V − ( x ) ≤ , then if V − ( x ) − H csup V − ( x ) ≤ there is nothing to prove. Otherwise,we suppose that V − ( x ) − H csup V − ( x ) > ε > . Then, without loss of generality, we can assumethat V − ( x ) − H csup V − ( x ) > ε > on B δ ( x ) . Next, we define t ′ = inf n t ≥ y ¯ x ( t ) / ∈ B δ ( x ) o . Let < ε < ε , T ≤ t ′ and consider ρ = + ∞ , i.e., no impulse for player − η . Furthermore, wepick a strategy u ε := ( τ εm , ξ εm ) m ≥ ∈ U for player − ξ which, due to DPP for V − , satisfies the13ollowing: V − ( x ) ≤ Z T ∧ τ ε f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt − c ( ξ ε ) exp ( − λτ ε )11 { τ ε ≤ T } + V − (cid:16) y x ( T ∧ τ ε ) (cid:17) exp (cid:16) − λ ( T ∧ τ ε ) (cid:17) + ε ≤ Z T ∧ τ ε f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt + exp ( − λτ ε ) (cid:18) V − (cid:16) y x ( τ ε ) (cid:17) − c ( ξ ε ) (cid:19) { τ ε ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT )11 { τ ε >T } + ε ≤ Z T ∧ τ ε f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt + H csup V − (cid:16) y x ( τ ε − ) (cid:17) exp ( − λτ ε )11 { τ ε ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT )11 { τ ε >T } + ε ≤ Z T ∧ τ ε f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt + V − (cid:16) y x ( τ ε − ) (cid:17) exp ( − λτ ε )11 { τ ε ≤ T } + V − (cid:16) y x ( T ) (cid:17) exp ( − λT )11 { τ ε >T } − ε exp ( − λτ ε )11 { τ ε ≤ T } + ε. Therefore, without loss of generality, we only need to consider strategy u ε ∈ U such that T < τ ε ,then we have V − ( x ) − ε ≤ Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt + V − (cid:16) y x ( T ) (cid:17) exp ( − λT ) . For T small enough, we have k y x ( T ) − x k → , from which we deduce V − (cid:16) y x ( T ) (cid:17) ≤ φ (cid:16) y x ( T ) (cid:17) + h V − ( x ) − φ ( x ) i , it follows, when ε goes to , that − exp ( − λT ) T V − ( x ) ≤ T Z T f (cid:16) y x ( t ) (cid:17) exp ( − λt ) dt + φ (cid:16) y x ( T ) (cid:17) − φ ( x ) T exp ( − λT ) . We use the fact that φ (cid:16) y x ( T ) (cid:17) − φ ( x ) = Z T b (cid:16) y x ( s ) (cid:17) .Dφ (cid:16) y x ( s ) (cid:17) ds, then we let T → to get λV − ( x ) − Dφ ( x ) .b ( x ) − f ( x ) ≤ . Finally, we get the subsolution property: max n min h λV − ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , V − ( x ) − H csup V − ( x ) i , V − ( x ) − H χinf V − ( x ) o ≤ . The supersolution property is proved analogously.14 .2 Viscosity Characterization of the New HJBI Quasi-Variational In-equality
This subsection is devoted to proving that both the lower and upper value functions of ourproblem are viscosity solutions to the new HJBI QVI, and we start by proving the followingtheorem.
Theorem 4.
Under assumptions H1 and H2 , a bounded and continuous function v is a viscositysupersolution to the classic HJBI QVI if and only if it is a viscosity supersolution to the newHJBI QVI.Proof. Assume first that v is a bounded and continuous viscosity supersolution to the new HJBIQVI. Then, for a function φ ∈ C ( R n ) and x ∈ R n such that v ( x ) = φ ( x ) and x is a localminimum point of v − φ , we have max n min h λv ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , F cinf Dφ ( x ) i , v ( x ) − H χinf v ( x ) o ≥ . If v ( x ) − H χinf v ( x ) ≥ then we are done with the first implication. Else we assume min h λv ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , F cinf Dφ ( x ) i ≥ . It follows that F cinf Dφ ( x ) ≥ . Let us first assume that v ∈ C ( R n ) , we then have for all ξ ∈ U ,for all x ∈ R n , F cinf Dv ( x ) ≥ , that is, for all ξ ∈ U , for all x ∈ R n , − Dv ( x ) .ξ ≥ − c ( ξ ) . Since v ( x ) − v ( x + ξ ) = Z − dds (cid:16) v ( x + sξ ) (cid:17) ds = Z − Dv ( x + sξ ) .ξds ≥ − c ( ξ ) . We then obtain for all ξ ∈ U , v ( x ) − h v ( x + ξ ) − c ( ξ ) i ≥ , which means that v ( x ) − H csup v ( x ) ≥ . Finally, v is a viscosity supersolution to the classic HJBI QVI when v ∈ C ( R n ) .We obtain the same result even v is not in C ( R n ) . It suffices to make the same regularization v ε ∈ C ∞ ( R n ) for v , as in the proof of the Theorem . in [19] (see also [24]), which convergesuniformly toward v in R n :Let θ be a positive function in C ∞ ( R n ) with supp θ ( x ) ⊂ B (0) , where B (0) is the open ballof center and radius , and R R n θ ( x ) dx = 1 . We then define for ε > the function θ ε by thefollowing: θ ε ( x ) = 1 ε n θ (cid:16) xε (cid:17) . R n the regularization v ε ( x ) = Z R n v ( y ) θ ε ( x − y ) dy. The function v ε is bounded, belongs to C ∞ ( R n ) and satisfies sup x ∈ R n (cid:12)(cid:12)(cid:12) v ε ( x ) − v ( x ) (cid:12)(cid:12)(cid:12) ≤ sup k x − y k≤ ε (cid:12)(cid:12)(cid:12) v ( x ) − v ( y ) (cid:12)(cid:12)(cid:12) , then it converges to v as ε goes to .In addition, for all x ∈ R n and since v is a viscosity supersolution to the new HJBI QVI, theregularization v ε satisfies F cinf Dv ε ( x ) ≥ δ ( ε ) , where δ ( ε ) goes to with ε .Therefore, by the same computation as in above, we get for all ξ ∈ U , v ε ( x ) − h v ε ( x + ξ ) − c ( ξ ) i ≥ δ ( ε ) , that is, when ε goes to , v ( x ) − h v ( x + ξ ) − c ( ξ ) i ≥ . Hence we get the desired result: v ( x ) − H csup v ( x ) ≥ . Assume now that v is a bounded and continuous viscosity supersolution to the classic HJBIQVI. And let x ∈ R n be a global minimum point of v − φ , where φ is a function in C ( R n ) and v ( x ) = φ ( x ) . We then have max n min h λv ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , v ( x ) − H csup v ( x ) i , v ( x ) − H χinf v ( x ) o ≥ . If v ( x ) − H χinf v ( x ) ≥ then we are done. Else we assume min h λv ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , v ( x ) − H csup v ( x ) i ≥ . Which gives v ( x ) − sup ξ ∈ U h v ( x + ξ ) − c ( ξ ) i ≥ , thus, for all ξ ∈ U , we get v ( x ) − v ( x + ξ ) + c ( ξ ) ≥ . Since for all ξ ∈ U , φ ( x ) − φ ( x + ξ ) + c ( ξ ) ≥ v ( x ) − v ( x + ξ ) + c ( ξ ) , then we have φ ( x ) − φ ( x + ξ ) + c ( ξ ) ≥ . We can then deduce, under assumption H2 , for all k > the following: φ ( x ) − φ ( x + kξ ) k ≥ − c ( kξ ) k ≥ − c ( ξ ) . k → , we get for all ξ ∈ U , − Dφ ( x ) .ξ + c ( ξ ) ≥ . Therefore, we obtain F cinf Dφ ( x ) ≥ . Finally, v is a viscosity supersolution to the new HJBI QVI. Theorem 5.
Under assumptions H1 and H2 , the lower and upper value functions are viscositysolutions to the new Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.Proof. We give the proof for the lower value function V − , similarly for V + . We first provethe viscosity subsolution property. We let φ be a function in C ( R n ) and x ∈ R n such that V − ( x ) = φ ( x ) and x is a local maximum of V − − φ . Since we have proved in Lemma 2 that V − ( x ) − H χinf V − ( x ) ≤ , then if F cinf Dφ ( x ) ≤ there is nothing to prove. Otherwise, for all x ∈ R n we assume that F cinf Dφ ( x ) > . We then get for all ξ ∈ U , inf ξ ∈ U h − Dφ ( x ) .ξ + c ( ξ ) i > . Then for all ξ ∈ U , for all x ∈ R n , − Dφ ( x ) .ξ > − c ( ξ ) . Since φ ( x ) − φ ( x + ξ ) = Z − dds (cid:16) φ ( x + sξ ) (cid:17) ds = Z − Dφ ( x + sξ ) .ξds > − c ( ξ ) , we get φ ( x ) − φ ( x + ξ ) + c ( ξ ) > , furthermore φ ( x ) − φ ( x + ξ ) + c ( ξ ) ≤ V − ( x ) − V − ( x + ξ ) + c ( ξ ) . Hence V − ( x ) − h V − ( x + ξ ) − c ( ξ ) i > . Thus, whenever F cinf Dφ ( x ) > we have V − ( x ) − H csup V − ( x ) > .Next, the same computation as in the proof of the viscosity subsolution sense for the classic HJBIQVI, Theorem 3, leads to the following viscosity subsolution property: max n min h λV − ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , F cinf Dφ ( x ) i , V − ( x ) − H χinf V − ( x ) o ≤ . For the prove of the viscosity supersolution property, since we have proved in Theorem 3 that V − is a viscosity solution to the classic HJBI QVI, then, due to the result in Theorem 4, thelower value function V − is a viscosity supersolution to the new HJBI QVI.17 Uniqueness of the Viscosity Solution of the New HJBIQuasi-Variational Inequality
We prove in the present section, via a comparison theorem, that the new HJBI QVI has aunique bounded and continuous solution in viscosity sense. As a consequence, the lower andupper value functions coincide, since they are both viscosity solutions to the new HJBI QVI.Hence the game has a value.We first give the following classical lemma which also appears in [31]:
Lemma 3.
Let v be a bounded uniformly continuous function and x ∈ R n such that: v ( x ) ≥ H χinf v ( x ) . Then there exists an element y in R n such that: ∃ δ > , ∀ x ∈ B δ ( y ) : v ( x ) < H χinf v ( x ) , where B δ ( y ) is the closed ball of center y and radius δ .Proof. Fix ε > and let x ∈ R n such that v ( x ) ≥ H χinf v ( x ) . Then there exists η ∈ V suchthat v ( x ) ≥ v ( x + η ) + χ ( η ) − ε, then we get for all η ∈ V , v ( x + η + η ) + χ ( η ) − v ( x + η ) ≥ v ( x + η + η ) + χ ( η ) + χ ( η ) − v ( x ) − ε ≥ χ ( η ) + χ ( η ) − χ ( η + η ) − ε. Thus, from assumption H1 and by letting ε → , we deduce v ( x + η ) < H χinf v ( x + η ) . Now we take y = x + η , then, since for all η ∈ V we have v ( y ) < v ( y + η ) + χ ( η ) − ε, we obtain, by uniform continuity of v where C v is the modulus of continuity, for all x ∈ R n , v ( x ) < v ( x + η ) + χ ( η ) + C v ( k x − y k ) − ε. Hence there exists δ > such that for all x ∈ B δ ( y ) we have v ( x ) − H χinf v ( x ) < . We next prove the following useful lemma. 18 emma 4.
Let v : R n → R be a bounded and continuous viscosity supersolution to the new HJBIQVI. If v ( x ) − H χinf v ( x ) < , then, for any µ , α and K such that < µ < , K > k f k ∞ /λ and < α < (1 − µ ) min (cid:16) inf ξ ∈ U c ( ξ ) , ( λK − k f k ∞ ) / k b k ∞ (cid:17) , the function v ∗ ( x ) = µv ( x ) + α p k x k +1 + K (1 − µ ) is a strict viscosity supersolution to thenew HJBI QVI.Proof. Let v be a bounded continuous viscosity supersolution to the new HJBI QVI, φ ∗ ∈ C ( R n ) and x ∈ R n be a local minimum point of v ∗ − φ ∗ such that v ( x ) − H χinf v ( x ) < .Then, for all x ∈ B δ ( x ) , where B δ ( x ) is the open ball of center x and radius δ > , we have µv ( x ) + α p k x k +1 − φ ∗ ( x ) ≤ µv ( x ) + α p k x k +1 − φ ∗ ( x ) , then v ( x ) − φ ∗ ( x ) − α p k x k +1 µ ≤ v ( x ) − φ ∗ ( x ) − α p k x k +1 µ , this inequality means that x is a local minimum point of v − φ , where φ ( x ) = (cid:16) φ ∗ ( x ) − α p k x k +1 (cid:17) /µ. Then, since v ( x ) − H χinf v ( x ) < and v is viscosity supersolution to the new HJBI QVI, we havethe following: min h λv ( x ) − Dφ ( x ) .b ( x ) − f ( x ) , F cinf Dφ ( x ) i ≥ , thus λv ( x ) − Dφ ( x ) .b ( x ) − f ( x ) ≥ and F cinf Dφ ( x ) ≥ . On one hand, we get λv ( x ) − µ Dφ ∗ ( x ) .b ( x ) + αxµ p k x k +1 .b ( x ) − f ( x ) ≥ , then λµv ( x ) − Dφ ∗ ( x ) .b ( x ) − µf ( x ) ≥ − α k b k ∞ , thus λv ∗ ( x ) − Dφ ∗ ( x ) .b ( x ) − f ( x ) ≥ λK (1 − µ ) − (1 − µ ) k f k ∞ − α k b k ∞ . Finally, since α < (1 − µ )( λK − k f k ∞ ) / k b k ∞ , we obtain λv ∗ ( x ) − Dφ ∗ ( x ) .b ( x ) − f ( x ) > . (12)On the other hand, since F cinf Dφ ( x ) ≥ , we get F cinf (cid:20) Dφ ∗ ( x ) µ − αxµ p k x k +1 (cid:21) ≥ , then inf ξ ∈ U (cid:20) − Dφ ∗ ( x ) .ξ + αx p k x k +1 .ξ + µc ( ξ ) (cid:21) ≥ ,
19t follows that inf ξ ∈ U h − Dφ ∗ ( x ) .ξ + c ( ξ ) + ( µ − c ( ξ ) i ≥ − α, from which we deduce, since µ − < , the following: inf ξ ∈ U h − Dφ ∗ ( x ) .ξ + c ( ξ ) + ( µ −
1) inf ξ ∈ U c ( ξ ) i ≥ inf ξ ∈ U h − Dφ ∗ ( x ) .ξ + c ( ξ ) + ( µ − c ( ξ ) i ≥ − α, therefore ( µ −
1) inf ξ ∈ U c ( ξ ) + inf ξ ∈ U h − Dφ ∗ ( x ) .ξ + c ( ξ ) i ≥ − α. Finally, from assumption H1 and since α < (1 − µ ) inf ξ ∈ U c ( ξ ) , we obtain F cinf Dφ ∗ ( x ) > . (13)Then the two strict inequalities (12) and (13) imply that min h λv ∗ ( x ) − Dφ ∗ ( x ) .b ( x ) − f ( x ) , F cinf Dφ ∗ ( x ) i > . Thus, we get that v ∗ is a strict viscosity supersolution to the new HJBI QVI.We are now in a position to prove the comparison theorem. Theorem 6. (Comparison Theorem) Under assumptions H1 and H2 , if u is a bounded andcontinuous viscosity subsolution to the new HJBI QVI and v is a bounded and continuous viscositysupersolution to the new HJBI QVI, then we have: ∀ x ∈ R n : u ( x ) ≤ v ( x ) . Proof.
Let u and v be a bounded and continuous viscosity subsolution and supersolution, re-spectively, to the new HJBI QVI. Our aim is to show, by contradiction, that u ≤ v .We denote by M = sup x ∈ R n (cid:16) u ( x ) − v ∗ ( x ) (cid:17) the maximal value of u − v ∗ , where v ∗ is defined as inLemma 4. We let R = (cid:16) k u k ∞ + k v k ∞ (cid:17) /α , then we have for all x ∈ R n such that k x k≥ R , u ( x ) ≤k u k ∞ +(1 − µ ) k v k ∞ ≤ v ∗ ( x ) , that is u ( x ) ≤ v ∗ ( x ) for all x ∈ R n \ B R (0) where B R (0) is the closed ball in R n of radius R centered at .Let us now assume that there exists ˆ x ∈ B R (0) , the open ball, such that M = u (ˆ x ) − v ∗ (ˆ x ) > , if it is not the case, i.e., M ≤ , then the proof is finished, will follow by letting µ → and α → . Step 1.
We can find x ∈ B R (0) and δ > such that sup x ∈ B δ ( x ) (cid:16) u ( x ) − v ∗ ( x ) (cid:17) ≥ u ( x ) − v ∗ ( x ) > , x ∈ B δ ( x ) , v ( x ) < H χinf v ( x ) , where B δ ( x ) is the closed ball of center x and radius δ .In fact, if v (ˆ x ) < H χinf v (ˆ x ) , then considering the continuity of u , v and H χinf we obtain the resultby taking x = ˆ x .Otherwise, we let v (ˆ x ) ≥ H χinf v (ˆ x ) , then, for some η ′ ∈ V , the result in Lemma 3 gives v (ˆ x + η ′ ) < H χinf v (ˆ x + η ′ ) , we then take x = ˆ x + η ′ to deduce ∃ δ > , ∀ x ∈ B δ ( x ) : v ( x ) < H χinf v ( x ) . (14)Furthermore, when v ∗ (ˆ x ) ≥ H χinf v ∗ (ˆ x ) , we fix ε > , then for α ∈ (0 , there exists η ′ ∈ V suchthat v ∗ (ˆ x ) ≥ v ∗ (ˆ x + η ′ ) + χ ( η ′ ) − αε, (15)which gives u (ˆ x + η ′ ) − v ∗ (ˆ x + η ′ ) ≥ u (ˆ x + η ′ ) + χ ( η ′ ) − v ∗ (ˆ x ) − αε ≥ u (ˆ x ) − v ∗ (ˆ x ) − αε, thus, for x = ˆ x + η ′ , we get u ( x ) − v ∗ ( x ) ≥ M − αε, (16)in the case where v ∗ (ˆ x ) < H χinf v ∗ (ˆ x ) it suffice to choose η ′ ∈ V for which (15) holds, then weproceed analogously to get (16).Therefore, by taking α sufficiently small, we get sup x ∈ B δ ( x ) (cid:16) u ( x ) − v ∗ ( x ) (cid:17) ≥ M > . As a consequence we consider M = sup x ∈ B δ ( x ) (cid:16) u ( x ) − v ∗ ( x ) (cid:17) . Step 2.
Let ε be a positive real number, ( x, y ) ∈ B δ ( x ) × B δ ( x ) and consider ψ ε the testfunction defined as follows: ψ ε : B δ ( x ) × B δ ( x ) → R ( x, y ) → ψ ε ( x, y ) = u ( x ) − v ∗ ( y ) − k x − y k ε . Let ( x m , y m ) be the maximal point of ψ ε and denote M ψ ε = max ψ ε ( x, y ) = ψ ε ( x m , y m ) . M ψ ε exists, since on the one hand, ψ ε is a bounded and continuous function on a bounded set, andon the other hand, it is negative in a neighborhood of the boundary k x k = δ or k y k = δ , and,by hypothesis, positive for some x = y . So, the search for the maximum can then be restrictedto a compact set B δ − γ ( x ) × B δ − γ ( x ) .Therefore u ( x m ) − v ∗ ( y m ) − k x m − y m k ε ≥ u ( x ) − v ∗ ( y ) − k x − y k ε . (17)21 Firstly, for x = x m , we get for all y ∈ B δ ( x ) , u ( x m ) − v ∗ ( y m ) − k x m − y m k ε ≥ u ( x m ) − v ∗ ( y ) − k x m − y k ε , then y m is a local minimal point of y → ( v ∗ − φ v ∗ )( y ) with φ v ∗ ( y ) = u ( x m ) − k x m − y k ε . From the inequality (14) and since y m ∈ B δ ( x ) we get v ( y m ) < H χinf v ( y m ) , in addition v is a viscosity supersolution to the new HJBI QVI, then by applying the resultin Lemma 4, we find the following: min h λv ∗ ( y m ) − D y φ v ∗ ( y m ) .b ( y m ) − f ( y m ) , F cinf D y φ v ∗ ( y m ) i > , thus, we obtain λv ∗ ( y m ) − D y φ v ∗ ( y m ) .b ( y m ) − f ( y m ) > and F cinf D y φ v ∗ ( y m ) > . (18)• Secondly, for y = y m , we get for all x ∈ B δ ( x ) , u ( x m ) − v ∗ ( y m ) − k x m − y m k ε ≥ u ( x ) − v ∗ ( y m ) − k x − y m k ε , then x m is a local maximal point of x → ( u − φ u )( x ) with φ u ( x ) = v ∗ ( y m ) + k x − y m k ε . Since u is a viscosity subsolution to the new HJBI QVI, we get u ( x m ) − H χinf u ( x m ) ≤ , and λu ( x m ) − D x φ u ( x m ) .b ( x m ) − f ( x m ) ≤ or F cinf D x φ u ( x m ) ≤ . (19)From (18) and (19), since F cinf D y φ v ∗ ( y m ) = F cinf D x φ u ( x m ) , we get λv ∗ ( y m ) − D y φ v ∗ ( y m ) .b ( y m ) − f ( y m ) > and λu ( x m ) − D x φ u ( x m ) .b ( x m ) − f ( x m ) ≤ . It follows that λ (cid:16) u ( x m ) − v ∗ ( y m ) (cid:17) + h D y φ v ∗ ( y m ) .b ( y m ) − D x φ u ( x m ) .b ( x m ) i + (cid:16) f ( y m ) − f ( x m ) (cid:17) ≤ . λ (cid:16) u ( x m ) − v ∗ ( y m ) (cid:17) + 2 k x m − y m k ε (cid:16) b ( y m ) − b ( x m ) (cid:17) + (cid:16) f ( y m ) − f ( x m ) (cid:17) ≤ . Since f and b are Lipschitz with constants C f and C b , respectively, we deduce λ (cid:16) u ( x m ) − v ∗ ( y m ) (cid:17) ≤ C b k x m − y m k ε + C f k x m − y m k . (20) Step 3.
Proving now that ∀ β > , ∃ ε > , ∀ ε ≤ ε : k x m − y m k ε ≤ β and showing thecontradiction.By taking x = y in (17) we get, for all x ∈ B δ ( x ) , u ( x ) − v ∗ ( x ) ≤ M ψ ε then < M ≤ M ψ ε .Let r = k u k ∞ + k v ∗ k ∞ then < M ψ ε ≤ r − k x m − y m k ε , it follows that k x m − y m k≤ εr .Since u is upper semi-continuous, we have ∀ β > , ∃ ε > , ∀ ε ≤ ε : u ( x m ) ≤ u ( y m ) + β, then u ( x m ) − v ∗ ( y m ) ≤ u ( y m ) − v ∗ ( y m ) + β ≤ M + β. Using M ≤ M ψ ε we get k x m − y m k ε ≤ β and M ≤ u ( x m ) − v ∗ ( y m ) , then from (20) we deduce λ (cid:16) u ( x m ) − v ∗ ( y m ) (cid:17) ≤ C b β + C f ε p β. By sending β to 0, we get u ( x m ) − v ∗ ( y m ) ≤ , from which yields for all x ∈ B δ ( x ) , u ( x ) − v ∗ ( x ) ≤ u ( x m ) − v ∗ ( y m ) ≤ , thus we get the contradiction M ≤ .Hence, by letting µ → and α → , we deduce for all x ∈ R n the desired comparison: u ( x ) ≤ v ( x ) . Theorem 7.
Under assumptions H1 and H2 , the new HJBI QVI has a unique bounded andcontinuous viscosity solution.Proof. Assume that v and v are two viscosity solutions to the new HJBI QVI. We first use v as a bounded and continuous viscosity subsolution and v as a bounded and continuous viscositysupersolution and we recall the comparison theorem. Then we change the role of v and v toget v ( x ) = v ( x ) for all x ∈ R n . Corollary 1.
Under assumptions H1 and H2 , the lower and upper value functions coincideand the value function V := V − = V + of the infinite horizon two-player zero-sum deterministicdifferential game is the unique viscosity solution to the new HJBI QVI. eferences [1] P. Azimzadeh, Zero-sum stochastic differential game with impulses, precommitment and un-restricted cost functions. Appl. Math. Optim., 1-32 (2017).[2] N. Barron, L.C. Evans and R. Jensen, Viscosity solutions of Isaacs’ equations and differentialgames with Lipschitz controls. J. Diff. Eqns., ¨ a user, Boston, (1997).[4] G. Barles, P. Bernhard and N. El Farouq, Deterministic minimax impulse control. Appl.Math. Optim., (3) 2102-2131 (2013).[10] M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans.Amer. Math. Soc., , 23-41 (2005).[14] S. Dharmatti, M. Ramaswamy, Zero-sum differential games involving hybrid controls. J.Optim. Theory Appl., , 75-102 (2006).[15] S. Dharmatti, A.J. Shaiju, Infinite dimensional differential games with hybrid controls. Proc.Indian Acad. Sci. Math., , 233-257 (2007).[16] B. El Asri, Deterministic minimax impulse control in finite horizon: the viscosity solutionapproach. ESAIM: Control Optim. Calc. Var., , 63-77 (2013) DOI: 10.1051/cocv/2011200.[17] B. EL Asri and S. Mazid, Zero-sum stochastic differential game in finite horizon involvingimpulse controls. Appl. Math. Optim., 1-33 (2018).2418] B. EL Asri and S. Mazid, Stochastic impulse control Problem with state and time depen-dent cost functions. Mathematical Control and Related Fields, 2020, 10 (4) : 855-875. doi:10.3934/mcrf.2020022[19] N. El Farouq, Degenerate first-order quasi-variational inequalities: an approach to approxi-mate the value function. SIAM J. Control Optim., (1972).[21] R.J. Elliott and N.J. Kalton, Cauchy problems for certain Issacs-Bellman equations andgames of survival. Trans. Amer. Math. Soc., (5) 773-797 (1984).[23] R. Isaacs, Differential games. A mathematical theory with applications to warfare and pur-suit, control and optimization. John Wiley & Sons, Inc., New York-London-Sydney, (1965).[24] P.L. Lions, Generalized solution of Hamilton-Jacobi equations. Pitman, Boston, (1982).[25] P.L. Lions, Optimal control for diffusion processes and Hamilton-Jacobi-Bellman equations,Part II. Comm. PDE,29