ASYMPTOTIC STABILITY

on the occasion of his 70th birthday

HAMILTON-JACOBI EQUATIONS WITH JUMPS:

ASYMPTOTIC STABILITY

AMIR MAHMOOD and SAIMA PARVEEN

The asymptotic stability of a global solution satisfying Hamilton-Jacobi equa- tions with jumps is analyzed in dependence on the strong dissipativity of the jump control function and using orbits of the differentiable flows to describe the corresponding characteristic system.

AMS 2000 Subject Classification: 35K20.

Key words: quasilinear Hamilton-Jacobi equations with jumps, asymptotic stabi- lity.

Asymptotic behavior of Hamilton-Jacobi (H-J) equations with jumps will be concentrated on revealing the significance of dissipativness property acting in the Hamiltonian H(x, u, p) =hp, g(x, u)i+L(x, u) and the jump functions h(x, u)∈R^m, (x, u)∈Rⁿ×[−1,1]. We analyze (H-J) equations of the form (1) ∂tu+H(x, u, ∂xu) = 0, t∈[tj, tj+1), x∈B(0,1)⊂Rⁿ,

where the jumps

u(tj, x) =u(t_j⁻, x) +hh(x, u(t_j⁻, x)),∆y(tj)i, j = 0,1,2, . . .

are defined by a sequence{t_j}_j≥0↑ ∞and a piecewise constant process{y(t) = y(t_j) ∈ R^m : t∈ [t_j, t_j+1), j ≥ 0, y(0) = 0}. The Cauchy method of charac- teristic systems allows to construct a global solution {(x(t, λ),b u(t, λ))b ∈Rⁿ× [−1,1] : t ∈[t_j, t_j+1), λ∈ Rⁿ, j ≥0} provided some weak dissipativity with respect to u ∈ [−1,1] of L and h is assumed. On the other hand, a global bounded solution for the H-J equation (1) with jumps is constructed combining u(t, λ)b ∈[−1,1] with a diffeomorphism{λ=ψ(t,·)∈ C¹(D⊂Rⁿ;Rⁿ) :t≥0}

which is piecewise smooth for t ∈ [tj, tj+1), j ≥ 0. In doing this we need to impose additional conditions which are strictly related with asymptotic sta- bility for both components of a solution {u(t, x) = u(t;b ψ(t, x)) :t ≥ 0} and

MATH. REPORTS11(61),4 (2009), 321–333

{p(t, x) =∂xu(t, x) :t≥0}. Here,L(u)∈R,h(u)∈R^mandu∈[−1,1] only de- pend on the unknown solution. When g(x, u) =α(u)bg(x) and bg∈ C¹(Rⁿ,Rⁿ) agrees with a nonlinear growth condition, the analysis of the asymptotic stabi- lity is given in Theorem 2.1, but the conclusion is only locally valid with respect tox∈D⊂Rⁿ. In Section 1 we give some auxiliary results regarding the global existence of the characteristic system solution. They imply that both compo- nents{ bu(t, λ), ∂_λbu(t, λ)

∈Rⁿ⁺¹ :t≥0,λ∈Rⁿ}are bounded. Also included are two results (Lemmas 1.5 and 1.7) where the construction of the smooth mapping {λ=ψ(t, x) :t ≥0, x ∈ D ⊆Rⁿ} is given. Usually, the construc- tion of such a smooth mapping involves a backward integral equation combined with a contractive mapping theorem when the original equationbx(t;λ) =xcan be rewritten as G(τb (t;λ))[λ] =x, τ(t;λ) = Rt

0 α(u(s;b λ))ds, where {G(σ)[λ] :b σ ∈ R, λ ∈ Rⁿ} is the global flow generated by gb ∈ C_b¹(Rⁿ;Rⁿ). As far as x(t;b λ) = x and ψ(t,x(t;b λ)) = λ for any t ≥ 0, x ∈ Rⁿ, λ ∈ Rⁿ, we get a direct verification that {u(t, x)^def=bu(t;ψ(t, x)) : t ≥ 0, x ∈ Rⁿ} is the solu- tion of the H-J equation with jumps. The situation changes when the vector fieldbg∈ C¹(Rⁿ,Rⁿ) is a nonlinear and unbounded one. It is analyzed in Theo- rem 2.1 pointing out that the contractive mapping theorem can be only locally applied (seex∈D= intB(x∗, γ)⊆Rⁿ). It implies that a nonstandard method must be used in order to get that {u(t, x)^def=bu(t;ψ(t, x)) :t≥0,x∈D⊆Rⁿ} is the solution of the H-J equation with jumps. The last theorem in Section 2 (see Theorem 2.2) tels us that in the case when g(x, u) =

l

P

k=1

αk(u)bgk(x) is a summation of several vector fields of the type occurring in Theorem 2.1, we need to impose a commutativity hypothesis.

1. SOME AUXILIARY RESULTS

We are given an increasing sequence {t_j}_j≥0, and a piecewise constant process

{y(t) =y(t_j)∈R^m :t∈[t_j, t_j+1), j≥0, y(0) = 0}

for which ∆y(t_j) = y(t_j)−y(t_j⁻), j ≥ 1, can be taken as a bounded value from R^m₊ and suitable for a fixed goal. Consider the H-J equation with jumps

(2)







∂tu(t, x) +H(x, u(t, x), ∂xu(t, x)) = 0, t∈[tj, tj+1), x∈Rⁿ, u∈R, u(t_j, x) =u(t_j⁻, x) +hh(x, u(t_j−, x)),∆y(t_j)i, j≥0,

u(0, x) =u0(x), u0∈Cb_b¹(Rⁿ⁺¹),

where H(x, u, p) = hp, g(x, u)i+L(x, u), g ∈ Cb_b¹(Rⁿ⁺¹;Rⁿ), L ∈ C¹(Rⁿ⁺¹) and h ∈ Cb_b¹(Rⁿ⁺¹). By Cb_b¹ we denote the space consisting of all continuously

differentiable functions f(x, u) : Rⁿ×R 7→ R^k, k = n,1, m, for which the partial derivatives ∂_if(x, u) = ∂_xif(x, u) and ∂_uf(x, u), i ∈ {1, . . . , n}, are bounded. To construct a global bounded solution for (2), we need to convince ourselves that the global solution

{(x(t, λ),b u(t, λ))b ∈Rⁿ⁺¹:t∈[t_j, t_j+1), λ∈Rⁿ⁺¹, j≥0}

exists and satisfies the corresponding characteristic system

(3)

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



 dxb

dt =g(x,b bu), x(0, λ) =b λ∈Rⁿ, t≥0, dub

dt =−L(x,b u),b t∈[t_j, t_j+1),

u(tb j;λ) =bu(t_j⁻;λ) +hh(bx(tj, λ),u(tb _j⁻;λ)),∆y(tj)i, j ≥0,

u(0;b λ) =u₀(λ), u₀∈ C_b¹(Rⁿ).

Without any dissipativity conditions onLandh, the standard analysis assume L∈Cb_b¹(Rⁿ⁺¹) and g∈Cb_b¹(Rⁿ⁺¹,Rⁿ), and yields a unique global solution

{z(t, λ) = (x(t, λ),b u(t, λ))b ∈Rⁿ⁺¹ :t≥0, λ∈Rⁿ}

of (3). On each interval t∈[t_j, t_j+1) we obtain a smooth function {z(t;λ)∈ Rⁿ⁺¹,t∈[t_j, t_j+1),λ∈Rⁿ} satisfying

dz

dt =Z(z), t∈[tj, tj+1), z(tj;λ) =z(t_j⁻;λ) +b(z(t_j⁻;λ),∆y(tj)), j ≥0, where z(0;λ) = (λ, u₀(λ)) =z₀(λ)∈Rⁿ⁺¹ and

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

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

Z(z)^def= g(x,b bu)

−L(bx,u)b

!

∈Rⁿ⁺¹, z= (x,b u)b ∈Rⁿ⁺¹,

b(z, ν) = 0

hh(z), νi

!

, for 0∈Rⁿ.

Using the piecewise smooth solution z(t, λ) = (bx(t, λ),bu(t, λ)), t ∈ [tj, tj+1), λ ∈ Rⁿ, j ≥ 0, of (3), we get the solution of the H-J equation (2) as a composition

(4) u(t, x)^def=u(t, ψ(t, x)),b t≥0, x∈Rⁿ,

where the smooth mapping {λ=ψ(t, x) ∈Rⁿ :t≥0, x∈ Rⁿ} is found as a unique solution fulfilling bx(t;λ) =x ∈ Rⁿ,t≥0, and

(5) ψ(0, x) =x, bx(t, ψ(t, x)) =x, ∀t≥0.

As far as{λ=ψ(t, x)∈Rⁿ:t≥0, x∈Rⁿ}is the unique solution of equation (5), we notice that

(6) ψ(t,x(t;b λ)) =λ, λ∈Rⁿ, t≥0,

and this will be the main ingredient supporting the idea that {u(t, x) : t ∈ [t_j, t_j+1), x ∈ Rⁿ, j ≥ 0} defined in (4) is the solution of the H-J equation (2). The construction of the mapping {λ=ψ(t, x) ∈ Rⁿ : t≥0, x∈ Rⁿ} is given using a bounded solution{bu(t;λ) :t≥0, λ∈Rⁿ}for which the gradient {∂_λu(t, x) :b t ≥0, λ∈ Rⁿ} is a bounded mapping. In this respect we recall that {u(t, x)b ∈R:y ∈[tj, tj+1), j ≥0, λ∈Rⁿ} is a piecewise smooth scalar function and {bx(t, λ)∈Rⁿ:t≥0, λ∈Rⁿ}is a smooth mapping satisfying

x(t, λ) =b bx(tj, λ) + Z t

tj

g(x(s, λ),b u(s, λ))ds,b t∈[tj, tj+1), j≥1,

bx(t, λ) =λ+ Z t

0

g(x(s, λ),b bu(s, λ))ds, t∈[0, t₁],

(7)









 dub

dt =−L(bx(t;λ),bu), t∈[t_j, t_j+1),

bu(tj;λ) =bu(t_j⁻;λ) +hh(bx(tj;λ),u(tb _j⁻;λ)),∆y(tj)i, j≥1, bu(0;λ) =u₀(λ), u₀ ∈ C_b¹(Rⁿ), sup|u₀(x)|=k₀ <1.

The weak dissipativity conditions (8)

(L(x,0) = 0, ∂uL(x, u)≥0, (x, u)∈Rⁿ×[−1,1],

h_i(x,0) = 0, ∂_uh_i(x, u)≤0, i∈ {1, . . . , m}, (x, u)∈Rⁿ×[−1,1], will be assumed.

Lemma 1.1. Consider g∈Cb¹_b(Rⁿ⁺¹;Rⁿ) and let L∈ C¹(Rⁿ⁺¹) and h∈ Cb_b¹(Rⁿ×[−1,1];R^m) satisfy condition (8). Let ∆y(tj)∈R^m+ be such that

−1≤

∂_uh(x, u),∆y(t_j)

≤0, (x, u)∈Rⁿ×[−1,1], j≥1.

Then the unique global solution { bx(t;λ),bu(t;λ)

: t ≥ 0, λ ∈ Rⁿ} of the characteristic system (7) satisfies

(|u(tb _j;λ)| ≤ |bu(t_j⁻;λ)|, |bu(t;λ)| ≤ |u(tb _j⁻;λ)|, t∈[t_j, t_j+1), j ≥1,

|u(t;b λ)| ≤ |u₀(λ)| ≤1, ∀t≥0, λ∈Rⁿ.

Remark 1.2. Takingu₀∈ C_b¹(Rⁿ) such that sup|u₀(x)|=K₀ <1, under the hypothesis of Lemma 1.1, we get a solution {u(t;b λ) : t ≥ 0, λ ∈ Rⁿ} of (7) fulfilling |bu(t;λ)| ≤ 1, t ≥ 0, λ ∈ Rⁿ. Now, we are interested to obtain a bounded solution for which {∂_λbu(t, λ) : t ≥ 0, λ ∈ Rⁿ} is bounded and

|∂_λu(t;λ)| ≤ |∂_λu0(λ)|,t≥0, λ∈Rⁿ. In this respect, we assume that

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

(a)L∈ C¹(R), L(0) = 0, ∂uL(u)≥0, u∈[−1,1];

(b)h_i∈ C¹([−1,1]), h_i(0) = 0, ∂_uh_i(u)≤0, u∈[−1,1], i∈ {1, . . . , m};

(c)g∈Cb¹_b(Rⁿ×[−1,1];Rⁿ)andu₀ ∈ C_b¹(Rⁿ)satisfies sup|u₀(x)|=K₀ <1.

(9)

Lemma 1.3. Consider H(x, u, p) = hp, g(x, u)i+L(u), where g and L fulfil (9). Leth(u)∈R^m and u0 ∈ C_b¹(R) be such that (9) are satisfied. Take

∆y(t_j)∈R^m+ such that

0≥ h∂_uh(u),∆y(tj)i ≥ −1, u∈[−1,1], j ≥1, and consider the global solution

{bu(t;λ) :t≥0, λ∈Rⁿ} satisfying (3). Then,

(|u(t;b λ)| ≤ |u₀(λ)| ≤1, t≥0, λ∈Rⁿ,

|∂_λu(t;b λ)| ≤ |∂_λu0(λ)|, t≥0, λ∈Rⁿ.

Remark 1.4. Lemmas 1.2 and 1.3 help us to conclude that looking for a bounded global solution {u(t;x) : t ≥ 0, x ∈ Rⁿ} for which the gra- dient {p(t, x) = ∂xu(t;x) : t ≥ 0, x ∈ Rⁿ} also is a bounded function, we need to assume some weak dissipativity condition for both L ∈ C¹(R) and h ∈ C¹([−1,1];R^m). Unfortunately, the same dissipativity conditions are not sufficient for obtaining the smooth mapping{λ=ψ(t, x)∈Rⁿ:t≥0, x∈Rⁿ} satisfying equations (5) and (6). We shall present two types of necessary con- ditions which lead us to the smooth global mapping {λ = ψ(t, x) : t ≥ 0, x∈Rⁿ}. ForL and g we assume that

(

(a)L∈ C¹(R), L(0) = 0, 0< γ≤∂uL(u), u∈[−1,1],

(b)g(x, u) =α(u)bg(x), α∈ C¹[−1,1]), bg∈ C_b¹(Rⁿ,Rⁿ), α(0) = 0.

(10)

Denote C₁ = max{|∂_uα(u)| : u ∈ [−1,1]}, C₂ = sup{|bg(x)| : x ∈ Rⁿ} and assume thatu₀ and h fulfil







(a)u0 ∈ C_b¹(Rⁿ),sup|u₀(x)|=K0 <1, sup|∂_xu0(x)|=K1; (b)¹_γC₁C₂K₁ ≤ρ, whereρ∈(0,1)is fixed;

(c)h_i∈ C¹([−1,1]), h_i(0) = 0, ∂_uh_i(u)≤0, u∈[−1,1], i∈ {1, . . . , m}.

(11)

Lemma1.5. ConsiderH(x, u, p) =hp, g(x, u)i+L(u)and the correspond- ing characteristic system (3) where g, L, u0 and hi, i∈ {1, . . . , m}, fulfil (10) and (11). Take ∆y(t_j)∈R^m+ sufficiently small such that

0≥ h∂_uh(u),∆y(tj)i ≥ −1, u∈[−1,1], j ≥1.

Then there exist a unique global solution {(x(t, λ),b u(t, λ)) :b t ≥ 0, λ ∈ Rⁿ} of (3) and a smooth mapping {λ=ψ(t, x) :t≥0, x∈Rⁿ} such that

(12) x(t;b ψ(t, x) =x, ψ(t,x(t, λ)) =b λ, ψ(0, x) =x, t≥0, x, λ∈Rⁿ, where |u(t, λ)| ≤b K0 <1 and |∂_λu(t, λ)| ≤b K1,t≥0, λ∈Rⁿ.

Remark 1.6. The unique global solution {λ = ψ(t, x) : t ≥ 0, x ∈ Rⁿ} given in Lemma 1.5 fulfils equation (12), ψi(t,x(t, λ)) =b λi, t ≥ 0, i ∈ {1, . . . , n}, where λ= (λ1, . . . , λn)∈Rⁿ. By a direct computation we get (13) d

dt[ψi(t,x(t;b λ))] =∂tψi(t,bx(t;λ))+h∂_xψ(t,x(t;b λ)), g(x(t, λ),b u(t;b λ))i= 0 for any t ≥ 0, i ∈ {1, . . . , n} and λ ∈ Rⁿ. In addition, for λ = ψ(t, x) we obtain bx(t, ψ(t, x)) = x, bu(t, ψ(t, x)) = u(t, x) and equation (13) becomes ψi(0, x) =xi and

(14) ∂_tψ_i(t, x)+h∂_xψ_i(t, x), g(x, u(t, x))i= 0, ∀t≥0, x∈Rⁿ, i∈ {1, . . . , n}, whereg(x, u) =α(u)bg(x). We are going to analyze a second type of conditions which leads us to a smooth mapping {λ=ψ(t, x) : t≥0, x∈Rⁿ} satisfying (14) and a solution u(t, x) = bu(t, ψ(t, x)), t≥0, x∈Rⁿ, of H-J equation (2) which is only asymptotically stable. In this respect, suppose L andg fulfil (15)

((a)L∈ C¹(R), L(0) = 0, ∂_uL(u)≥0, u∈[−1,1],

(b)g(x, u) =α(u)bg(x), α∈ C¹[−1,1]), bg∈ C_b¹(Rⁿ,Rⁿ), α(0) = 0.

Denote C₁ = max{|∂_uα(u)|:u∈[−1,1]},C₂ = sup{|bg(x)|:x∈Rⁿ}and take u0 and h such that



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

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



(a)u0∈ C¹_b(Rⁿ),sup|u₀(x)|=K0, sup|∂_uu0(x)|=K1; (b)h_i∈ C¹([−1,1]), h_i(0) = 0, ∂_uh_i(u)≤0, i∈ {1, . . . , m}

and

m

P

i=1

∂_uh_i(u)≤ −δ <0, u∈[−1,1];

(c) 2dC₁C₂K₁≤ρ, where ρ∈(0,1) is fixed and d= max

j≥0(t_j+1−t_j).

(16)

Lemma 1.7. Consider H(x, u, p) = hp, g(x, u)i+L(u), where g and L fulfil (15). Let u0 and hi, i ∈ {1, . . . , m}, be such that conditions (16) are satisfied. Take ∆y(tj)∈R^m+ satisfying the inequalities

−1≤ h∂_uh(u),∆y(t_j)i ≤ −1

2, u∈[−1,1], j≥1.

Then a smooth mapping {λ=ψ(t, x) :t≥0, x∈Rⁿ} satisfying the equations x(t, ψ(t, x)) =b x, ψ(0, x) =x, ψ(t,x(t, λ)) =b λ, t≥0, x, λ∈Rⁿ, does exist where{(x(t;b λ),u(t;b λ)) :t≥0, λ∈Rⁿ}is the global solution of (3) and |bu(t, λ)| ≤1, t≥0, λ∈Rⁿ.

2. MAIN RESULTS

The first theorem concerns the local asymptotic stability when the vector field g(x, u) = α(u)bg(x), and bg ∈ C¹(Rⁿ;Rⁿ) agrees with a nonlinear growth condition. The second theorem given here analyzes the asymptotic stability property when the vector field g(x, u) =α1(u)bg1(x) +α2(u)gb2(x) contains two commuting vector fields bg₁,bg₂ ∈ C¹(Rⁿ;Rⁿ). Asymptotic stability of the H-J equation ∂tu+H(x, u, ∂xu) = 0 is concerned. It will be the goal of the next two theorems. We are going to construct a local asymptotically stable solution for the H-J equation (2). In this respect, for some u∗ ∈R, x∗ ∈Rⁿ fixed we should assume

(17)

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

(a)L∈ C¹(R), L(u∗) = 0, ∂_uL(u)≥0, u∈[a, b];

(b)g(x, u) =α(u)bg(x), α∈ C¹[a, b]), α(u∗) = 0;

(c)bg∈ C¹(Rⁿ;Rⁿ), bg(x∗) = 0, where a=u∗−1, b=u∗+ 1. Denote

c₁= max{|∂_xα(u)|:u∈[a, b]}, c₂ = max{|g(x)|b :x∈B(x∗, γ)}, where γ >0 is fixed. Take u₀ and h such that

(18)

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

(a)u0 ∈ C_b¹(Rⁿ), sup|u₀(x)|=K0<1,sup|∂_xu0(x)|=K1; (b)hi ∈ C¹([a, b]), hi(u∗) = 0, ∂hi(u)≤0

and

m

P

i=1

∂_uh_i(u)≤δ <0, i∈ {1, . . . , m}, u∈[a, b];

(c) 2dc1c2K1 ≤ρ,whereρ∈(0,1) andd= max

j≥0(tj+1−tj) satisfies 2dc1c2 ≤γ.

Theorem 2.1. Consider H(x, u, p) = hp, g(x, u)i+L(u), where g and L satisfy (17) and let u₀, h be such that (18) are verified. Take ∆y(t_j) ∈R^m+

for which

−1≤ h∂_uh(u),∆y(t_j)i ≤ −1

2, u∈[a, b], j≥0.

Then there exists an asymptotic stable solution

{u(t, x)∈[a, b] :t∈[t_j, t_j+1), x∈B(x∗, γ), j≥0}

of the H-J equation with jumps (2)such that u(0, x) =u∗+u0(x),

|u(t, x)−u∗| ≤ ¹₂j

, t∈[t_j, t_j+1), x∈B(x∗, γ), j≥0,

|∂_iu(t, x)| ≤ L_iK₁ 1−ρ

1 2

j

, t∈[t_j, t_j+1), x∈B(x∗, γ), j ≥0, i∈ {1, . . . , n}, for some constant Li>0.

Proof. By hypothesis, the conclusion of Lemma 1.3 holds. Using (18) and Lemma 1.2, by a direct computation we obtain the estimates

|u(t;b λ)−u∗| ≤ |u₀(λ)| ≤K₀ <1,|∂_λbu(t, λ)| ≤ |∂_λu₀(λ)|, t≥0, λ∈Rⁿ, (19) |u(t;b λ)−u∗| ≤ |u(tb j;λ)−u∗| ≤ ¹₂j

K0, t∈[tj, tj+1), λ∈Rⁿ, j≥0,

|∂_λu(t;b λ)| ≤ |∂_λu(tb j;λ)| ≤ ¹₂j

K1, t∈[tj, tj+1), λ∈Rⁿ, j≥0.

This time, the smooth mapping {λ=ψ(t, x)} will be found as a unique solu- tion of the integral equation

(20) λ=Gb −τ(t;λ)

[x], τ(t;λ) = Z t

0

α u(s;b λ)

ds, t≥0, x∈B(x∗, γ)⊆Rⁿ. The vector field bg∈ C¹(Rⁿ;Rⁿ) is a nonlinear unbounded one and the corres- ponding local flow {G(σ)[x] :b σ ∈ [−β, β], x ∈ B(x∗, γ)} is constructed for β = 2dC1 satisfying

(21) βc₂≤γ (see (18, c)).

A direct computation shows that

|τ(t, λ)| ≤

∞

X

j=0

Z tj+1

tj

Z 1 0

|∂_uα u∗+θ(bu(t;λ)−u∗) dθ|

|u(t, λ)b −u∗|dt

≤dC₁

∞

X

j=0

1 2

j

= 2dC₁ =β, ∀t≥0, λ∈Rⁿ. Using (21) we easily get that the right hand side

(22) V(t, x;λ)^def=G(−τb (t;λ))[x] =x− Z 1

0

bg G(−θτb (t;λ)) [x]dθ

τ(t;λ) in (20) satisfies

(23) |V(t, x;λ)−x∗| ≤ |x|+ 2dC1C2 =|x−x∗|+βC2≤2γ

for any t ≥ 0, x ∈ B(x∗, γ) and λ ∈ Rⁿ. Here, we used the fact that the local flow {G(σ)[x] :b σ ∈ [−β, β], x ∈ B(x∗, γ)}, is bounded and G(σ)[x]b ∈ B(x∗,2γ) for anyσ∈[−β, β],x∈B(x∗, γ).It can be easily seen, noticing that {y_k(σ, x) : σ ∈ [−β, β], x ∈B(x∗, γ)}k≥0 defining the local flow is uniformly bounded and verifies

y_k(σ, x)∈B(x∗,2γ), σ∈[−β, β], x∈B(x∗, γ), k ≥0.

With this notation, the integral equation (20) can be written asλ=V(t, x;λ), where the smooth mappingV(t, x;λ) is a contractive application with respect

toλ∈Rⁿ. We then computeM(t, x;λ) =∂λV(t, x;λ) =bg(V(t, x;λ))∂λτ(t;λ), where (see (20))

∂λτ(t, λ) = Z t

0

∂uα bu(s;λ)

∂λbu(s;λ)ds.

Using (19) we get

(24) |∂_λτ(t, λ)| ≤C₁K₁

∞

X

j=0

1 2

j

= 2dC₁K₁

and from (23) and (24) we obtain that |M(t, x;λ)| ≤2dC1C2K1 =ρ ∈(0,1) for any t ≥ 0, x ∈ B(x∗, γ) and λ ∈ Rⁿ. Applying the contractive mapping theorem we obtain a smooth mapping {λ = ψ(t, x) ∈ B(x∗,2γ) : t ≥ 0, x∈B(x∗, γ)} as the unique solution of the integral equation

(25) ψ(t, x) =V(t, x;ψ(t, x)), t≥0, x∈B(x∗, γ).

Define

(26) u(t, x) =u(t, ψ(t, x)),b t≥0, x∈B(x∗, γ).

To prove that {u(t, x) :t≥0, x∈B(x∗, γ)} defined in (26) is the solution of the H-J equation (2) we need to show that the smooth function satisfying (25) is a solution of the H-J equations

(27)

(∂tψ(t, x) +∂xψ(t, x)g(x, u(t, x)) = 0, t≥0, x∈intB(x∗, γ), ψ(0, x) =x.

In this respect, for et ∈ [tj, tj+1) and xe ∈ intB(x∗, γ) fixed define {x(s;b x),e u(s;b x) :e s∈[et,et+ε]⊆[tj, tj+1)} as the solution of the characteristic system (28)





 dbx

ds =g(x,b u),b bx(et;ex) =x,e dbu

ds =−L(u),b bu(et,x) =e u(eb t;ψ(et,ex)) =u(et,x)e

satisfyingx(s,b ex)∈B(x∗, γ), s∈[et,et+ε). Notice that the solution of (28) can be obtained as the restriction to [et,et+ε) of the global solution{bx(s;λ),bu(s;λ) : s≥0} satisfying the original characteristic system













 dbx

ds =g(x,b u),b bx(0;λ) =λ, dbu

ds =−L(bu), u(tb _j, λ) =bu(t_j⁻;λ) +hh(bu(t_j⁻;λ)),∆y(t_j)i, s∈[tj, tj+1), j≥0,

withu(0;b λ) =u0(λ) +u∗ forλ=ψ(et,ex) fixed. We now use the representation bx(s,x) =e G(τb (s;ψ(et,x)))[ψ(ee t,x)],e s∈[et,et+ε],

and replacing x by {x(s,b ex) :s∈[et,et+ε)} into (22) we get

V(s,x(s,b ex);ψ(et,ex)) =G(−τb (s;ψ(et,ex)))◦G(τb (s;ψ(et,ex)))[ψ(et,ex)] = (29)

=ψ(et,x) = const., se ∈[et,et+ε).

Differentiating with respect to s, from (29) we obtain the H-J equations (30) ∂tV(et,ex;ψ(et,x)) +e ∂xV(et,ex;ψ(et,x))g(e x, u(ee t,x)) = 0e

at s =et, where (et,x)e ∈[t_j, t_j+1)×intB(x∗, γ) was arbitrarily fixed. On the other hand, differentiating with respect to tand x, from (25) we obtain (31)

(∂_tψ(t, x) = [I_n−M(t, x;ψ(t, x))]⁻¹[∂_tV(t, x;λ)](λ=ψ(t, x)),

∂iψ(t, x) = [In−M(t, x;ψ(t, x))]⁻¹[∂iV(t, x;λ)](λ=ψ(t, x)), i∈ {1, . . . , n}.

Combining (30) and (31) we conclude that the H-J equations (27) are valid.

Therefore, the piecewise smooth scalar function

{u(t, x) =bu(t;ψ(t, x)) :t∈[t_j, t_j+1), x∈intB(x∗, γ), j≥0}

fulfils the original H-J equations with jumps (2). In addition, we get u(t, x)∈[a, b] and |u(t, x)−u∗| ≤

1 2

j

, t∈[t_j, t_j+1), j≥0

for any x ∈ B(x∗, γ) and as far as ∂iu(t, x) = h∂_λbu(t;ψ(t, x)), ∂iψ(t, x)i we easily see (see (19) and (31)) that

|∂_iu(t, x)| ≤Li

K1

1−ρ 1

2 j

, t∈[tj, tj+1), x∈B(x∗, γ), j≥0, where Li = max{|∂_xiG(σ)[x]|b : σ ∈ [−β, β], x ∈ B(x∗, γ)} for i∈ {1, . . . , n}.

The proof is complete.

Comment. The result given in Theorem 2.1 relies essentially on the special structure we have assumed for the vector field g(x, u) = α(u)bg(x), gb∈ C¹(Rⁿ,Rⁿ), α∈ C¹([a, b]), wherea=u∗−1,b=u∗+ 1 andα(u∗) = 0. A relaxation of this hypothesis allows vector fields

g(x, u) =

l

X

k=1

α_k(u)bg_k(x), α_k ∈ C¹([a, b]), bg_k∈ C¹(Rⁿ;Rⁿ),

where αk(u∗) = 0, k ∈ {1, . . . , l}, for some u∗ ∈ R. To get an asymptotic stable solution, we must assume, in addition, that {bg₁, . . . ,bg_l} ⊆ C¹(Rⁿ;Rⁿ) commute using the standard Lie product. The second theorem of this paper is

encompassing the details we need in order to get an asymptotic stable solution when the vector field g(x, u) = α₁(u)bg₁(x) +α₂(u)bg₂(x) is defined by α_k ∈ C¹([a, b]), bgk ∈ C¹(Rⁿ;Rⁿ), k = 1,2, where αk(u∗) = 0, k ∈ {1,2}, for some u∗ ∈R fixed. Everywhere in what follows, we take a domainD=B(x∗, γ)⊂ Rⁿ, where x∗ ∈ Rⁿ and γ > 0 are fixed. Denote the standard Lie product [bg1,bg2] ={{∂_xgb1}gb2− {∂_xbg2}bg1},x∈Rⁿ and leta=u∗−1,b=u∗+ 1, where u∗∈Ris fixed. Using the Hamiltonian function

H(x, u, p) =hp, g(x, u)i+L(u), g(x, u) =α₁(u)bg₁(x) +α₂(u)bg₂(x), we assume that {bg₁,bg₂} ⊆ C¹(Rⁿ;Rⁿ) andL∈ C¹(R) satisfy

(32)







(a)L(u∗) = 0, ∂uL(u)≥0, u∈[a, b];

(b)α_k(u∗) = 0, k = 1,2;

(c) [bg₁,bg₂](x) = 0, for anyx∈B(x∗,3γ).

The corresponding (H-J) equations with jumps are







∂_tu+H(x, u, ∂_xu) = 0, t∈[t_j, t_j+1), x∈D=B(x∗, γ), u(tj, x) =u(t_j⁻, x) +h(u(t_j⁻, x))∆y(tj), j≥0, x∈D, u(0, x) =u∗+u0(x), x∈D,

where the scalar jump function h∈ C¹([a, b]) and ∆y(tj) =y(tj)−y(t_j⁻)≥0, j ≥ 1, are associated with the scalar piecewise constant process {y(t) ≥ 0 : y(0) = 0}. Define

(

C1 = max{|∂_uα1(u)|+|∂_uα2(u)|:u∈[a, b]}, C2 = max{|bg1(x)|+|bg2(x)|:x∈B(x∗,3γ)}.

Take u0 and h such that

(33)















(a)u₀∈ C¹_b(Rⁿ), sup|u₀(x)|=K₀ <1 and sup|∂_xu₀(x)|=K₁; (b)h∈ C¹([a, b]), h(u∗) = 0, ∂uh(u)≤δ <0, u∈[a, b];

(c) 2dC₁C₂K₁=ρ∈(0,¹₂), whered= max

j≥0(t_j+1) and β= 2dC₁ satisfies βC₂ ≤ ^γ₂.

Theorem2.2. ConsiderH(x, u, p) =hp, g(x, u)i+L(u), whereg(x, u) = α₁(u)bg₁(x) +α₂(u)bg₂(x) and L(u) fulfil (32). Letu₀ and h be such that (33) are satisfied. Take ∆y(tj)≥0 such that

(34) −1≤∂_uh(u)∆y(t_j)≤ −1

2, u∈[a, b], j≥1.

Then there exists a global solution{u(t, x) :t≥0, x∈D}of the H-J equations with jumps (2) which is asymptotically stable such that

(35)

(|u(t, x)−u∗| ≤ ¹₂j

, t∈[tj, tj+1), x∈D, j≥0,

|∂_iu(t, x)| ≤L_i1−ρ^{K1 1}₂j

, t∈[t_j, t_j+1), x∈D, j≥0, i∈ {1, . . . , n}, for some constant Li>0, where K1>0 and ρ∈(0,1)are given in (33).

Proof. We use the same arguments as in the proof of Theorem 2.1.

By hypothesis, the conclusions of Lemma 1.5 regarding the global solution {bu(t;λ) :t≥0,λ∈Rⁿ}hold. The equations





 dbu

dt =−L(u),b bu(t_j⁻;λ) +h(u(tb _j⁻;λ))∆y(t_j), t∈[t_j, t_j+1), j≥0, u(0;b λ) =u∗+u0(λ), λ∈Rⁿ,

(|u(t;b λ)−u∗| ≤ |u₀(λ)| ≤K₀<1, t≥0, λ∈Rⁿ,

|∂_λu(t;b λ)| ≤ |∂_λu₀(λ)|, t≥0, λ∈Rⁿ

are satisfied. In addition, using (33 (a), (b)) and (34), by a direct computa- tion we get

(|∂_λbu(t;λ)| ≤ ¹₂j

K₁, t∈[t_j, t_j+1), j≥0, λ∈Rⁿ,

|u(t;b λ)−u∗| ≤ |u(tb j;λ)−u∗| ≤ ¹₂j

, t∈[tj, tj+1), j≥0, λ∈Rⁿ. This time, {bx(t;λ) : t≥0, λ∈B(x∗,2γ) ⊆Rⁿ} is the global solution of the characteristic system





 dbx

dt(t;λ) =α₁(u(t;b λ))bg₁(x(t;b λ)) +α₂(u(t;b λ))bg₂(x(t;b λ)), t≥0, bx(0;λ) =λ∈B(x∗,2γ).

As far as [bg₁,bg₂](x) = 0,∀x∈B(x∗,3γ), we may and do write bx(t;λ) as (36) x(t;b λ) =Gb₁(τ₁(t, λ))◦Gb₂(τ₂(t, λ))[λ], t≥0, λ∈B(x∗,2γ)

provided {bx(t;λ) ∈B(x∗,3γ) : t ≥0, λ∈ B(x∗,2γ)}. Here {Gbk(tk)[λ] :tk ∈ [−β, β], λ ∈ B(x∗,2γ+ ^γ₂)} is the local flow generated by bg_k ∈ C¹(Rⁿ;Rⁿ).

Since

τ_k(t, λ) = Z t

0

α_k(bu(s;λ))ds=

= Z t

0

Z 1 0

∂_uα_k(u∗+θ[u(s;b λ)−u∗])dθ

(u(s;b λ)−u∗)ds, k∈ {1,2}, we get (see (2) and (33, c))

|τ_k(t, λ)| ≤C1

Z 1 0

|bu(s;λ)−u∗|ds≤C1

∞

X

j=0

Z tj+1

tj

1 2

j

ds≤2dC1 =β

for any t≥0, λ ∈Rⁿ, k ∈ {1,2}, where βC2 ≤ ^γ₂. This shows that the flow {bx(t;λ) : t ≥0, λ∈ B(x∗,2γ)} defined in (36) fulfils bx(t, λ) ∈ B(x∗,3γ) and the equations

x=bx(t;λ) =Gb1(−τ₁(t;λ))◦Gb2(−τ₂(t;λ))[x], x∈B(x∗, γ) =D, λ∈B(x∗,2γ), are equivalent to

(37) λ=Gb2(−τ₂(t;λ))◦Gb1(−τ₁(t;λ))[x], λ∈B(x∗,2γ), x∈D.

Denote

V(t, x;λ) =Gb₂(−τ₂(t;λ))◦Gb₁(−τ₁(t;λ))[x] =Gb₁(−τ₁(t;λ))◦Gb₂(−τ₂(t;λ))[x]

for t ≥ 0, x ∈ B(x∗, γ), λ ∈ B(x∗,2γ). To prove that the smooth mapping V(t, x;λ) is a contraction with respect to λ∈B(x∗,2γ), we compute

M(t, x;λ) =∂_λV(t, x;λ) =−[bg₁(V(t, x;λ))∂_λτ₁(t;λ)+bg₂(V(t, x;λ))∂_λτ₂(t;λ)].

Using (2) and (33, c) we obtain

|M(t, x;λ)| ≤ρ∈ 0,¹₂

for anyt≥0, x∈B(x∗, γ), λ∈B(x∗,2γ).

This allows to use the contractive mapping theorem for the integral equations (37) and to get the smooth solution λ = ψ(t, x) = lim

k→∞λk(t, x) ∈ B(x∗,2γ) satisfying







λ₀(t, x) =x∗, |λ_k(t, x)−x∗| ≤ 1

1−ργ ≤2γ, ∀k≥0, ψ(t, x) =V(t, x;ψ(t, x)), t≥0, x∈B(x∗, γ).

Define {u(t, x) = bu(t, ψ(t, x)) :t∈[t_j, t_j+1), x∈ B(x∗, γ), j ≥0}. Using the same argument as in the proof of Theorem 2.1, we get (35) and the proof is complete.

REFERENCES

[1] B. Iftimie, I. Molnar and C. Vˆarsan,Solutions of some elliptic equations associated with a piecewise continuous process. Rev. Roumaine Math. Pures Appl.53(2008),4, 323–338.

Received 15 March 2009 GC University

Abdus Salam School of Mathematical Science 68 B, New MuslimTown

54600 Lahore, Pakistan