Variational Solutions of Stationary Hamilton-Jacobi Equations

Vasile Iftode

Abstract This work deals with the stationary Hamilton-Jacobi equation F (B^{∗}ψ_{y}) − (Ay, ψ_{y}) = g in the class of convex continuous functions ψ on a real Hilbert space H. After obtaining an symptotic result, the existence, the uniqueness and a Galerkin approximation of the solution to the above equation are established.

Mathematics Subject Classification: 49L20, 49A15 Key words: Hamilton-Jacobi equations, variational solutions

Let H and U be two real Hilbert spaces with the scalar products (·, ·) and < ·, · >, respectively. The norms of H and U will be denoted by | · | and | · |_{U}, respectively. We define the function ψ^{∞} : H → R, (1)

ψ^{∞}(y) = inf {∫_{0}^{∞}(g(x(t)) + h(u(t)))dt; x′ = Ax + Bu, x(0) = y; u ∈ L^{1}(R^{+}; U)}

in the following hypotheses:

(i) A is the infinitesimal generator of a C_{0}-semigroup of contractions on H i.e., A is m-dissipative on H; B is a linear continuous operator from U to H.

(ii) The function g : H → R belongs to K ∩ C_{Lip}^{1}(H) and satisfies the following two conditions:

(2)  if g(y_{n}) → 0 for some sequences {y_{n}} then y_{n} → 0;
(3)  g(e^{−(λI−A)t}y) ∈ L^{1}(R^{+}) for λ > 0 and all y ∈ H.

Here e^{−(λI−A)t} is the semigroup generated by the operator −λI + A, K is the closed convex cone of C(H) consisting of all convex functions φ : H → R which satisfy the condition φ(y) ≥ φ(0) = 0 for all y ∈ H and C_{Lip}^{1}(H) is the space of all φ ∈ C^{1}(H) such that the Fréchet derivative φ^{(1)} is Lipschitz on H. (iii) F : U → R is a convex function which is bounded on bounded subsets and belongs to C^{1}(U). We associate the function h : U → R̄, defined by

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 33-38 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

h(u) = sup{− < p, u > −F (p); p ∈ U} i.e., h(u) = F^{∗}(−u),

to the function F. Then the assumption (iii) implies lim_{|u|_{U} →∞} \frac{h(u)}{|u|_{U}} = +∞. Further

assume that F and F^{∗} are strictly convex. (iv) ψ_{0} : H → R is a function which belongs to K ∩ C_{Lip}^{1}(H).

Under the above hypotheses, we see that Ψ^{∞} < +∞ on H and for every y ∈ H the infimum defining Ψ^{∞}(y) is attained in a unique pair (x^{∗}, u^{∗}). Moreover, the function Ψ^{∞} is convex and lower semicontinuous on H. Since it is everywhere finite we conclude that it is continuous on H.

Now we shall prove the following asymptotic result. Theorem 1. Let assumptions (i), (ii), (iii) and (iv) be satisfied. Then the solution Ψ to the problem

(4)  \{ \begin{matrix}ψ_{t} + F (B^{∗}ψ_{y}) − (Ay, ψ_{y}) = g; t ≥ 0, y ∈ D(A) \\ ψ(0, y) = ψ_{0}(y), y ∈ H \end{matrix}

satisfies

(5)  lim_{t→∞} ψ(t, y) = ψ^{∞}(y) for all y ∈ H.

We have denoted by Ψ_{t} and Ψ_{y} the partial derivatives (in a generalized sense) of the function Ψ with respect to t and y, respectively. Here B^{∗} is the adjoint of B. Proof. The solution Ψ to the Cauchy problem (4) is given by (see [1], p.54)

(6)  \begin{matrix}ψ(t, y) = inf \{∫_{0}^{t}(g(x(s)) + h(u(s)))ds + ψ_{0}(x(t)); x′ = Ax + Bu, \\ x(0) = y; u ∈ L^{1}(0, t; U)\} \end{matrix}

for t ≥ 0, y ∈ H. This yields

(7)  \begin{aligned}∫_{0}^{t}(g(x^{t}(s)) + h(u^{t}(s)))ds + ψ_{0}(x^{t}(t)) & = & ψ(t, y) ≤ ∫_{0}^{t}(g(x^{∗}(s))+ \\ & + & h(u^{∗}(s)))ds + ψ_{0}(x^{∗}(t)) \end{aligned}

where (x^{∗}, u^{∗}) is the optimal pair in Problem (1) and (x^{t}, u^{t}) in (6), i.e.,

\{ \begin{matrix}(x^{t})′ = Ax^{t} + Bu^{t} on [0, t] \\ x^{t}(0) = y. \end{matrix}

Letting t → +∞, we see by (7) that

(8)  lim sup_{t→∞} ψ(t, y) ≤ ψ^{∞}(y).

We set

ũ^{t}(s) = \{ \begin{matrix}u^{t}(s) & for & 0 ≤ s ≤ t \\ 0 & for & s > t \end{matrix}

and

x̃^{t}(s) = e^{As}y + ∫_{0}^{s} e^{A(s−τ)}Bũ^{t}(τ)dτ.

Let some sequence t_{n} → ∞ and ley {x̃^{t_{n}}} and {ũ^{t_{n}}} be such that

(9)  ψ^{∞}(y) ≤ ∫_{0}^{∞} (g(x̃^{t_{n}}(s)) + h(ũ^{t_{n}}(s))) ds ≤ ψ^{∞}(y) + \frac{1}{n},
(x̃^{t_{n}})′ = Ax̃^{t_{n}} + Bũ^{t_{n}} on R^{+}
x̃^{t_{n}}(0) = y.

Since |x̃^{t_{n}}(s)| ≤ C (|y| + ∫_{0}^{s} ∥ũ^{t_{n}}(τ)∥ dτ), ∀s ≥ 0 and g(x̃^{t_{n}}(s)) ≥ g(0)−|∂g(0)| |x̃^{t_{n}}(s)|,

∀s ≥ 0 by (9) it follows that ∫_{0}^{∞} h(ũ^{t_{n}}(s))ds ≤ C (∫_{0}^{∞} ∥ũ^{t_{n}}(s)∥ ds + 1) where C is

independent of n. The latter implies that {ũ^{t_{n}}} is weakly compact in L^{1}(R^{+}; U) (by virtue of the Dunfort-Pettis theorem). Hence on some sequence t_{n} → ∞, ũ^{t_{n}} → ũ weakly in L^{1}(R^{+}; U) and x̃^{t_{n}}(s) → x̃(s) weakly in H for every s ≥ 0, where x̃ is the mild solution to

\{ \begin{matrix}x′ = Ax + Bũ on R^{+} \\ x(0) = y. \end{matrix}

Since the functional (x, u) → ∫_{0}^{T} (g(x) + h(u))dt is weakly lower semicontinuous on

every C([0, T]; H) × L^{1}(0, T; U) (because it is convex and lower semicontinuous), it follows by (7) and (8) that

∫_{0}^{∞}(g(x̃^{t}(s)) + h(ũ^{t}(s)))ds ≤ lim sup_{t→∞} ψ(t, y) ≤ ψ^{∞}(y)

which obviously implies (5) as claimed. □

Now let us consider the stationary Hamilton-Jacobi equation

(10)  F (B^{∗}ψ_{y}) − (Ay, ψ_{y}) = g, ∀y ∈ D(A)

in the real Hilbert space H.

By a solution to (10) we shall mean a function ψ : H → R which is Gâteaux differentiable and staisfies Eq. (10) for all y ∈ D(A).

Equation (10) is relevant in the calculus of variations with infinite horizon and in nonlinear analysis. The main result is Theorem 2. Let assumptions (i), (ii) part (3) and (iii) be satisfied. Then Eq. (10) has at least one solution ψ^{∞} ∈ K. If also condition (2) holds then the solution ψ^{∞} to Eq.

(10) is unique. Proof. For every t ≥ 0, the unique optimal pair (x^{∗}, u^{∗}) in Problem (1) is also an optimal pair for the finite horizon control problem (11)

\begin{aligned}ψ^{∞}(y) & = & inf \{∫_{0}^{t}(g(x(s)) + h(u(s)))ds + ψ^{∞}(x(t)); x′ = Ax + Bu, x(0) = y\} \\ & = & ∫_{0}^{t}(g(x^{∗}(s)) + h(u^{∗}(s)))ds + ψ^{∞}(x^{∗}(t)). \end{aligned}

By the maximum principle ([1], p. 13), for every t ≥ 0 there exists p^{t} ∈ C([0, t]; H) which satisfies the system

\{ \begin{matrix}(p^{t})′ + A^{∗}p^{t} ∈ ∂g(x^{∗}), & 0 ≤ s ≤ t \\ B^{∗}p^{t} ∈ ∂h(u^{∗}), & 0 ≤ s ≤ t \\ p^{t}(t) ∈ −∂ψ^{∞}(x^{∗}(t)) & . \end{matrix}

(Here ∂ is the subdifferentiation symbol). Next by (11) we see that

(12)  ψ^{∞}(x^{∗}(t)) = ∫_{t}^{∞}(g(x^{∗}(s)) + h(u^{∗}(s)))ds, ∀t ≥ 0.

Assume that y ∈ D(A). Then x^{∗} is right differentiable at t = 0 and \frac{d^{+}x^{∗}}{dt} (0) = Ay + Bu^{∗}(0). Since

\frac{ψ^{∞}(x^{∗}(t)) − ψ^{∞}(y)}{t} →^{t→0} (η, \frac{d^{+}x^{∗}}{dt} (0))

where η is an element of ∂Ψ^{∞}(y), it follows by (12) that

(13)  (η, Ay + Bu^{∗}(0)) + g(y) + h(u^{∗}(0)) = 0

where u^{∗}(0) = ∇h^{∗}(−B^{∗}η) and η ∈ ∂ψ^{∞}(y) (we may take η = −p^{t}(0)). By using (13) and the conjugacy formula, we get

F (B^{∗}η) − (Ay, η) = g(y).

To conclude the proof of existence it sufficies to show that Ψ^{∞} is Gâteaux differentiable, i.e., ∂Ψ^{∞} is single valued.

To this end we define the operator Γ : H → H, Γy = −p(0) where p ∈ C([0, T]; H) is the solution to the system (T is fixed)

(14)  \{ \begin{matrix}x′ = Ax + Bu, & 0 ≤ t ≤ T \\ p′ + A^{∗}p ∈ ∂g(x), & 0 ≤ t ≤ T \\ x(0) = y, p(T) ∈ −∂Ψ^{∞} & (x(T)) \end{matrix}

which, by virtue of the maximum principles is equivalent with the control problem

(15)  inf {∫_{0}^{T} (g(x) + h(u))dt + Ψ^{∞}(x(T)); x′ = Ax + Bu, x(0) = y} .

Since F and F^{∗} are strictly convex, so is h^{∗} and therefore there exists a unique optimal pair (x, u) for the problem (15). Since F is Gâteaux differentiable, ∂F^{∗} and therefore ∂h is single valued. Thus there exists a unique B^{∗}p = ∂h(u) which satisfies the system (14) and therefore Γ is single valued. On the other hand, we see that Γy ∈ ∂Ψ^{∞}(y). To prove that Γ = ∂Ψ^{∞} it suffices to show that the range R(I + Γ) is all of H. To this end we consider the equation y + Γy = w which is equivalent to

(16)  \{ \begin{matrix}x′ = Ax + Bu, & 0 ≤ t ≤ T \\ p′ + A^{∗}p ∈ ∂g(x), & 0 ≤ t ≤ T \\ x(0) = y, p(0) = y − w \\ p(T) ∈ −∂Ψ^{∞}(x(T)) \\ B^{∗}p ∈ ∂h(u), & 0 ≤ t ≤ T \end{matrix}

and again by virtue of the maximum principle it is equivalent to the control problem

inf \{∫_{0}^{T} (g(x) + h(u))dt + Ψ^{∞}(x(T)) + \frac{1}{2}|x(0) − w|^{2}; x′ = Ax + Bu\}

which clearly admits at least one solution which is also a solution to (16). Hence Γ = ∂Ψ^{∞}, and ∂Ψ^{∞} = (Ψ^{∞})′ is single valued as claimed.

As for uniqueness we consider the differential equation

(17)  \{ \begin{matrix}x′ = Ax − B∇h^{∗}(B^{∗}Ψ_{x}^{0}(x)), t ≥ 0 \\ x(0) = y \end{matrix}

where Ψ^{0} is any solution to Eq. (10). For each y ∈ D(A), Eq. (17) has a unique solution x̃ ∈ W_{loc}^{1,∞}(R^{+}; H). We set ũ = −∇h^{∗}(B^{∗}Ψ_{x}^{0}(x)) and take the inner product of (17) by −Ψ_{x}^{0}(x̃). Since Ψ^{0} is a solution to Eq. (10) we get

\frac{d}{dt}Ψ^{0}(x̃(t)) − F (B^{∗}Ψ_{x}^{0}(x̃(t))) + g(x̃(t)) + (ũ(t), −B^{∗}Ψ_{x}^{0}(x^{∗}(t))) = 0

a.e. t ≥ 0. By using the conjugacy formula, the latter becomes

\frac{d}{dt}Ψ^{0}(x̃(t)) + g(x̃(t)) + h(ũ(t)) = 0 a.e. t ≥ 0

and therefore

(18)  Ψ^{0}(y) = Ψ^{0}(x̃(t)) + ∫_{0}^{t}(g(x̃(s)) + h(ũ(s)))ds, ∀t ≥ 0.

Finally

(19)  Ψ^{0}(y) = ∫_{0}^{∞}(g(x̃(s)) + h(ũ(s)))ds ≥ Ψ^{∞}(y)

where Ψ^{∞} is given by (1). (To get (19) we have used the fact that x̃(t_{n}) → 0 for some t_{n} → ∞ because g(x̃) ∈ L^{1}(R^{+}) and g satisfies condition (2)).

Now let (x^{∗}, u^{∗}) be the optimal pair for Problem (1) (y ∈ D(A)). Again by Eq.

(10) we have

\frac{d}{dt}Ψ^{0}(x^{∗}(t)) = (x^{′∗}(t), Ψ_{x}^{0}(x^{∗}(t))) = (Ax^{∗}(t) + Bu^{∗}(t), Ψ_{x}^{0}(x^{∗}(t))) =
= F (B^{∗}Ψ_{x}^{0}(x^{∗}(t))) − g(x^{∗}(t)) − (u^{∗}(t), −B^{∗}Ψ_{x}^{0}(x^{∗}(t))) ≥
≥ −h(u^{∗}(t)) − g(x^{∗}(t)) a.e. t > 0

and therefore

Ψ^{0}(x^{∗}(t)) + ∫_{0}^{t}(h(u^{∗}(s)) + g^{∗}(x^{∗}(s)))ds ≥ Ψ^{0}(y), ∀t ≥ 0.

Since g(x^{∗}) ∈ L^{1}(R^{+}) it follows that lim_{t_{n}→∞} x^{∗}(t_{n}) → 0 for some sequence t_{n} → ∞

and therefore lim_{t_{n}→∞} Ψ^{0}(x^{∗}(t_{n})) = Ψ^{0}(0) = 0. Hence Ψ^{0}(y) ≤ Ψ^{∞}(y) for all y ∈ D(A).

Along with (19) the latter implies that Ψ^{∞} = Ψ^{0} as claimed. □

Now we shall briefly discuss a Galerkin approximation of the stationary Hamilton-Jacobi equation (10). Assume that the hypotheses (i)-(iii) hold. It will be more convenient to regard A as a linear continuous operator from the space V ⊂ H to its dual V′ ⊃ H. Following a well-known scheme (see for instance [1], [2], [3]), the internal approximations of the spaces V and U, of the operators A and B^{∗} and of the functions F and g will be defined. Thus on the finite dimensional space V_{h}, Eq. (10) becomes

(20)  F_{h}(B_{h}^{∗}Ψ_{y}^{h}(y_{h})) − (A_{h}y_{h}, Ψ_{y}^{h}(y_{h}))_{h} = g_{h}(y_{h}) for all y_{h} ∈ V_{h}.

By Theorem 2, Eq. (20) has a generalized solution Ψ^{h} : V_{h} → R given by

Ψ^{h}(y_{h}) = inf {∫_{0}^{∞}(g_{h}(x_{h}(t)) + F_{h}^{∗}(−u_{h}(t)))dt}

where x_{h}, u_{h} satisfy the system

\{ \begin{matrix}x′_{h} = A_{h}x_{h} + B_{h}u_{h} a.e. t > 0 \\ x_{h}(0) = y_{h}. \end{matrix}

As regards the convergence of the solutions Ψ^{h} to a solution Ψ to Eq. (10), we have the following result Theorem 3. For h → 0 we have Ψ^{h}(y_{h}) → Ψ(y) for all y ∈ V, where Ψ is the variational solution to Eq. (10) given by (1). The proof is similar (making obvious changes) to the proofs of the analogous results in [1] and [3]. Acknowledgements. A version of this paper was presented at the Third Conference of Balkan Society of Geometers, Workshop on Electromagnetic Flows and Dynamics, July 31 - August 3, 2000, University POLITEHNICA of Bucharest, Romania.

References

[1] V. Barbu and G. Da Prato, Hamilton-Jacobi Equations in Hilbert Spaces, Pitman, Research Notes in Mathematics, 86 (1983).

[2] R. Glowinski, J.L. Lions an R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

[3] V. Iftode, Approximation of dual Hamilton-Jacobi equations, Rivista di Mat. Pura Appl., 15 (1994), 57-75.

University Politehnica of Bucharest, Department of Mathematics I

Splaiul Independentei 313, 77206 Bucharest, Romania

e-mail:iftode@mathem.pub.ro