Introduction to Hamilton-Jacobi Equations and Periodic Yu-Yu Liu NCKU Math August 22, 2012 Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 1 / 15
H-J equations H-J equations A Hamilton-Jacobi equation is a first order equation { H(Du, u, x) = 0, x Ω u = 0, x Ω May not have C 1 solutions { u = 1, x ( 1, 1) u( 1) = u(1) = 0 Solutions are defined in viscosity sense. [Crandall-Lions 83] Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 2 / 15
Viscosity Solution 1 H-J equations Given u C(Ω) and x Ω. Define the super-differential and sub-differential of u at x: D + u(x) = {p R n : u(y) u(x) + p (y x) + o( y x ), y x} D u(x) = {p R n : u(y) u(x) + p (y x) + o( y x ), y x} u is a viscosity subsolution if H(p, u(x), x) 0, x Ω, p D + u(x). u is a viscosity supersolution if H(p, u(x), x) 0, x Ω, p D u(x). u C(Ω) is a viscosity solution if u is both a subsolution and supersolution. Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 3 / 15
Viscosity Solution 2 H-J equations u USC(Ω) is a viscosity subsolution if for any φ C 1 (Ω) such that u φ reaches maximum at x 0 and u(x 0 ) = φ(x 0 ), then H(Dφ(x 0 ), φ(x 0 ), x 0 ) 0. u LSC(Ω) is a viscosity supersolution if for any φ C 1 (Ω) such that u φ reaches minimum at x 0 and u(x 0 ) = φ(x 0 ), then H(Dφ(x 0 ), φ(x 0 ), x 0 ) 0. Viscosity solutions of 2nd order PDE [Crandall-Ishii-Lions 92] Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 4 / 15
H-J equations Viscosity Solution 3 (Vanishing Viscosity) If u ɛ is the smooth solution of H(Du ɛ, u ɛ, x) = ɛ u ɛ and u ɛ u locally uniformly as ɛ 0, then u is a viscosity solution. u(x) = 1 x is the unique viscosity solution of { u = 1, x ( 1, 1) u( 1) = u(1) = 0 (Regularity) Suppose that H(p, u, x) is coercive in variable p: lim H(p, u, x) = + uniformly in u, x p + then the viscosity solution is locally Lipschitz continuous. Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 5 / 15
Optimal Control Optimal Control Consider the ODE with control: { y (s) = f (y(s), α(s)), t < s < T y(t) = x x R n : initial point at time t. T : terminal time. f : R n A R n bounded and Lipschitz. A: compact subset in R m α( ) A: set of the admissible control: A = {α : [t, T ] A α( ) is measurable}. Control problem: find α( ) which optimizes the cost functional: C x,t [α( )] = T t r(y(s), α(s))ds + g(y(t )), y( ) solves the ODE. r : R n A R: running cost. g : R n R: terminal cost. Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 6 / 15
H-J-B equation Optimal Control Define the value function as u(x, t) = inf C x,t[α( )] α( ) A (Hamilton-Jacobi-Bellman equation) u is the unique viscosity solution of the terminal value problem: { ut + min a A {f (x, a) Du + r(x, a)} = 0 in Rn (0, T ) u(x, T ) = g(x) on R n {t = T }. Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 7 / 15
G-equation Optimal Control Choose A = B sl (0) R n, f (x, a) = V (x) + a R n, r(x, a) 0: min a B sl (0) {( V (x) + a) p} = V (x) p s L p The viscosity solution of { ut V (x) Du s L Du = 0 in R n (0, T ) u(x, T ) = g(x) on R n {t = T } is given by u(x, t) = inf g(y(t )), α( ) A where the infimum is over all trajectories y : [t, T ] R n satisfying { y (s) = V (y(s)) + α(s), t < s < T y(t) = x and the control α(s) s L. Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 8 / 15
For the media involving microscopic self-repeating environments, the process to extract the macroscopic average out. Heat conduction in composite material: { ( A ɛ u ɛ ) = f, x Ω u ɛ = 0, x Ω A ɛ = A( x ɛ ) Rn n : thermal conductivity tensor A(y): 1-periodic, uniformly positive definite of elliptic PDE: as ɛ 0, u ɛ ū (in some sense) the solution of { ( Ā ū ) = f, x Ω ū = 0, x Ω Ā: homogenized conductivity tensor Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 9 / 15
1D case 1D problem: d ( ( x ) ) du ɛ a = f, 0 < x < L dx ɛ dx u ɛ (0) = u ɛ (L) = 0 a(y): 1-periodic, 0 < α a(y) β <. As ɛ 0, u ɛ ū the solution of d ( ā dū ) = f, 0 < x < L dx dx ū(0) = ū(l) = 0 where ā is the harmonic mean of a(y): ( 1 ) 1 1 ā = a(y) dy 0 Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 10 / 15
of H-J Periodic homogenization of Hamilton-Jacobi equation [Lions-Papanicolaou-Varadhan 86]: ( ut ɛ + H Du ɛ, x ) = 0 ɛ As ɛ 0, u ɛ ū the solution of H: effective Hamiltonian ū t + H(Dū) = 0 (i) H(p, y) is continuous and periodic in y (ii) H(p, y) is bounded in y for bounded p (iii) H(p, y) is coercive in p uniformly in y: H(p, y) as p Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 11 / 15
Asymptotic Expansion Two-scale asymptotic expansion: u ɛ (x, t) = u 0 (x, t) + ɛu 1 ( x, x ɛ, t ) + x: slow variable, y = x ɛ : fast variable u 1 (x, y, t): periodic in y Leading order: u ɛ t = u 0 t + ɛu 1 t + Du ɛ = D x u 0 + D y u 1 + ɛd x u 0 u 0 t + H(D x u 0 + D y u 1, y) = 0 should be independent of variable y Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 12 / 15
Cell Problem (Cell Problem) Given any P R n, find unique number H = H(P) such that the equation has a periodic solution v(y). H(P + Dv, y) = H, y T n For λ > 0, let v (λ) be the unique periodic viscosity solution of H(P + Dv (λ), y) = λv (λ), y T n Due to the coercivity, u (λ) is Lipschitz continuous uniformly in λ. As λ 0, λu (λ) H uniformly in y and u (λ) u the viscosity solution of the cell problem. Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 13 / 15
Result For ɛ > 0, assume that u ɛ is the unique viscosity solution of { u ɛ t + H ( Du ɛ, x ) ɛ = 0 u ɛ. (x, 0) = g(x) Then as ɛ 0, u ɛ converges uniformly to ū the unique viscosity solution of the following effective equation: { ūt + H(Dū) = 0, ū(x, 0) = g(x) where H is given by the cell problem. Proof: perturbed test function method [Evans 89] of nonlinear 2nd order PDE [Evans 92] Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 14 / 15
Computing Effective Hamiltonian So solve the cell problem numerically, consider the evolution equation [Qian 03]: { vt + H(P + Dv, y) = 0 in T n (0, ) v(x, 0) = 0 on T n {t = 0}. Due to the coercivity of H, the effective Hamiltonian can be approximated as: v(x, t) H = lim, t + t which converges uniformly in x. Yu-Yu Liu (NCKU Math) H-J equation and August 22, 2012 15 / 15