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Center for Uncertainty Quantification An A Posteriori Error Estimate for Symplectic Euler Approximation of Optimal Control Problems

Jesper Karlsson, Stig Larsson, Mattias Sandberg, Anders Szepessy, Raul Tempone

unpublished

This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns Symplectic Euler solutions of the Hamil-tonian system connected with the optimal control problem. The error representation has a leading order term consisting of an error density that is computable from Symplectic Euler solutions. Under an assumption of the pathwise convergence of the approximate dual
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... proximate dual function as the maximum time step goes to zero, we prove that the remainder is of higher order than the leading error density part in the error representation. With the error representation , it is possible to perform adaptive time stepping. We apply an adaptive algorithm originally developed for ordinary differential equations. 1. Optimal Control The optimal control problem is to minimize the functional T 0 h(X(t), α(t)) dt + g(X(T)), (1) with given functions h : R d × B → R and g : R d → R, with respect to the state variable X : [0, T ] → R d and the control α : [0, T ] → B, with control set, B, a subset of some Euclidean space, R d , such that the ODE constraint, X (t) = f (X(t), α(t)), 0 < t ≤ T, X(0) = x 0 , (2) is fulfilled. This optimal control problem can be solved (globally) using the Hamilton-Jacobi-Bellman (HJB) equation u t + H(x, u x) = 0, x ∈ R d , 0 ≤ t < T, u(·, T) = g(·), x ∈ R d , (3) with u t and u x denoting the time derivative and spatial gradient of u, respectively, and the Hamiltonian, H : R d × R d → R, defined by H(x, λ) := min α∈B λ · f (x, α) + h(x, α) , (4) and value function u(x, t) := inf X:[t,T ]→R d , α:[t,T ]→B T t h(X(s), α(s)) ds + g(X(T)) , (5) where X (s) = f (X(s), α(s)), t < s ≤ T, X(t) = x. The global minimum to the optimal control problem (1)-(2) is thus given by u(x 0 , 0). If the Hamiltonian is sufficiently smooth, the bi-characteristics to the HJB equation (3) are given by the following Hamilto-nian system: X (t) = H λ (X(t), λ(t)), 0 < t ≤ T, X(0) = x 0 , −λ (t) = H x (X(t), λ(t)), 0 ≤ t < T, λ(T) = g x (X(T)), (6) where H λ , H x , and g x denote gradients with respect to λ and x, respectively, and the dual variable, λ : [0, T ] → R d , satisfies λ(t) = u x (X(t), t) along the characteristic. It can be solved numerically using the Symplectic Euler method: X n+1 − X n = ∆t n H λ (X n , λ n+1), n = 0,. .. , N − 1, X 0 = x 0 , λ n − λ n+1 = ∆t n H x (X n , λ n+1), n = 0,. .. , N − 1, λ N = g x (X N), (7) with 0 = t 0 < t 1 <. .. < t N = T , ∆t n := t n+1 − t n , and X n , λ n ∈ R d , see [5, 4]

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