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By Daniel A. Murray

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48 FABIO CAMILLI AND MAURIZIO FALCONE Step O. Given W O E IRN , define W1,0 = Rl (WO) E IRNl W 2,0 = R2(W O) E IRN2 = O. Compute for r = 1,2, and set n Step i. Step 2. Compute for r = 1,2 W r,n+l _ { i - Wr,n+l/2 i , . {W1,n+l/2 W},n+l/2} mIn i '. i E Ir i E 10 . Step 3. Check a stopping criterion. IF it is satisfied THEN STOP ELSE Increase n by 1 and GO TO Step i. Notice that the definition of D guarantees W/,n+l = W i2,n+1 for each i E 10 . The above algorithm allows to split the computations in the subdomains making a link at the end of each iteration (Step 2).

We start recalling some preliminary results. 1) corresponding to the control a E A. Whenever this will be possible without ambiguity we will adopt the simplified notations yx(t) or y(t) instead of yx(t, a(t)). We want to minimize Jx with respect to the controls in Ax so that, to have a meaningful problem, we need to assume at least that for any x E IT. 1) is not sufficient in general to guarantee the continuity of v(x) in IT. This is due to the structure of the multivalued map x 1--+ Ax. 2) 3ft> 0 : '

ROFMAN, On deterministic control problems: an approximation procedure for the optimal cost (part I and II), SIAM J. Control and Optimization 23 (1985), pp. 242-285. [IK] H. ISHII AND S. KOIKE, A new formulation of state constraints problems for first-order PDEs, preprint 1993. [Lo] P. LORETI, Some properties of constrained viscosity solutions of HamiltonJacobi-Bellman equations, SIAM J. Control and Optimization 25 (1987). [LT] P. LORETI AND M. E. TESSITORE, Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations, J.

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