We compare a general controlled diffusion process with a deterministic system where a second controller drives the disturbance against the first controller. We show that the two models are equivalent with respect to two properties: the viability (or controlled invariance, or weak invariance) of closed smooth sets, and the existence of a smooth control Lyapunov function ensuring the stabilizability of the system at an equilibrium.
Mots clés : controlled diffusion, robust control, differential game, invariance, viability, stabilization, viscosity solution, optimality principle
@article{COCV_2008__14_2_343_0, author = {Cesaroni, Annalisa and Bardi, Martino}, title = {Almost sure properties of controlled diffusions and worst case properties of deterministic systems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {343--355}, publisher = {EDP-Sciences}, volume = {14}, number = {2}, year = {2008}, doi = {10.1051/cocv:2007053}, mrnumber = {2394514}, zbl = {1133.93036}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007053/} }
TY - JOUR AU - Cesaroni, Annalisa AU - Bardi, Martino TI - Almost sure properties of controlled diffusions and worst case properties of deterministic systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 343 EP - 355 VL - 14 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007053/ DO - 10.1051/cocv:2007053 LA - en ID - COCV_2008__14_2_343_0 ER -
%0 Journal Article %A Cesaroni, Annalisa %A Bardi, Martino %T Almost sure properties of controlled diffusions and worst case properties of deterministic systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 343-355 %V 14 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007053/ %R 10.1051/cocv:2007053 %G en %F COCV_2008__14_2_343_0
Cesaroni, Annalisa; Bardi, Martino. Almost sure properties of controlled diffusions and worst case properties of deterministic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 2, pp. 343-355. doi : 10.1051/cocv:2007053. http://www.numdam.org/articles/10.1051/cocv:2007053/
[1] On the support of Wiener functionals, Asymptotic problems in probability theory: Wiener functionals and asymptotics (Sanda/Kyoto, 1990), Pitman Res. Notes Math. Ser 284, Longman Sci. Tech., Harlow (1993) 3-34. | MR | Zbl
, and ,[2] The viability theorem for stochastic differential inclusions. Stochastic Anal. Appl 16 (1998) 1-15. | MR | Zbl
and ,[3] Characterization of stochastic viability of any nonsmooth set involving its generalized contingent curvature. Stochastic Anal. Appl 21 (2003) 955-981. | MR | Zbl
and ,[4] Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkäuser, Boston (1997). | MR | Zbl
and ,[5] Almost sure stabilizability of controlled degenerate diffusions. SIAM J. Control Optim 44 (2005) 75-98. | MR | Zbl
and ,[6] Propagation of maxima and strong maximum principle for viscosity solution of degenerate elliptic equations. I: Convex operators. Nonlinear Anal 44 (2001) 991-1006. | MR | Zbl
and ,[7] Propagation of maxima and strong maximum principle for fully nonlinear degenerate elliptic equations. II: Concave operators. Indiana Univ. Math. J 52 (2003) 607-627. | MR | Zbl
and ,[8] Invariant sets for controlled degenerate diffusions: a viscosity solutions approach, in Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W.M. McEneaney, G.G. Yin and Q. Zhang Eds., Birkhäuser, Boston (1999) 191-208. | MR | Zbl
and ,[9] A geometric characterization of viable sets for controlled degenerate diffusions. Set-Valued Anal 10 (2002) 129-141. | MR | Zbl
and ,[10] -optimal control and related minimax design problems. A dynamic game approach, 2nd edn., Birkhäuser, Boston (1995). | MR | Zbl
and ,[11] Hölder norms and the support theorem for diffusions. Ann. Inst. H. Poincaré Probab. Statist 30 (1994) 415-436. | Numdam | MR | Zbl
, and ,[12] Robust control approach to option pricing, including transaction costs, in Advances in dynamic games, Ann. Internat. Soc. Dynam. Games 7, Birkhäuser, Boston (2005) 391-416. | MR
,[13] Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris Sér. I Math 327 (1998) 17-22. | MR | Zbl
, , and ,[14] A differential game with two players and one target. SIAM J. Control Optim 34 (1996) 1441-1460. | MR | Zbl
,[15] Stability properties of controlled diffusion processes via viscosity methods. Ph.D. thesis, University of Padova (2004).
,[16] A converse Lyapunov theorem for almost sure stabilizability. Systems Control Lett 55 (2006) 992-998. | MR | Zbl
,[17] User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc 27 (1992) 1-67. | MR | Zbl
, and ,[18] Invariance of stochastic control systems with deterministic arguments. J. Diff. Equ 200 (2004) 18-52. | MR | Zbl
and ,[19] Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Sect. B (N.S.) 13 (1977) 99-125. | Numdam | MR | Zbl
,[20] Controlled Markov Process and Viscosity Solutions. Springer-Verlag, New York (1993). | MR | Zbl
and ,[21] Robust nonlinear control design. State-space and Lyapunov techniques. Birkäuser, Boston (1996). | MR | Zbl
and :[22] On the existence of optimal controls. SIAM J. Control Optim 28 (1990) 851-902. | MR | Zbl
and ,[23] Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech 10 (1995) 1981-2004. | MR
, ,[24] A simple proof of the support theorem for diffusion processes, Séminaire de Probabilités, XXVIII, Lect. Notes Math 1583, Springer, Berlin (1994) 36-48. | Numdam | MR | Zbl
and ,[25] Differential game-theoretic thoughts on option pricing and transaction costs. Int. Game Theory Rev 2 (2000) 209-228. | MR | Zbl
,[26] Dynamic programming for stochastic target problems and geometric flow. J. Eur. Math. Soc 4 (2002) 201-236. | MR | Zbl
and ,[27] A stochastic representation for the level set equations. Comm. Part. Diff. Equ 27 (2002) 2031-2053. | MR | Zbl
and ,[28] Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control. Optim 31 (1993) 604-623. | MR | Zbl
,[29] Stability of dynamical systems with competitive controls: the degenerate case. J. Math. Anal. Appl 191 (1995) 428-449. | MR | Zbl
,[30] control of nonlinear systems: differential games and viscosity solutions. SIAM J. Control Optim 34 (1996) 1071-1097. | MR | Zbl
,[31] Equivalence between nonlinear control problems and existence of viscosity solutions of Hamilton-Jacobi-Isaacs equations. Appl. Math. Optim 39 (1999) 17-32. | MR | Zbl
,[32] On the support of diffusion processes with applications to the strong maximum principle, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, Univ. California Press, Berkeley (1972) 333-359. | MR | Zbl
and ,[33] On degenerate elliptic-parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math 25 (1972) 651-713. | MR | Zbl
and ,[34] On the gap between deterministic and stochastic ordinary differential equations. Ann. Probability 6 (1978) 19-41. | MR | Zbl
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