Almost sure properties of controlled diffusions and worst case properties of deterministic systems

Cesaroni, Annalisa; Bardi, Martino

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 2, pp. 343-355

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.

Classification : 93D09, 93E15, 49L25, 49N70
Keywords: controlled diffusion, robust control, differential game, invariance, viability, stabilization, viscosity solution, optimality principle

[1] S. Aida, S. Kusuoka and D. Stroock, 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. | Zbl | MR

[2] J.-P. Aubin and G. Da Prato, The viability theorem for stochastic differential inclusions. Stochastic Anal. Appl 16 (1998) 1-15. | Zbl | MR

[3] J.-P. Aubin and H. Doss, Characterization of stochastic viability of any nonsmooth set involving its generalized contingent curvature. Stochastic Anal. Appl 21 (2003) 955-981. | Zbl | MR

[4] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkäuser, Boston (1997). | Zbl | MR

[5] M. Bardi and A. Cesaroni, Almost sure stabilizability of controlled degenerate diffusions. SIAM J. Control Optim 44 (2005) 75-98. | Zbl | MR

[6] M. Bardi and F. Da Lio, Propagation of maxima and strong maximum principle for viscosity solution of degenerate elliptic equations. I: Convex operators. Nonlinear Anal 44 (2001) 991-1006. | Zbl | MR

[7] M. Bardi and F. Da Lio, Propagation of maxima and strong maximum principle for fully nonlinear degenerate elliptic equations. II: Concave operators. Indiana Univ. Math. J 52 (2003) 607-627. | Zbl | MR

[8] M. Bardi and P. Goatin, 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. | Zbl | MR

[9] M. Bardi and R. Jensen, A geometric characterization of viable sets for controlled degenerate diffusions. Set-Valued Anal 10 (2002) 129-141. | Zbl | MR

[10] T. Başar and P. Bernhard, $H^{\infty }$-optimal control and related minimax design problems. A dynamic game approach, 2nd edn., Birkhäuser, Boston (1995). | Zbl | MR

[11] G. Ben Arous, M. Grădinaru and M. Ledoux, Hölder norms and the support theorem for diffusions. Ann. Inst. H. Poincaré Probab. Statist 30 (1994) 415-436. | Zbl | MR | Numdam

[12] P. Bernhard, 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] R. Buckdahn, S. Peng, M. Quincampoix and C. Rainer, Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris Sér. I Math 327 (1998) 17-22. | Zbl | MR

[14] P. Cardaliaguet, A differential game with two players and one target. SIAM J. Control Optim 34 (1996) 1441-1460. | Zbl | MR

[15] A. Cesaroni, Stability properties of controlled diffusion processes via viscosity methods. Ph.D. thesis, University of Padova (2004).

[16] A. Cesaroni, A converse Lyapunov theorem for almost sure stabilizability. Systems Control Lett 55 (2006) 992-998. | Zbl | MR

[17] M.C. Crandall, H. Ishii and P.L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc 27 (1992) 1-67. | Zbl | MR

[18] G. Da Prato and H. Frankowska, Invariance of stochastic control systems with deterministic arguments. J. Diff. Equ 200 (2004) 18-52. | Zbl | MR

[19] H. Doss, Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Sect. B (N.S.) 13 (1977) 99-125. | Zbl | MR | Numdam

[20] W.H. Fleming and H.M. Soner, Controlled Markov Process and Viscosity Solutions. Springer-Verlag, New York (1993). | Zbl | MR

[21] R.A. Freeman and P.V. Kokotovic: Robust nonlinear control design. State-space and Lyapunov techniques. Birkäuser, Boston (1996). | Zbl | MR

[22] U.G. Haussmann and J.P. Lepeltier, On the existence of optimal controls. SIAM J. Control Optim 28 (1990) 851-902. | Zbl | MR

[23] Y. Lin, E.D. Sontag, Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech 10 (1995) 1981-2004. | MR

[24] A. Millet and M. Sanz-Solé, A simple proof of the support theorem for diffusion processes, Séminaire de Probabilités, XXVIII, Lect. Notes Math 1583, Springer, Berlin (1994) 36-48. | Zbl | MR | Numdam

[25] G.J. Olsder, Differential game-theoretic thoughts on option pricing and transaction costs. Int. Game Theory Rev 2 (2000) 209-228. | Zbl | MR

[26] H.M. Soner and N. Touzi, Dynamic programming for stochastic target problems and geometric flow. J. Eur. Math. Soc 4 (2002) 201-236. | Zbl | MR

[27] H.M. Soner and N. Touzi, A stochastic representation for the level set equations. Comm. Part. Diff. Equ 27 (2002) 2031-2053. | Zbl | MR

[28] P. Soravia, Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control. Optim 31 (1993) 604-623. | Zbl | MR

[29] P. Soravia, Stability of dynamical systems with competitive controls: the degenerate case. J. Math. Anal. Appl 191 (1995) 428-449. | Zbl | MR

[30] P. Soravia, ${\mathscr{H}}_{\infty }$ control of nonlinear systems: differential games and viscosity solutions. SIAM J. Control Optim 34 (1996) 1071-1097. | Zbl | MR

[31] P. Soravia, Equivalence between nonlinear ${\mathscr{H}}_{\infty }$ control problems and existence of viscosity solutions of Hamilton-Jacobi-Isaacs equations. Appl. Math. Optim 39 (1999) 17-32. | Zbl | MR

[32] D.W. Stroock and S.R.S. Varadhan, 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. | Zbl | MR

[33] D.W. Stroock and S.R.S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math 25 (1972) 651-713. | Zbl | MR

[34] H.J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations. Ann. Probability 6 (1978) 19-41. | Zbl | MR