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Archive of Issues

Russia Ufa
Year
2014
Issue
2
Pages
29-42
 Section Mathematics Title On deterministic approach to solution of stochastic optimal control problem with controlled diffusion Author(-s) Ismagilov N.S.a Affiliations Ufa State Aviation Technical Universitya Abstract We consider an optimal control problem for a one-dimensional process driven by stochastic differential equation, which has both drift and diffusion coefficients controlled, diffusion being linear in control $$dx(t) = b(t,x(t),u(t))\,dt + \sigma(t,x(t))u(t)\,dW(t), \quad x(0) = x_0,$$ where $x(t)$ is the state variable, $u(t)$ is the control variable and $W(t)$ is the Wiener process. We prove a theorem which gives a structure of solution for the considered differential equation as a superposition of functions $x(t) = \Phi(t,u(t)W(t) + y(t))$, where $\Phi(t,v)$ is the known function, which is completely determined by the diffusion coefficient $\sigma(t,x)$ and is independent of control, and $y(t)$ is the solution to the pathwise-deterministic measure-driven differential equation $$dy(t) = B(t,y(t),u(t))\,dt - W(t)\,du(t).$$ The revealed structure of the solution enables us to consider a new pathwise-deterministic impulsive optimal control problem with the state variable $y(t)$ which is equivalent to the original stochastic optimal control problem. Pathwise problems may have anticipative solutions, so we propose a method that makes it possible to build nonanticipative optimal solutions. The basic idea of the method is to modify cost functional in new pathwise problem with special integral term, which guarantees nonanticipativity of solutions. Keywords stochastic optimal control, stochastic differential equations, deterministic approach, pathwise optimization, optimal impulsive control UDC 519.21, 517.977 MSC 93E20, 49K45, 60H30, 49N25 DOI 10.20537/vm140202 Received 29 October 2013 Language Russian Citation Ismagilov N.S. On deterministic approach to solution of stochastic optimal control problem with controlled diffusion, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 29-42. References Wets R.J.B. On the relation between stochastic and deterministic optimization, Control Theory, Numerical Methods and Computer Systems Modelling. Lecture Notes in Economics and Mathematical Systems, Berlin: Springer-Verlag, 1975, vol. 107, pp. 350-361. Rockafellar R.T., Wets R.J.B. Nonanticipativity and $L$$1$ -martingales in stochastic optimization problems, Mathematical Programming Study, 1973, vol. 6, pp. 170-187. Davis M.H.A. Anticipative LQG control, IMA Journal of Mathematical Control and Information, 1989, vol. 6, no. 3, pp. 259-265. Davis M.H.A., Burstein G. A deterministic approach to stochastic optimal control, with application to anticipative control, Stochastics and Stochastics Reports, 1992, vol. 40, no. 3-4, pp. 203-256. Ismagilov N.S., Nasyrov F.S. On deterministic approach to stochastic optimal control problem, Vestnik UGATU, 2013, vol. 17, no. 5, pp. 38-43 (in Russian). Nasyrov F.S. Lokal'nye vremena, simmetrichnye integraly i stokhasticheskii analiz (Local times, symmetric integrals and stochastic analysis), Moscow: Fizmatlit, 2011, 212 p. Miller B.M., Rubinovich E.Ya. Optimizatsiya dinamicheskikh sistem s impul’snymi upravleniyami (Optimization of dynamic systems with pulse control), Moscow: Nauka, 2005, 429 p. Protter P. Stochastic integration and differential equations, Berlin: Springer, 2004, 415 p. Dykhta V.A., Samsonyuk O.N. Optimal'noe impul'snoe upravlenie s prilozheniyami (Optimal impulse control with applications), Moscow: Fizmatlit, 2000, 256 p. Arutyunov A.V., Karamzin D.Yu., Pereira F.L. On constrained impulsive control problems, Journal of Mathematical Sciences, 2010, vol. 165, no. 6, pp. 654-688. Arutyunov A.V., Karamzin D.Yu., Pereira F. Pontryagin's maximum principle for constrained impulsive control problems, Nonlinear Analysis, 2012, vol. 75, no. 3, pp. 1045-1057. Oksendal B. Stochastic differential equations, Berlin-Heidelberg: Springer, 2003, 360 p. Mordukhovich B.S. Variational analysis and generalized differentiation, Berlin: Springer, 2006, 579 p. Full text