R.O. Mastaliyev.
First-and second-order necessary optimality conditions in control problems described by stochastic differential equations

The authors investigate the problem of optimal control of nonlinear stochastic systems, the mathematical model of which is given by the Ito stochastic differential equation. Under the assumption of openness of the control area by means of the first and second variations of the quality functional, the first-and second-order necessary optimality conditions are derived. In a special case, a stochastic analogue of the Legendre-Clebsch condition is obtained from the necessary second-order optimality condition. Finally, the optimality of controls that are special in the classical sense is investigated and multipoint necessary optimality conditions for singular (in the indicated sense) controls are established.

Keywords: Stochastic control problem, Optimal control, Variations of quality functional, Necessary condition, Stochastic analogue of the Euler equation, Stochastic analogue of the Legendre-Clebsch condition
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